![Introduction
The first impulse to study non-associative structures came in the first decades of the
20th century from the foundation of geometry, in particular from the investigation of
coordinate systems of non-Desarguesian planes. An additional interest forW. Blaschke
to treat loops and quasigroups systematically came from topological questions of dif-
ferential geometry, in particular from the topological behavior of geodesic foliations
[11]. R. Baer [7], A. A. Albert [4], [5], and R. H. Bruck [17] established the theory
of quasigroups and loops as an independent algebraic theory. For Baer the geometry
associated with a loop remains an important tool, for Bruck the theory of loops is a
part of general algebra [18]. Albert prefers to consider translations of a loop and to
see a loop as a section in the group generated by them. The development of the theory
of loops and quasigroups in the last 50 years was continued in the spirit of these three
directions. An important representative for the study of loops, quasigroups and their
associated geometry as abstract objects is V. D. Belousov [9]. The investigation of
loops within the framework of topological algebra, topological geometry and differ-
ential geometry gained importance by the work of A. I. Malcev [89], Κ. H. Hofmann
[47], H. Salzmann [120] and M. A. Akivis [3], The usefulness of analytic methods for
the theory of loops is shown in the work of L. V. Sabinin [116]. All these branches of
the theory of loops are collected and documented in [21].
Our aim here is to consider the theory of loops as a part of group theory; this means
to treat loops as sharply transitive sections in groups. Hence the group theoretical point
of view predominates the methods of non-associative algebra. We restrict our attention
to such classes of groups in which the simple objects are classified (e.g. finite groups,
algebraic groups, Lie groups). Since an essential part of our work will deal with
sharply transitive sections in Lie groups we shall use systematically also differential
geometric methods. From incidence structures we shall substantially use 3-nets which
are the geometries associated with loops; they are the most important tool for problems
concerning isotopism classes of loops. The local version of differentiable 3-nets are
3-webs which are coordinatized by local differentiable loops. Since the theory of
3-webs is systematically studied and the application of results on local loops can be
used in the global theory, the local point of view of 3-web geometry and the theory of
differentiable loops belong to the arsenal of our methods.
Binary operations " · " Μ χ Μ Μ on a set Μ with the property that for given
a, b e Μ the equations a · χ = b and y • a = b are uniquely solvable correspond to
sharply transitive subsets in transformation groups generated by these subsets. Indeed
the left transformations λα : χ ax are bijections and for given y, ζ e Μ there
exists precisely one left translation λα with Aa (j) — z. In Μ there is an element 1](https://image.slidesharecdn.com/88614-250726132104-773001ef/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-18-320.jpg)
![2 Introduction
for which 1-jc = jc-1 = jcif and only if the set of left translations contains the
identity. Sets with a binary operation " · " which correspond to sharply transitive sets
of transformations containing the identity are called loops.
If (L, ·) is a loop then the left translations χ ax (a e L) generate a permu-
tation group G on the set L. For non-associative L the group G is not sharply
transitive on Μ and the stabilizer Η of 1 e L in G is different from the identity.
The mapping σ : a λα : L —> G is & section with respect to the natural projec-
tion π : G G/Η. One may identify L with the factor space G/H and transport
the multiplication. Hence, the theory of loops coincides with the theory of triples
(G, Η, σ), where G is a group, Η a subgroup containing no normal non-trivial sub-
group of G and σ a section G/H —• G such that a(G/H) acts sharply transitively on
the left cosets xH, χ e G, and generates G. According to Baer [7], the set a ( G / H )
of representatives is sharply transitive on the factor set G/H if and only if σ (G/H) is
a set of representatives in G for every subgroup conjugate to Η. In [101] and [102] it
is stressed that it is always possible and fruitful to carry over the loop properties into
properties of the section σ.
If C denotes the category of topological spaces, differentiable manifolds, or of
algebraic varieties, then a loop is a C-loop if its multiplication, left division and right
division are C-morphisms. In the theory of triples (G, Η, σ), where G is a Lie group,
Η a closed subgroup and σ . G/H ^ G a C-morphism, the triple (G, Η,σ) defines
a C-loop multiplication by the rule
(χι H) • (X2H) = σ(χιΗ)χ2Η.
The first attempt to deal with differentiable and analytic loops was to follow the
ideas of Sophus Lie and to classify analytic loops by their tangential objects. In the
last 30 years this research program has been applied successfully to differentiable
Moufang loops (they are automatically analytic, cf. [102]) by Kuz'min, Kerdman and
Nagy. By their results the theory of differentiable Moufang loops has been carried
up to the level of the theory of Lie groups. Since the Hausdorff-Campbell formula
works also for binary Lie algebras, the theory of diassociative analytic local loops
may be treated successfully using binary Lie algebras the structure of which has been
determined by A. N. Grishkov. After this progress the attention turned to the class
of differentiable Bol loops. Here as well the investigations proceeded in the spirit of
Sophus Lie. With any analytic Bol loop L there is associated a Bol algebra B(L) such
that two Bol loops are locally isomorphic if and only if the corresponding Bol algebras
are isomorphic ([91], XII.8.12. Proposition). Hence the analytic local Bol loops can
be classified by the Bol algebras. But at this point a crucial difference from the theory
of Lie groups and Lie Moufang loops comes to light. Whereas any local Lie group or
local Lie Moufang loop may be embedded into a global one, this fact fails to be true for
local analytic Bol loops (and hence much more so for analytic local loops in general).
Already the classification of local analytic 2-dimensional Bol loops by Ivanov ([61],
[62], [63]) and the classification of global differentiable 2-dimensional loops show
the great difference between the varieties of differentiable global and local Bol loops.](https://image.slidesharecdn.com/88614-250726132104-773001ef/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-19-320.jpg)
![Introduction 3
Moreover, in Section 17 we exemplify by a class of analytic loops that the infinitesimal
behavior of a loop does not determine its global properties. Hence the investigation of
global differentiable loops cannot be reduced to that of local loops, and the procedure
to investigate suitable sections in Lie groups seems to us the only feasible method
for the classification of differentiable global loops for which the group topologically
generated by left translations is a Lie group. This restriction seems relatively mild to us
since differentiable loops with some weak associativity conditions have this property
(e.g. Bol loops, see [91], Proposition XII.2.14 and [102]).
With any loop L there is associated an incidence structure, called a 3-net Ν (cf. [8]);
if L is differentiable or analytic then Ν is also differentiable or analytic. Conversely,
to a 3-net Ν there corresponds a full class of isotopic loops, and with each point of Ν
taken as origin we may associate a coordinate loop defining the multiplication for the
points of a line through the origin graphically. Two coordinate loops of a net Ν are
isomorphic if and only if their points of origin are in the same orbit of the collineation
group Θ of T
V which preserves the directions ([8], p. 50).
Any identity in a loop corresponds to a configuration in the associated 3-net,
and configurations in 3-nets yield identities in some coordinate loops. The 3-nets
associated with Bol loops are of particular importance. For every line of a distinguished
pencil in such a net there exists an involutory collineation fixing this line pointwise.
These reflections generate a group Γ acting transitively on the distinguished pencil.
If the Bol loop is differentiable then Γ is a Lie group which induces the structure
of an affine symmetric space on the distinguished pencil. In an algebraic setting this
symmetric space is a left distributive groupoid which is called the core of the Bol loop.
These relations allow us to apply the rich theory of symmetric spaces as well as the
results on left distributive quasigroups in order to classify wide classes of Bol loops.
3-webs are incidence geometries associated with differentiable local loops; their
pencils of lines form 3 foliations. If a 3-web is associated with a differentiable local
Bol loop then for every leaf of one of the 3 foliations there exists a local reflection.
The group generated by these local reflections induces an affine locally symmetric
space on the manifold of leaves of this foliation; this connects the theory of local
differentiable Bol loops with the classical theory of locally symmetric spaces.
Our contribution has two parts. The first part contains the foundations of our new
methods and their applications to the extension theory of Bol loops, to the algebraic
theory of symmetric spaces and to the Lie theory of smooth loops. Moreover, we
classify strongly 2-divisible finite, differentiable and algebraic Bruck loops, and we
clarify the role of compactness of the group topologically generated by the left trans-
lations of a compact loop. In the second part we apply our methods to the topological
and differentiable loops on manifolds of small dimension.
In the Sections 1, 3 and 4 we develop the foundations for our point of view,
describe the interactions between the properties of sharply transitive sections and
the corresponding loops, discuss the relations between configurations in 3-nets and
corresponding identities in the coordinate loops and show what the local version of
these concepts is in the case of differentiable objects; at this instant the importance of](https://image.slidesharecdn.com/88614-250726132104-773001ef/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-20-320.jpg)
![4 Introduction
differentiable foliations and 3-webs comes into light. Using the methods of algebraic
topology we prove that any topological loop and hence any topological 3-net on a
connected topological manifold is orientable.
In Section 2 we thoroughly analyze the structure of proper loops which are ex-
tensions of groups by groups. In contrast to the obstructions for study of general
extensions of loops, the special case of extension theory developed in Section 2 al-
lows a transparent description by group theoretical methods. If we work in the category
of loops realized on topological or differentiable manifolds then our basic assumption
is that the group G topologically generated by the left translations is a Lie group. In
Section 5 we study differentiable loops such that the sharply transitive section cor-
responding to L is (locally) covered by 1-parameter subgroups. This class (locally)
coincides with the opposite loops of geodesic loops with respect to a uniquely deter-
mined affine connection with vanishing curvature. Special cases of these loops are
the opposite loops of the geodesic (local) loops of reductive homogeneous spaces and
the differentiable (local) Bol loops. Sections 6 to 13 are devoted to a thorough study
of Bol loops, their analytic and algebraic properties as well as to their relations to the
classical theory of affine symmetric spaces. Here we pay attention to the global as
well as to the local Bol loops. In Section 6 some representations of local Bol loops
are given as sections in Lie groups and their relations to Lie triple systems and Bol
algebras. In Section 7 the geometric version of isotopism classes of Bol loops, the
Bol nets, are investigated. Of special interest for us is the group of collineations of a
Bol net as well as some of its subgroups, for instance the group generated by the Bol
reflections.
The core of a Bol loop introduced and studied in Section 9 is a symmetric space in
the sense of Loos, and it may be used in the theory of groups of exponent 3. (Cf. Corol-
lary 9.7). The closest relation between a local Bol loop L and its local core takes place
if the Bol algebra of L is a Lie triple system. For a global Bol loop this is the case if the
core is a symmetric quasigroup and the Bol loop L satisfies the automorphic inverse
property. Then the Bol loop L is a left Α-loop, and to any element χ there exists pre-
cisely one element y with χ — y2
. These loops are called strongly 2-divisible Bruck
loops. Differentiable strongly 2-divisible loops having the left inverse property are
already Bruck loops if they are left Α-loops and satisfy the automorphic inverse prop-
erty. Differentiable Bol loops are always locally strongly 2-divisible. G. Glauberman
proved [36] that any strongly 2-divisible Bruck loop L can be embedded into the group
G generated by the left translations of L and that the multiplication of the embedded
loop L is induced by the multiplication of G. We generalize this embedding to the
class of strongly 2-divisible Bol loops. Moreover, using this construction we classify
strongly 2-divisible differentiable connected Bruck loops in Section 11. They corre-
spond in a unique way to pairs (G, σ), where G is a connected Lie group and σ is an
involutory automorphism of G such that the subgroup centralized by σ contains no
non-trivial normal subgroup of G, the exponential mapping from the (— l)-eigenspace
m of σ on the Lie algebra g into G is a diffeomorphism and expm generates G. Our
systematic study of strongly 2-divisible Bruck loops in algebraic groups (Section 12),](https://image.slidesharecdn.com/88614-250726132104-773001ef/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-21-320.jpg)
![Introduction 5
as well as of Bruck loops associated with symmetric quasigroups over groups (Sec-
tion 10) is motivated by the increasing importance of this class of loops in algebra
and geometry. One of the reasons for this is the classification problem of sharply
2-transitive permutation groups. Any such group corresponds in a unique way to a
so-called near domain which is an algebraic structure (F, + , ·; 0, 1) with two oper-
ations: with respect to the addition (F, + ; 0) is a Bruck loop such that χ + χ = 0
implies χ = 0, with respect to the multiplication (F {0}, ·; 1) is a group and the
left distributive law holds (cf. [66], [75]). Till now, the only known examples of near
domains are the near fields where the additive structure is an abelian group. Although
this question motivated many interesting contributions, the existence of strongly 2-
divisible Bruck loops having a sharply transitive group of automorphisms remains an
interesting and important problem (cf. e.g. [76], [77], [68]).
For differentiable Bol loops it is no restriction to assume that the group topolog-
ical^ generated by the left translations is a Lie group. We prove in Section 9 that
the category of real analytic Bol loops coincides with the category of connected topo-
logical Bol loops for which the group topologically generated by the left translations
is a connected, locally compact, locally connected and finite-dimensional topological
group (cf. [92]). As a consequence we obtain that any closed subloop of an analytic
Bol loop is analytic. For differentiable Moufang loops the group topologically gen-
erated by all left and right translations is a Lie group. In constrast to this, there are
examples of differentiable Bol loops having a Lie group as the group topologically
generated by all left and right translations as well as those for which this is not the
case (cf. Sections 7 and 22).
In Section 13 we consider local Bol loops associated with the same symmetric
space and ask under what conditions two such local Bol loops are locally isotopic. If
the symmetric space is compact and irreducible then the answer is affirmative if one
excludes the symmetric space on the 7-sphere as well as the Grassmannian manifold
of 3-planes in the 8-dimensional real vector space. We prove here also a generalization
of a theorem of A. Fomenko [29] that if the group G topologically generated by the
left translations of a simply connected differentiable (local) Bol loop L is reductive
then L is the direct product of a reductive Lie group by a direct product of proper
Bol loops Li not having any connected non-trivial normal subloop. Moreover, the
symmetric space associated with L(- is irreducible and G decomposes in an analogous
way as L into factors.
In Section 14 we give for loops generalizations of the semidirect products of
groups. These constructions allow us to find examples of compact differentiable
connected loops having a compact Lie group as the group topologically generated by
the left translations. These examples of loops are realized on products on 7-spheres,
7-dimensional real projective spaces and of spaces of compact Lie groups; in them the
associativity law is strongly violated. But it is not possible to discover examples with
good associativity properties besides the Scheerer extensions studied in Sections 2 and
15. The reason for this are the theorems which give a full classification of compact
differentiable Bol loops. These loops are repeated extensions of groups by groups and](https://image.slidesharecdn.com/88614-250726132104-773001ef/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-22-320.jpg)
![6 Introduction
by Moufang loops; the theory of these extensions is developed in Sections 2 and 15.
It follows in particular that every connected compact differentiable simple Bol loop is
a Moufang loop. These results proved in Section 16 belong to the main achievements
of the first part of our work. The following theorem has the same quality: Every
compact connected topological loop is a Moufang loop if the group topologically
generated by all left and right translations is a compact Lie group. According to [55]
the topological loops such that the group topologically generated by all their left and
right translations is a compact Lie group are precisely those which have an invariant
uniformity. In contrast to this, at the end of Section 14, we prove that there are many
local Bol loops having an interpretation as local sections in compact Lie groups.
The result in [121] shows that the assumption of compactness for the group G
topologically generated by the left translations of a topological loop is very restrictive.
If G is a compact quasi-simple Lie group then it must be locally isomorphic to SOs(K).
If G is not quasi-simple, then Scheerer describes in [121] the global sections in G.
It was his description which put us in the position to classify compact differentiable
Bol loops. The group topologically generated by the left translations of such a loop
is a reductive compact Lie group having at least 2 quasi-simple factors; in this case
one of these two factors is locally isomorphic to SOs(M). It seems to us that the
most interesting objects for the study of compact topological or differentiable loops
which are not Bol loops but which have a compact connected Lie group as the group
topologically generated by the left translations are the loops homeomorphic to the
(η — 1)-dimensional projective space or to the (η — l)-sphere, where η e {4, 8} since
several natural algebraic or analytic assumptions force these loops to be classical. Till
now no sharply transitive continuous section in the group SOs (M) respectively S04.(R)
is known which does not correspond to the Moufang loop or to the group of octonions
or of quaternions of norm one, respectively. But for η = 4 or η — 8 we have been able
to find loops diffeomorphic to the (η — 1)-dimensional projective space, respectively
to the (η — 1)-dimensional sphere such that the groups topologically generated by the
left, right and by all translations coincide; they are Lie groups isomorphic to PSL„ (M),
or to SL„(R), respectively (cf. Theorem 29.3).
In Section 17 we deal with topological loops homeomorphic to R", admitting a
sharply transitive subgroup Ν of the group topologically generated by the left transla-
tions and give examples in which the group G topologically generated by all left and
right translations is a nilpotent Lie group; in these cases Ν is not noraial in G. If in
contrast to this we assume that Ν is a normal subgroup of the group G topologically
generated by all left and right translations then G (which must be a Lie group) is not
nilpotent. Whereas any 2-dimensional loop in this class must be a group, in dimen-
sion 3 there are already interesting examples of proper loops of this type. The main
result in this section is the statement that for any simple compact Lie algebra g the
multiplication χ ο y = χ + y + [χ, >>] defines a loop L on g such that the group G
topologically generated by the left translations coincides with the group generated by
all translations of L and that G is isomorphic to a semidirect product G = Τ χ Η.
The normal subgroup Τ in this semidirect product is the group Kn
of all translations](https://image.slidesharecdn.com/88614-250726132104-773001ef/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-23-320.jpg)
![8 Introduction
The exceptional status of the hyperbolic plane loop and of its isotopy class is based
upon the fact that it is associated with the hyperbolic plane geometry and generalizes
in a direct way the vector group of the euclidean plane. We treat the hyperbolic
plane loop in Section 22. In particular, we characterize it there within the class
of strongly left alternative 2-dimensional connected topological loops by the group
topologically generated by the left translations, as well as within the 2-dimensional
connected topological Bruck loops by the fixed point free action of the inner mapping
group.
In the second part of our work we aim at a classification of connected differentiable
Bol loops of dimension 2 and the determination of the full collineation group of all
4-dimensional differentiable 3-nets which have a Bol loop among their coordinate
loops. This goal is achieved in Sections 23 and 24. In view of the results of Section 8
this is also the classification of connected 2-dimensional topological Bol loops having
a locally compact group topologically generated by their left translations.
The determination of the 2-dimensional differentiable Bol loops L is based on the
fact that with any such L there is associated a 2-dimensional symmetric space and
that these spaces as well as the groups Σ generated by their reflections are classified
([34]). Since the groups Σ are related by means of the collineation groups of the
corresponding nets to the groups G generated by the left translations of the coordinate
loops which are Bol loops, the groups Σ allow us to determine possible candidates
for the groups G. It turns out that the connected Lie groups G are of dimension 3
or 4. This fact motivated us to classify, in Section 23, all differentiable strongly left
alternative connected loops having a 3-dimensional Lie group as the group topologi-
cally generated by their left translations. Furthermore we characterize the left A-loops
among them; beside the hyperbolic plane loop they correspond, up to isomorphism, to
the 3-dimensional connected Lie groups having precisely two 1-dimensional normal
subgroups.
The 2-dimensional differentiable connected Bol loops are dominated by Bruck
loops: every such Bol loop is isotopic to a Bruck loop. There exist precisely two
2-dimensional connected differentiable Bruck loops. Both of them are closely related
to metric plane geometries and their groups of motions. One of them is the geodesic
loop of the hyperbolic plane. The other one, which is also homeomorphic to R2
, is the
geodesic loop of the symmetric space realized on the manifold of lines of positive slope
in the pseudo-euclidean affine plane. The left, right as well as the middle nucleus of the
hyperbolic plane loop is trivial, the left and the middle nucleus of the pseudo-euclidean
plane loop is trivial but its right nucleus is isomorphic to ®
L
It is remarkable that the behavior of the right translations of the 2-dimensional
differentiable connected Bol loops L differs fundamentally from that of left translations
fundamentally: the group topologically generated by the right translations of L cannot
be a Lie group.
The 4-dimensional differentiable Bol nets attracted our interest since they carry all
information about the isotopism classes of 2-dimensional differentiable Bol loops. We
calculated the groups topologically generated by the Bol reflections and determined](https://image.slidesharecdn.com/88614-250726132104-773001ef/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-25-320.jpg)
![valuable things which they managed to take along with them. Tuti showed
the skirt that was torn while she was dishonored. Saarnaza is 40 years old
and Tuti—50. The Cossacks tore from the sufferers their silver head-
ornaments.
(4) Nubara Krikoryanz, 70-75 years old, mother of Ovanes
Krikoryanz. She corroborated all the testimony given by the priest, and
added the following: “I was violated by five Cossacks. It was dark in the
room. The Cossacks, entering the room, lit a match, which was soon
extinguished. Seeing that I was a woman, the Cossacks seized me and
violated me, one after another. It was at midnight. The Cossacks plundered
our house. The wife of Ovanes was hiding in the mountains with others, and
only thanks to this circumstance she escaped disgrace.”
APPENDIX B
THE REPLY TO THE CROWN SPEECH BY THE FIRST DUMA,
1906[24]
Your Majesty: In a speech addressed to the representatives of the people it
pleased your Majesty to announce your resolution to keep unchanged the
decree by which the people were assembled to carry out legislative
functions in coöperation with their monarch. The State Duma sees in this
solemn promise of the monarch to the people a lasting pledge for the
strengthening and the further development of legislative procedure in strict
conformity with constitutional principles. The State Duma, on its side, will
direct all its efforts toward perfecting the principles of national
representation and will present for your Majesty’s confirmation a law for
national representation, based, in accordance with the manifest will of the
people, upon principles of universal suffrage.
Your Majesty’s summons to us to coöperate in a work which shall be
useful to the country finds an echo in the hearts of all the members of the
State Duma. The State Duma, made up of representatives of all classes and
all races inhabiting Russia, is united in a warm desire to regenerate Russia
and to create within her a new order, based upon the peaceful coöperation
of all classes and races, upon the firm foundation of civic liberty.
But the State Duma deems it its duty to declare that while present
conditions exist, such reformation is impossible.](https://image.slidesharecdn.com/88614-250726132104-773001ef/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-40-320.jpg)
Loops in Group Theory and Lie Theory Péter Nagy
- 1. Loops in Group Theory and Lie Theory Péter Nagy - PDF Download (2025) https://ebookultra.com/download/loops-in-group-theory-and-lie- theory-peter-nagy/ Visit ebookultra.com today to download the complete set of ebooks or textbooks
- 2. Here are some recommended products for you. Click the link to download, or explore more at ebookultra.com Mixed Methods Research Merging Theory with Practice 1st Edition Sharlene Nagy Hesse-Biber https://ebookultra.com/download/mixed-methods-research-merging-theory- with-practice-1st-edition-sharlene-nagy-hesse-biber/ Ethics Theory and Practice Jacques P. Thiroux https://ebookultra.com/download/ethics-theory-and-practice-jacques-p- thiroux/ Group Theory for High Energy Physicists 1st Edition Rafique https://ebookultra.com/download/group-theory-for-high-energy- physicists-1st-edition-rafique/ Eye Movement Theory Interpretation and Disorders Theory Interpretation and Disorders 1st Edition Dominic P. Anderson https://ebookultra.com/download/eye-movement-theory-interpretation- and-disorders-theory-interpretation-and-disorders-1st-edition-dominic- p-anderson/
- 3. Group Theory A Physicist s Survey 1st Edition Pierre Ramond https://ebookultra.com/download/group-theory-a-physicist-s-survey-1st- edition-pierre-ramond/ The Representation Theory of the Symmetric Group Gordon Douglas James https://ebookultra.com/download/the-representation-theory-of-the- symmetric-group-gordon-douglas-james/ Group Analytic Therapists at Work 1st Edition Amélie Noack https://ebookultra.com/download/group-analytic-therapists-at-work-1st- edition-amelie-noack/ The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics 1st Edition A.N. Vasil'Ev https://ebookultra.com/download/the-field-theoretic-renormalization- group-in-critical-behavior-theory-and-stochastic-dynamics-1st-edition- a-n-vasilev/ Simple Lie Algebras over Fields of Positive Characteristic Volume I Structure Theory Helmut Strade https://ebookultra.com/download/simple-lie-algebras-over-fields-of- positive-characteristic-volume-i-structure-theory-helmut-strade/
- 5. Loops in Group Theory and Lie Theory Péter Nagy Digital Instant Download Author(s): Péter Nagy, Karl Strambach ISBN(s): 9783110170108, 3110170108 Edition: Reprint 2011 ed. File Details: PDF, 12.80 MB Year: 2002 Language: english
- 6. de Gruyter Expositions in Mathematics 35 Editors Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Columbia University, New York R.O.Wells, Jr., Rice University, Houston
- 7. de Gruyter Expositions in Mathematics 1 The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann, J. D. Lawson, J. S. Pym (Eds.) 2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues 3 The Stefan Problem, A. M. Meirmanov 4 Finite Soluble Groups, K. Doerk, T. O. Hawkes 5 The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin 6 Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin 7 Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, Μ. V. Zaicev 8 Nilpotent Groups and their Automorphisms, Ε. I. Khukhro 9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug 10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini 11 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao 12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub 13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Nazarov, B. A. Plamenevsky 14 Subgroup Lattices of Groups, R. Schmidt 15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, P. H. Tiep 16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti, A. J. Sommese 17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno 18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig 19 Blow-up in Quasilinear Parabolic Equations, A. A. Samarskii, V.A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov 20 Semigroups in Algebra, Geometry and Analysis, Κ. H. Hofmann, J. D. Lawson, Ε. B. Vinberg (Eds.) 21 Compact Projective Planes, H. Salzmann, D. Betten, Τ. Grundhöf er, Η. Hühl, R. Löwen, M. Stroppel 22 An Introduction to Lorentz Surfaces, Τ. Weinstein 23 Lectures in Real Geometry, F. Broglia (Ed.) 24 Evolution Equations and Lagrangian Coordinates, A. M. Meirmanov, V. V. Pukhnachov, S. I. Shmarev 25 Character Theory of Finite Groups, B. Huppert 26 Positivity in Lie Theory: Open Problems, J. Hilgert, J. D. Lawson, K.-H. Neeb, Ε. B. Vinberg (Eds.) 27 Algebra in the Stone-Cech Compactification, N. Hindman, D. Strauss 28 Holomorphy and Convexity in Lie Theory, K.-H. Neeb 29 Monoids, Acts and Categories, M. Kilp, U. Knauer, Α. V. Mikhalev 30 Relative Homological Algebra, Edgar E. Enochs, Overtoun M. G. Jenda 31 Nonlinear Wave Equations Perturbed by Viscous Terms, Viktor P. Maslov, Petr P. Mosolov 32 Conformal Geometry of Discrete Groups and Manifolds, Boris N. Apanasov 33 Compositions of Quadratic Forms, Daniel B. Shapiro 34 Extension of Holomorphic Functions, Marek Jarnicki, Peter Pflug
- 8. Loops in Group Theory and Lie Theory by Peter Τ. Nagy Karl Strambach W DE _G Walter de Gruyter · Berlin · New York 2002
- 9. Authors Peter Τ. Nagy Institute of Mathematics University of Debrecen P.O.B. 12 4010 Debrecen, Hungary nagypeti@math.klte.hu Karl Strambach Mathematisches Institut der Universität Erlangen-Nürnberg Bismarckstr. Υ/τ 91054 Erlangen, Germany strambach@mi. uni-erlangen. de Mathematics Subject Classification 2000: 22-02, 20-02; 20N05, 20G20, 22E60, 51A20, 51A25, 51H20, 53C30, 53C35, 57S10, 57S15, 57S20 Key words: Loops, quasigroups, Lie groups, Lie transformation groups, Lie algebras, tangent alge- bra, symmetric spaces, 3-net, 3-web, configurations, collineation groups, Bol loops, Bruck loops, Moufang loops, topological translation planes, quasifields © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress — Cataloging-in-Publication Data Nagy, Peter Tibor. Loops in group theory and Lie theory / Peter T. Nagy, Karl Stram- bach. p. cm ISBN 3-11-017010-8 1. Loops (Group theory) 2. Lie groups. I. Strambach, Karl. II. Title. QA 174.2 .N34 2001 512'.2-dc21 2001042383 Die Deutsche Bibliothek — Cataloging-in-Publication Data Nagy, Peter T.: Loops in group theory and Lie theory / by Peter T. Nagy ; Karl Stram- bach. - Berlin ; New York : de Gruyter, 2002 (De Gruyter expositions in mathematics ; 35) ISBN 3-11-017010-8 © Copyright 2002 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typesetting using the authors' TgX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Cover design: Thomas Bonnie, Hamburg.
- 10. Preface In this book the theory of loops is viewed as a part of group theory. Special attention is paid to topological, differentiable, and algebraic loops which are treated using Lie groups, algebraic groups and symmetric spaces. The topic of this book requires some patience with computations in non-associative structures; the basic notions needed as well as facts from the general theory of loops are collected in Section 1.1. The reader is expected to have a basic knowledge of group theory and some fa- miliarity with Lie groups, Lie algebras and differential geometry of homogeneous spaces. For the new material complete proofs are given. For the needed known results, the proofs of which require extensive preparation not concerning our topic, precise available references are quoted. We tried to organize the sections of this book in such a way that they are readable mostly independently from each other. But e.g. the sections 5,6, 8,9,10,11 and 13 as well as 2, 9, 10, 14, 15 and 16 treat closely related topics from the same point of view and with similar methods. The second part of this book is mainly devoted to examine the possibilities in small dimensions; the variety of constructed examples shows that a classification is feasible only for loops with some weak associativity condition. The authors are grateful to the Volkswagen-Stiftung (RiP-Program at Oberwol- fach), to the Paul Erdös Summer Research Center for Mathematics, to Deutscher Akademischer Austauschdienst and to the Hungarian Ministry of Education for the partial support of this project. Special thanks are due to A. Figula, T. Grundhöfer, Ο. Η. Kegel, G. P. Nagy, A. Schleiermacher and V. Zambelli who read the whole or some parts of this manuscript and gave us many useful suggestions. We also want to mention a workshop at the Palacky University of Olomouc supported by GACR 201/99/0265 of J. Mikes. Finally we thank L. Kozma for the preparation of the book in final form.
- 12. Notation In principle, we try to avoid in the text symbols and abbreviations, in particular if they are not of standard use in the literature. As usual Z, R, C denote the integer, real and complex numbers, respectively. H means the (real) quaternions and Ο the (real) octonions (Cayley numbers). If (Μ, ·) is a loop then χ · (y · z) is often denoted by λ: · yz- If G is a group and Μ is a subset of G then (M) is the subgroup generated by M. If Μ is a subset in a topological space S then Μ is the closure of Μ in S. If Η is a subgroup of G then we write Η < G and G/H is the set of left cosets of Η in G. G' means mainly the commutator subgroup of G. For the classical groups we use the standard notation as e.g. GL„(^T), PGL„(A"), SOn{K), PSL2 (/0, PSpn (K), PSU„(/0 for a suitable field K. For the Lie algebras of the classical groups we use the corresponding notation gln, $on, on, $un, etc. If the algebraic structures M and M2 are isomorphic we write Mi = M2. If φ : A —> ß is a map then the image of an element χ e A is usually denoted by φ(χ); but sometimes we write χ φ or χφ. For structures M and Mi the structure M χ Mi is given as usual by {C*i,*2);e M,x2 e M2}. Let Gi and G2 be groups and φ : G — G 2 be a homomorphism. Then we distinguish between the following two subgroups of G χ Gi'· Gl X <p(Gi) = {{χ,φ{χΐ)) Xl,X2 € G l ) and (ΰι,φ(ΰι)) = {(χ,φ(χ)); χ G Gj} . Ν χ Κ denotes a semidirect product of the normal subgroup Ν by the group Κ. For two sets A and Β the relation A c Β means that Λ is a (not necessarily proper) subset of Β. The differentiability class Cr (r — 1 , . . . , 00) is the class of r-times continuously differentiable maps and manifolds, Cw denotes the class of real analytic maps and manifolds.
- 14. Contents Preface ν Notation vii Introduction 1 Parti General theory of transitive sections in groups and the geometry of loops 1 Elements of the theory of loops 13 1.1 Basic facts on loops 13 1.2 Loops as sections in groups 17 1.3 Topological loops and differentiable loops 29 2 Scheerer extensions of loops 42 3 Nets associated with loops 53 4 Local 3-nets 60 5 Loop-sections covered by 1-parameter subgroups and geodesic loops 65 6 Bol loops and symmetric spaces 80 7 Bol nets 95 8 Strongly topological and analytic Bol loops 100 9 Core of a Bol loop and Bruck loops 102 9.1 Core of a Bol loop 102 9.2 Symmetric spaces on differentiable Bol loops 113 10 Bruck loops and symmetric quasigroups over groups 120
- 15. χ Contents 11 Topological and differentiable Bruck loops 129 12 Bruck loops in algebraic groups 143 13 Core-related Bol loops 150 14 Products and loops as sections in compact Lie groups 166 14.1 Pseudo-direct products 166 14.2 Crossed direct products 168 14.3 Non-classical differentiable sections in compact Lie groups 170 14.4 Differentiable local Bol loops as local sections in compact Lie groups 173 15 Loops on symmetric spaces of groups 174 15.1 Basic constructions 174 15.2 A fundamental reduction 182 15.3 Core loops of direct products of groups 186 15.4 Scheerer extensions of groups by core loops 190 16 Loops with compact translation groups and compact Bol loops 194 17 Sharply transitive normal subgroups 208 Part II Smooth loops on low dimensional manifolds 18 Loops on 1-manifolds 235 19 Topological loops on 2-dimensional manifolds 249 20 Topological loops on tori 256 21 Topological loops on the cylinder and on the plane 262 21.1 2-dimensional topological loops on the cylinder 262 21.2 Non-solvable left translation groups 264 22 The hyperbolic plane loop and its isotopism class 276 23 3-dimensional solvable left translation groups 289 23.1 The loops L(a) and their automorphism groups 290 23.2 Sharply transitive sections in £2 x K. 298 23.3 Sections in the 3-dimensional non-abelian nilpotent Lie group 308 23.4 Non-existence of strongly left alternative loops 312
- 16. Contents xi 24 4-dimensional left translation group 317 25 Classification of differentiable 2-dimensional Bol loops 321 26 Collineation groups of 4-dimensional Bol nets 329 27 Strongly left alternative plane left A-loops 335 28 Loops with Lie group of all translations 338 29 Multiplicative loops of locally compact connected quasifields 344 29.1 2-dimensional locally compact quasifields 345 29.2 Rees algebras Qe 346 29.3 Mutations of classical compact Moufang loops 348 Bibliography 351 Index 359
- 18. Introduction The first impulse to study non-associative structures came in the first decades of the 20th century from the foundation of geometry, in particular from the investigation of coordinate systems of non-Desarguesian planes. An additional interest forW. Blaschke to treat loops and quasigroups systematically came from topological questions of dif- ferential geometry, in particular from the topological behavior of geodesic foliations [11]. R. Baer [7], A. A. Albert [4], [5], and R. H. Bruck [17] established the theory of quasigroups and loops as an independent algebraic theory. For Baer the geometry associated with a loop remains an important tool, for Bruck the theory of loops is a part of general algebra [18]. Albert prefers to consider translations of a loop and to see a loop as a section in the group generated by them. The development of the theory of loops and quasigroups in the last 50 years was continued in the spirit of these three directions. An important representative for the study of loops, quasigroups and their associated geometry as abstract objects is V. D. Belousov [9]. The investigation of loops within the framework of topological algebra, topological geometry and differ- ential geometry gained importance by the work of A. I. Malcev [89], Κ. H. Hofmann [47], H. Salzmann [120] and M. A. Akivis [3], The usefulness of analytic methods for the theory of loops is shown in the work of L. V. Sabinin [116]. All these branches of the theory of loops are collected and documented in [21]. Our aim here is to consider the theory of loops as a part of group theory; this means to treat loops as sharply transitive sections in groups. Hence the group theoretical point of view predominates the methods of non-associative algebra. We restrict our attention to such classes of groups in which the simple objects are classified (e.g. finite groups, algebraic groups, Lie groups). Since an essential part of our work will deal with sharply transitive sections in Lie groups we shall use systematically also differential geometric methods. From incidence structures we shall substantially use 3-nets which are the geometries associated with loops; they are the most important tool for problems concerning isotopism classes of loops. The local version of differentiable 3-nets are 3-webs which are coordinatized by local differentiable loops. Since the theory of 3-webs is systematically studied and the application of results on local loops can be used in the global theory, the local point of view of 3-web geometry and the theory of differentiable loops belong to the arsenal of our methods. Binary operations " · " Μ χ Μ Μ on a set Μ with the property that for given a, b e Μ the equations a · χ = b and y • a = b are uniquely solvable correspond to sharply transitive subsets in transformation groups generated by these subsets. Indeed the left transformations λα : χ ax are bijections and for given y, ζ e Μ there exists precisely one left translation λα with Aa (j) — z. In Μ there is an element 1
- 19. 2 Introduction for which 1-jc = jc-1 = jcif and only if the set of left translations contains the identity. Sets with a binary operation " · " which correspond to sharply transitive sets of transformations containing the identity are called loops. If (L, ·) is a loop then the left translations χ ax (a e L) generate a permu- tation group G on the set L. For non-associative L the group G is not sharply transitive on Μ and the stabilizer Η of 1 e L in G is different from the identity. The mapping σ : a λα : L —> G is & section with respect to the natural projec- tion π : G G/Η. One may identify L with the factor space G/H and transport the multiplication. Hence, the theory of loops coincides with the theory of triples (G, Η, σ), where G is a group, Η a subgroup containing no normal non-trivial sub- group of G and σ a section G/H —• G such that a(G/H) acts sharply transitively on the left cosets xH, χ e G, and generates G. According to Baer [7], the set a ( G / H ) of representatives is sharply transitive on the factor set G/H if and only if σ (G/H) is a set of representatives in G for every subgroup conjugate to Η. In [101] and [102] it is stressed that it is always possible and fruitful to carry over the loop properties into properties of the section σ. If C denotes the category of topological spaces, differentiable manifolds, or of algebraic varieties, then a loop is a C-loop if its multiplication, left division and right division are C-morphisms. In the theory of triples (G, Η, σ), where G is a Lie group, Η a closed subgroup and σ . G/H ^ G a C-morphism, the triple (G, Η,σ) defines a C-loop multiplication by the rule (χι H) • (X2H) = σ(χιΗ)χ2Η. The first attempt to deal with differentiable and analytic loops was to follow the ideas of Sophus Lie and to classify analytic loops by their tangential objects. In the last 30 years this research program has been applied successfully to differentiable Moufang loops (they are automatically analytic, cf. [102]) by Kuz'min, Kerdman and Nagy. By their results the theory of differentiable Moufang loops has been carried up to the level of the theory of Lie groups. Since the Hausdorff-Campbell formula works also for binary Lie algebras, the theory of diassociative analytic local loops may be treated successfully using binary Lie algebras the structure of which has been determined by A. N. Grishkov. After this progress the attention turned to the class of differentiable Bol loops. Here as well the investigations proceeded in the spirit of Sophus Lie. With any analytic Bol loop L there is associated a Bol algebra B(L) such that two Bol loops are locally isomorphic if and only if the corresponding Bol algebras are isomorphic ([91], XII.8.12. Proposition). Hence the analytic local Bol loops can be classified by the Bol algebras. But at this point a crucial difference from the theory of Lie groups and Lie Moufang loops comes to light. Whereas any local Lie group or local Lie Moufang loop may be embedded into a global one, this fact fails to be true for local analytic Bol loops (and hence much more so for analytic local loops in general). Already the classification of local analytic 2-dimensional Bol loops by Ivanov ([61], [62], [63]) and the classification of global differentiable 2-dimensional loops show the great difference between the varieties of differentiable global and local Bol loops.
- 20. Introduction 3 Moreover, in Section 17 we exemplify by a class of analytic loops that the infinitesimal behavior of a loop does not determine its global properties. Hence the investigation of global differentiable loops cannot be reduced to that of local loops, and the procedure to investigate suitable sections in Lie groups seems to us the only feasible method for the classification of differentiable global loops for which the group topologically generated by left translations is a Lie group. This restriction seems relatively mild to us since differentiable loops with some weak associativity conditions have this property (e.g. Bol loops, see [91], Proposition XII.2.14 and [102]). With any loop L there is associated an incidence structure, called a 3-net Ν (cf. [8]); if L is differentiable or analytic then Ν is also differentiable or analytic. Conversely, to a 3-net Ν there corresponds a full class of isotopic loops, and with each point of Ν taken as origin we may associate a coordinate loop defining the multiplication for the points of a line through the origin graphically. Two coordinate loops of a net Ν are isomorphic if and only if their points of origin are in the same orbit of the collineation group Θ of T V which preserves the directions ([8], p. 50). Any identity in a loop corresponds to a configuration in the associated 3-net, and configurations in 3-nets yield identities in some coordinate loops. The 3-nets associated with Bol loops are of particular importance. For every line of a distinguished pencil in such a net there exists an involutory collineation fixing this line pointwise. These reflections generate a group Γ acting transitively on the distinguished pencil. If the Bol loop is differentiable then Γ is a Lie group which induces the structure of an affine symmetric space on the distinguished pencil. In an algebraic setting this symmetric space is a left distributive groupoid which is called the core of the Bol loop. These relations allow us to apply the rich theory of symmetric spaces as well as the results on left distributive quasigroups in order to classify wide classes of Bol loops. 3-webs are incidence geometries associated with differentiable local loops; their pencils of lines form 3 foliations. If a 3-web is associated with a differentiable local Bol loop then for every leaf of one of the 3 foliations there exists a local reflection. The group generated by these local reflections induces an affine locally symmetric space on the manifold of leaves of this foliation; this connects the theory of local differentiable Bol loops with the classical theory of locally symmetric spaces. Our contribution has two parts. The first part contains the foundations of our new methods and their applications to the extension theory of Bol loops, to the algebraic theory of symmetric spaces and to the Lie theory of smooth loops. Moreover, we classify strongly 2-divisible finite, differentiable and algebraic Bruck loops, and we clarify the role of compactness of the group topologically generated by the left trans- lations of a compact loop. In the second part we apply our methods to the topological and differentiable loops on manifolds of small dimension. In the Sections 1, 3 and 4 we develop the foundations for our point of view, describe the interactions between the properties of sharply transitive sections and the corresponding loops, discuss the relations between configurations in 3-nets and corresponding identities in the coordinate loops and show what the local version of these concepts is in the case of differentiable objects; at this instant the importance of
- 21. 4 Introduction differentiable foliations and 3-webs comes into light. Using the methods of algebraic topology we prove that any topological loop and hence any topological 3-net on a connected topological manifold is orientable. In Section 2 we thoroughly analyze the structure of proper loops which are ex- tensions of groups by groups. In contrast to the obstructions for study of general extensions of loops, the special case of extension theory developed in Section 2 al- lows a transparent description by group theoretical methods. If we work in the category of loops realized on topological or differentiable manifolds then our basic assumption is that the group G topologically generated by the left translations is a Lie group. In Section 5 we study differentiable loops such that the sharply transitive section cor- responding to L is (locally) covered by 1-parameter subgroups. This class (locally) coincides with the opposite loops of geodesic loops with respect to a uniquely deter- mined affine connection with vanishing curvature. Special cases of these loops are the opposite loops of the geodesic (local) loops of reductive homogeneous spaces and the differentiable (local) Bol loops. Sections 6 to 13 are devoted to a thorough study of Bol loops, their analytic and algebraic properties as well as to their relations to the classical theory of affine symmetric spaces. Here we pay attention to the global as well as to the local Bol loops. In Section 6 some representations of local Bol loops are given as sections in Lie groups and their relations to Lie triple systems and Bol algebras. In Section 7 the geometric version of isotopism classes of Bol loops, the Bol nets, are investigated. Of special interest for us is the group of collineations of a Bol net as well as some of its subgroups, for instance the group generated by the Bol reflections. The core of a Bol loop introduced and studied in Section 9 is a symmetric space in the sense of Loos, and it may be used in the theory of groups of exponent 3. (Cf. Corol- lary 9.7). The closest relation between a local Bol loop L and its local core takes place if the Bol algebra of L is a Lie triple system. For a global Bol loop this is the case if the core is a symmetric quasigroup and the Bol loop L satisfies the automorphic inverse property. Then the Bol loop L is a left Α-loop, and to any element χ there exists pre- cisely one element y with χ — y2 . These loops are called strongly 2-divisible Bruck loops. Differentiable strongly 2-divisible loops having the left inverse property are already Bruck loops if they are left Α-loops and satisfy the automorphic inverse prop- erty. Differentiable Bol loops are always locally strongly 2-divisible. G. Glauberman proved [36] that any strongly 2-divisible Bruck loop L can be embedded into the group G generated by the left translations of L and that the multiplication of the embedded loop L is induced by the multiplication of G. We generalize this embedding to the class of strongly 2-divisible Bol loops. Moreover, using this construction we classify strongly 2-divisible differentiable connected Bruck loops in Section 11. They corre- spond in a unique way to pairs (G, σ), where G is a connected Lie group and σ is an involutory automorphism of G such that the subgroup centralized by σ contains no non-trivial normal subgroup of G, the exponential mapping from the (— l)-eigenspace m of σ on the Lie algebra g into G is a diffeomorphism and expm generates G. Our systematic study of strongly 2-divisible Bruck loops in algebraic groups (Section 12),
- 22. Introduction 5 as well as of Bruck loops associated with symmetric quasigroups over groups (Sec- tion 10) is motivated by the increasing importance of this class of loops in algebra and geometry. One of the reasons for this is the classification problem of sharply 2-transitive permutation groups. Any such group corresponds in a unique way to a so-called near domain which is an algebraic structure (F, + , ·; 0, 1) with two oper- ations: with respect to the addition (F, + ; 0) is a Bruck loop such that χ + χ = 0 implies χ = 0, with respect to the multiplication (F {0}, ·; 1) is a group and the left distributive law holds (cf. [66], [75]). Till now, the only known examples of near domains are the near fields where the additive structure is an abelian group. Although this question motivated many interesting contributions, the existence of strongly 2- divisible Bruck loops having a sharply transitive group of automorphisms remains an interesting and important problem (cf. e.g. [76], [77], [68]). For differentiable Bol loops it is no restriction to assume that the group topolog- ical^ generated by the left translations is a Lie group. We prove in Section 9 that the category of real analytic Bol loops coincides with the category of connected topo- logical Bol loops for which the group topologically generated by the left translations is a connected, locally compact, locally connected and finite-dimensional topological group (cf. [92]). As a consequence we obtain that any closed subloop of an analytic Bol loop is analytic. For differentiable Moufang loops the group topologically gen- erated by all left and right translations is a Lie group. In constrast to this, there are examples of differentiable Bol loops having a Lie group as the group topologically generated by all left and right translations as well as those for which this is not the case (cf. Sections 7 and 22). In Section 13 we consider local Bol loops associated with the same symmetric space and ask under what conditions two such local Bol loops are locally isotopic. If the symmetric space is compact and irreducible then the answer is affirmative if one excludes the symmetric space on the 7-sphere as well as the Grassmannian manifold of 3-planes in the 8-dimensional real vector space. We prove here also a generalization of a theorem of A. Fomenko [29] that if the group G topologically generated by the left translations of a simply connected differentiable (local) Bol loop L is reductive then L is the direct product of a reductive Lie group by a direct product of proper Bol loops Li not having any connected non-trivial normal subloop. Moreover, the symmetric space associated with L(- is irreducible and G decomposes in an analogous way as L into factors. In Section 14 we give for loops generalizations of the semidirect products of groups. These constructions allow us to find examples of compact differentiable connected loops having a compact Lie group as the group topologically generated by the left translations. These examples of loops are realized on products on 7-spheres, 7-dimensional real projective spaces and of spaces of compact Lie groups; in them the associativity law is strongly violated. But it is not possible to discover examples with good associativity properties besides the Scheerer extensions studied in Sections 2 and 15. The reason for this are the theorems which give a full classification of compact differentiable Bol loops. These loops are repeated extensions of groups by groups and
- 23. 6 Introduction by Moufang loops; the theory of these extensions is developed in Sections 2 and 15. It follows in particular that every connected compact differentiable simple Bol loop is a Moufang loop. These results proved in Section 16 belong to the main achievements of the first part of our work. The following theorem has the same quality: Every compact connected topological loop is a Moufang loop if the group topologically generated by all left and right translations is a compact Lie group. According to [55] the topological loops such that the group topologically generated by all their left and right translations is a compact Lie group are precisely those which have an invariant uniformity. In contrast to this, at the end of Section 14, we prove that there are many local Bol loops having an interpretation as local sections in compact Lie groups. The result in [121] shows that the assumption of compactness for the group G topologically generated by the left translations of a topological loop is very restrictive. If G is a compact quasi-simple Lie group then it must be locally isomorphic to SOs(K). If G is not quasi-simple, then Scheerer describes in [121] the global sections in G. It was his description which put us in the position to classify compact differentiable Bol loops. The group topologically generated by the left translations of such a loop is a reductive compact Lie group having at least 2 quasi-simple factors; in this case one of these two factors is locally isomorphic to SOs(M). It seems to us that the most interesting objects for the study of compact topological or differentiable loops which are not Bol loops but which have a compact connected Lie group as the group topologically generated by the left translations are the loops homeomorphic to the (η — 1)-dimensional projective space or to the (η — l)-sphere, where η e {4, 8} since several natural algebraic or analytic assumptions force these loops to be classical. Till now no sharply transitive continuous section in the group SOs (M) respectively S04.(R) is known which does not correspond to the Moufang loop or to the group of octonions or of quaternions of norm one, respectively. But for η = 4 or η — 8 we have been able to find loops diffeomorphic to the (η — 1)-dimensional projective space, respectively to the (η — 1)-dimensional sphere such that the groups topologically generated by the left, right and by all translations coincide; they are Lie groups isomorphic to PSL„ (M), or to SL„(R), respectively (cf. Theorem 29.3). In Section 17 we deal with topological loops homeomorphic to R", admitting a sharply transitive subgroup Ν of the group topologically generated by the left transla- tions and give examples in which the group G topologically generated by all left and right translations is a nilpotent Lie group; in these cases Ν is not noraial in G. If in contrast to this we assume that Ν is a normal subgroup of the group G topologically generated by all left and right translations then G (which must be a Lie group) is not nilpotent. Whereas any 2-dimensional loop in this class must be a group, in dimen- sion 3 there are already interesting examples of proper loops of this type. The main result in this section is the statement that for any simple compact Lie algebra g the multiplication χ ο y = χ + y + [χ, >>] defines a loop L on g such that the group G topologically generated by the left translations coincides with the group generated by all translations of L and that G is isomorphic to a semidirect product G = Τ χ Η. The normal subgroup Τ in this semidirect product is the group Kn of all translations
- 24. Introduction 7 of the affine space over the vector space g, where Κ — R, unless g is the unitary Lie algebra su„(C) or the symplectic Lie algebra su„(H); in the last two cases Κ is the field of complex numbers or of quaternions, respectively; the group Η is isomorphic to the connected component of the group GL/(AT), where I is the dimension over Κ of the minimal representation of g, and acts irreducibly on g. A further statement which is valid for any finite-dimensional loop L having a Lie group G as the group topologically generated by its left translations is Theorem 20.1. There we describe explicitly the structure of G if L is homeomorphic to the «-dimensional torus. We want to stress that, in general, for coverings G of Lie groups which are topologically generated by the left translations of a connected topological loop there are no topo- logical loops having G as the group topologically generated by their left translations. We confirm this phenomenon in Section 19 showing that no proper covering of the group PSL2(M) can be a group topologically generated by the left translations of a 2-dimensional topological loop. Many of the examples of loops admitting a sharply transitive normal subgroup Ν in the group generated by left translations may be realized in the category of algebraic loops and algebraic groups. From the topological point of view the simplest topological loops L are those which are realized on one-dimensional topological manifolds and which have a locally compact group G as the group topologically generated by their left translations. We shall deal with this class in Section 18. We show that for any proper loop L of this type the group G is a covering of the group PSL2OR), and we describe all loops L which are coverings of differentiable loops homeomorphic to the circle and have only trivial centre. In this class there are proper loops satisfying the left inverse property; but any topological left Α-loop as well as any monassociative loop in this class is already one of the 2 one-dimensional Lie groups. The main part of Sections 19 to 22 is devoted to topological loops on 2-dimensional manifolds having a locally compact group G as the group topologically generated by their left translations. Any such loop is homeomorphic to the torus, to the cylinder or to the plane. The abundance of the examples of these loops constructed there shows that a complete classification of such loops cannot be expected. But they may be described explicitly if they are coverings of loops realized on the torus or the cylinder and if G is not solvable. This is shown in Sections 19 and 20. If the group G topologically gen- erated by the left translations of a 2-dimensional topological loop L is a quasi-simple Lie group then it is isomorphic to the group PSL2(K). For this group we give explicit descriptions of some sharply transitive sections and show that there exists one among them consisting only of parabolic elements. Moreover, if the group topologically generated by the left translations of a topological connected 2-dimensional loop is a non-solvable Lie group then the group topologically generated by all left and right translations cannot be a Lie group. In contrast to this we give in Section 28 examples of 2-dimensional connected differentiable loops having a solvable Lie group topologi- cally generated by their left translations for which the group topologically generated by all left and right translations is also a Lie group.
- 25. 8 Introduction The exceptional status of the hyperbolic plane loop and of its isotopy class is based upon the fact that it is associated with the hyperbolic plane geometry and generalizes in a direct way the vector group of the euclidean plane. We treat the hyperbolic plane loop in Section 22. In particular, we characterize it there within the class of strongly left alternative 2-dimensional connected topological loops by the group topologically generated by the left translations, as well as within the 2-dimensional connected topological Bruck loops by the fixed point free action of the inner mapping group. In the second part of our work we aim at a classification of connected differentiable Bol loops of dimension 2 and the determination of the full collineation group of all 4-dimensional differentiable 3-nets which have a Bol loop among their coordinate loops. This goal is achieved in Sections 23 and 24. In view of the results of Section 8 this is also the classification of connected 2-dimensional topological Bol loops having a locally compact group topologically generated by their left translations. The determination of the 2-dimensional differentiable Bol loops L is based on the fact that with any such L there is associated a 2-dimensional symmetric space and that these spaces as well as the groups Σ generated by their reflections are classified ([34]). Since the groups Σ are related by means of the collineation groups of the corresponding nets to the groups G generated by the left translations of the coordinate loops which are Bol loops, the groups Σ allow us to determine possible candidates for the groups G. It turns out that the connected Lie groups G are of dimension 3 or 4. This fact motivated us to classify, in Section 23, all differentiable strongly left alternative connected loops having a 3-dimensional Lie group as the group topologi- cally generated by their left translations. Furthermore we characterize the left A-loops among them; beside the hyperbolic plane loop they correspond, up to isomorphism, to the 3-dimensional connected Lie groups having precisely two 1-dimensional normal subgroups. The 2-dimensional differentiable connected Bol loops are dominated by Bruck loops: every such Bol loop is isotopic to a Bruck loop. There exist precisely two 2-dimensional connected differentiable Bruck loops. Both of them are closely related to metric plane geometries and their groups of motions. One of them is the geodesic loop of the hyperbolic plane. The other one, which is also homeomorphic to R2 , is the geodesic loop of the symmetric space realized on the manifold of lines of positive slope in the pseudo-euclidean affine plane. The left, right as well as the middle nucleus of the hyperbolic plane loop is trivial, the left and the middle nucleus of the pseudo-euclidean plane loop is trivial but its right nucleus is isomorphic to ® L It is remarkable that the behavior of the right translations of the 2-dimensional differentiable connected Bol loops L differs fundamentally from that of left translations fundamentally: the group topologically generated by the right translations of L cannot be a Lie group. The 4-dimensional differentiable Bol nets attracted our interest since they carry all information about the isotopism classes of 2-dimensional differentiable Bol loops. We calculated the groups topologically generated by the Bol reflections and determined
- 26. Other documents randomly have different content
- 27. CHAPTER XXII THE TREND Whither? The future of Russia—Why the revolution has not yet succeeded—Probable outcome of the struggle—Inevitableness of eventual overthrow of present régime—Attitude of foreign Powers— The Russian people during the period of rebellion—Effect upon national character—The Czar and the people—The Czar and the world —What we may expect. Say not the struggle naught availeth, The labor and the wounds are vain, The enemy faints not, nor faileth, And as things have been they remain. If hopes were dupes, fears may be liars; It may be, in yon smoke conceal’d, Your comrades chase e’en now the fliers, And, but for you, possess the field. For while the tired waves, vainly breaking, Seem here no painful inch to gain, Far back, through creeks and inlets making, Comes silent, flooding in, the main. And not by eastern windows only, When daylight comes, comes in the light; In front, the sun climbs slow, how slowly! But westward, look, the land is bright! Arthur Hugh Clough. HEN the troubled year 1906 ended, the shadow of reaction began to deepen over the Russian empire. One by one the granted liberties and promised reforms of the manifesto of October, 1905, were being revoked and recalled. Early in 1907 the second Duma met, struggled through a brief existence, and was dissolved by the magic word of the Czar. Discouragement then possessed the people—a sense of heartbreaking hopelessness. To the men and women who had borne the heat
- 28. and burden of the struggle it seemed as if all the efforts and the sacrifices, the lives surrendered to the cause of liberalism and progress, had been in vain. The world at large passed hasty judgment: “The revolution has petered out.” The announcement that a new Duma would be convened in the late autumn of 1907 sounded hollow, for the new election laws, which disenfranchised millions of peasants, promises so completely to devitalize the results of the elections at the very outset, that the whole institution of parliamentarism seems reduced to a mere shell. The results of my observations lead me to accept this period of stagnation and temporary inactivity as a matter of course, a natural phenomenon, consistent and compatible with the mighty struggle in which the Russian nation is now plunged. At the beginning of this book I pointed out that the periods of great revolutions are seldom brief. M. Leroy-Beaulieu said to Tolstoi that Russia’s struggle might continue fifty years. Even that, it seems to me, is a comparatively short time for the working out of all the changes which Russia must undergo before she will be brought to the standard of modern civilization. The political phases of the situation are secondary to the vital social and economic changes which are working out. The ideas of a nation, as well as the customs of a great people and the forms of an ancient government, are all in the flux. Decades must necessarily elapse before such vast renovation is completed. And in the meantime the movement making for this renovation remains of world-wide importance, palpitating as it does with human interest, and involving as it does the concern of a substantial amount of the world’s commercial interest. France, Germany, Austria, England, and America all have business and commercial associations in Russia which are affected by the development or retardation of industrial and agricultural Russia. The intellectual influence of the philosophy of the revolution is equally universal, watched closely by Germany, and Austria, and France, and ultimately destined to touch the uttermost parts of the world. So was it in France—to a greater degree, perhaps, shall this be true of Russia. Precisely as there cannot be mountains without valleys, or flow without ebb, so there cannot be revolution without counter-revolution, or progress without reaction. In the manifesto of October, 1905, Czar Nicholas II said:
- 29. “We charge our government to carry out our inflexible will as follows: “1. To establish an unshakable foundation of the civic liberties of the population, such as inviolability of the person, liberty of conscience, of speech, of meetings, and of unions.... “3. To lay as an unchangeable rule that no law can enter into force without the approval of the imperial Duma; and that the representatives of the people should be entitled to an effective control over the executive power....” All the world knows how speedily every one of these glorious promises was swept aside. The “inflexible will” of the present Emperor of Russia is the most anarchistic influence in the world to-day. It submits to no discipline, it bows to no law, refuses to remember even through brief days most solemn pledges made to the Russian people before the world, and nonchalantly acquiesces in the careless breaking of even God’s laws. The government of Russia to-day rests not on law, or order, or right, but on might, militarism, and simon-pure terrorism. In Appendix D may be found the report of Captain Pietuchow on the Siedlce pogrom, in which is quoted the following utterance of Colonel Tichanowsky: “We must set against the terrorism of the revolution a still more frightful terrorism.” And this is what the officials of czardom are doing to-day. And the terrorism of the government is not only a “more frightful terrorism” than the “terrorism of the revolution,” it is the most frightful and the most monstrous terrorism of modern times, because the forms of government are converted into the tools of absolute lawlessness, and the victims of this terror are often the helpless among the people of the empire—women and girls thrown to the lust of Cossacks, old men and children the marks of police brutality. In the chapter on governmental terrorism, and in the appendix, there is adduced overwhelming evidence, and proof, of official complicity and governmental connivance with this terrorism. Beside the terrorism, the brutality and the ruthlessness of the Russian government, and the soldiers and officials acting in the name of the Russian government, the most heinous offenses of the people pale into insignificance. Individuals are human, and there comes a snapping-point when the sturdiest intellect can no longer beat back frenzy. But a government! A government, surely, cannot be exonerated on these grounds. Madness, desperation, passion should never possess the government of a
- 30. great empire. If it does, then is the incapability of that government amply proven, and its fall deservedly imminent. After the dissolution of the second Duma the Moscow “Viedomosti,” a reactionary organ, printed the following: The population of Russia amounts to some 150,000,000 souls. But in the revolution not more than 1,000,000 are inclined to take any active part. Were these 1,000,000 men and women shot down or massacred, there would still remain 149,000,000 inhabitants of Russia, and this would be quite sufficient to insure the greatness and prosperity of the Fatherland. I myself heard a prominent Russian officer coolly advocate the immediate execution of two million men and women judiciously chosen from every section of the empire, in order to stamp out the movement toward constitutionalism! As for the attitude of the Czar himself I have a conception which is based on careful observation, but which may be at variance with popular opinion in America. I believe that the Czar considers himself a God- ordained autocrat. I believe that he aspires to hand over to his heir and successor as absolute an autocracy as he inherited from his fathers. Elsewhere I have quoted a remark said to have been made by the Czar in 1906 to the effect that he believed “Russia could go for twenty years more without a constitution, and he purposed to do all he could to guide Russia back to where it was before the manifesto of October, 1905.” Everything that has transpired in Russia since these words were spoken points to their truth. The manifesto was wrung from the Czar by the sudden tide of revolution which for once caught the government unprepared. The granting of the constitution was like oil upon troubled waters. But as soon as the government had recovered from the shock it sustained through the revolutionary activity culminating in the general strike, it began quietly to take back everything that had been promised. The first Duma elections were seriously menaced, then on the eve of the meeting of the parliament its powers were substantially reduced. During the sessions of that body insults and rebukes were heaped upon it, and finally it was disbanded. The elections for the second Duma were still more seriously restricted, and although Duma number two was in many respects an advance upon the first Duma it was presently dissolved upon a ridiculous
- 31. pretext. It will be no surprise if the career of Duma number three is quite as short as that of the others, and if at the dissolution of it the government will say, in effect: “We have now experimented with parliamentary government, and the people of the country have shown their unpreparedness for self- government”—with the announcement of an indefinite postponement of further Duma experiments. This is practically what happened in Turkey. And in Russia itself, one hundred and fifty years ago, a similar incipient experiment was made. If this should occur now the world may well believe that the Russian government never had the faintest intention of introducing parliamentary government at this time. As for M. Stolypin—I believe him to be a shrewd, able administrator. I do not believe for a moment that he has liberal sympathies. In this I consciously take issue with many able writers, and even old and tried Russian correspondents. A member of the Constitutional Democratic Party, a deputy in the first Duma, a prominent university professor, who sat on a commission with M. Stolypin, and who had unusual opportunities for studying the premier, said to me: “I believe M. Stolypin to be the strongest man the government has, but a fanatic
- 32. Nicholas W. Tchaykovsky “Father of the Russian Revolution” of reaction.” I would not use the word “fanatic,” but I do believe him to be a devoted champion of reaction and autocracy. At the same time, he appreciates the desirability of appearing before the world in the rôle of a would-be reformer. No modern statesman has watched the press of the world more closely than he, and none has been quicker to trim his sails according to the weather indications that he has there discerned. M. Stolypin, besides being a clever and able minister, is also a brave man. And withal he is blessed with a charming and gracious personality, and it is through the irresistible influence of his polished and cosmopolitan manners that he so diplomatically throws dust in the eyes of the world
- 33. through the correspondents and business representatives of different countries who from time to time are accorded interviews with him. It remains true, however, in spite of his grace and affability, that previous to his administration women and young girls and boys of sixteen and seventeen were not hanged and shot for “suspected” revolutionary activity. It was M. Stolypin who inaugurated the field courts-martial which endeavor to confuse petty civil offenses with revolutionary crimes, thus affording an excuse for hundreds of executions. An Associated Press despatch from St. Petersburg under date of July 23, 1907, read as follows: From many quarters come reports of summary executions under the new regulations for the military district courts, which went into force Saturday. These regulations undo the work of the recent Duma, which abolished the notorious reign of the drumhead court-martial. Under them only seventy-two hours are permitted to elapse between indictment and execution, including the appeal to the Military Court of Cassation, whereas a fortnight was permitted under the old régime. These courts, too, have jurisdiction in all provinces, whereas the old drumhead courts could act only in provinces that had been placed in a state of extraordinary defense. At Kieff yesterday five sappers were executed, and to-day another sapper was sentenced to death. Three peasants have been executed at Moscow, another at Warsaw, and at Yekaterinoslaff three workmen have been put to death. At Riga a young man, named Berland, went into a clothing-store, chose an overcoat, and then started for the door. When asked to settle his bill, he drew a revolver, covered the clerk, and got away. He was captured and sentenced to death. Another young man, named Danbe, was sentenced to death at Riga for the theft of $5, and two girl accomplices, aged 12 and 20 years, were sentenced to exile and hard labor for life. I quote this telegram because the Associated Press has never been suspected of pro-revolutionary proclivities so far as I know, and because it indicates the true character of M. Stolypin and his non-temporizing administration. In thus emphasizing the offenses—not to say crimes—of the present government, I doubtless lay myself open to the charge of anti-governmental bias, yet I believe I am neither guilty of this charge nor blind to the faults, weaknesses, and mistakes of the revolutionary movement. My endeavor has
- 34. been to present a true picture of Russia to-day, and of the struggle going on there as I have witnessed it. Yet I must point out once more that the responsibility of a government is necessarily of a more serious nature than that of individuals who are the victims of governmental and official lawlessness, and whose life and environment, in spite of all they might do, is made insufferable through the corruption, inefficiency, and general immorality of the officials who are set to rule and to administer the land. There is a terrible menace, a grave danger, it seems to me, in this prolonged struggle. Where all standards of public and private morality are shaken—where rulers Catherine Breshkovsky The first woman ever sentenced to hard labor in the mines of Kara. After spending 23 years in prison and in Siberia she escaped, and after making a visit to America in behalf of her countrymen she has returned once more to her hazardous work in the heart of Russia, where she is now at work disguised as a peasant
- 35. and lawgivers are arch lawbreakers—the characters of the individuals living under such a régime must suffer. And alas, for the rising generation! When one thinks on these things the prophecy of Tolstoi has greatest weight— perhaps the seer in this, as in so many other things, is right, and Russia will continue to go from bad to worse, until the whole people awake in the very bottom of the abyss, and then, and then only, will they turn to God as their only hope of salvation. If the public opinion of the world would cry out against foreign bankers periodically advancing money to the present government to maintain its grip at the very throat of the people, governmental concessions would have to be granted. As it is, the people of Russia feel themselves pitted not only against their own government which has all of the machinery of the army and police to support it, but also against the financial interests of Europe and the rest of the world. The mere moral sympathy of America is not much of an offset to a French loan, or an Anglo-Russian alliance, unless it results in preventing American bankers from advancing American money to perpetuate the existing régime. These foreign loans are a terrible discouragement to the Russian people. Whenever the people reach the point where they believe their government will be obliged to yield certain fundamental human rights, through sheer inability to longer feed the forces of reaction, and to pay for the upkeep of the army, then the foreign bankers spring to the rescue. In Russia I do not look for any voluntary “grant” of liberties or freedom from czardom. I believe that, however much one may desire constitutional reform, the Russian people will eventually obtain their liberties through fighting for them. I foresee a long, long struggle. Since October, 1905, the Russian people have advanced enormously, and the Duma experiments, handicapped as they were, have yet proved immense educational influences; they have served to arouse the whole people to what may be, and to awaken within them a realization of what sooner or later must be. On this count alone the value of these short-lived parliaments must not be underrated. The Russian people now understand their own situation as they never have grasped it before. They have not merely lost faith in the Czar, they have learned that the trouble with Russia
- 36. to-day is that it suffers a blight, and that blight is autocracy, which in its very essence is incompatible with modern civilization, and that while the obliteration of autocracy may be a long task, the only escape from their present bondage is the accomplishment of this task. And the period of the struggle making for this end will be recorded in history as the Russian Revolution.
- 37. APPENDICES A—Caucasian testimony; B—The Duma’s Reply to the Throne Speech; C —M. Lopuchin’s letter to M. Stolypin; D—Report on Siedlce pogrom; E —Notes on Wages and Cost of Living. APPENDIX A TRANSLATION OF A FEW PAGES OF TESTIMONY FROM A WHOLE VOLUME OF SIMILAR EVIDENCE COLLECTED BY A SOCIETY OF TIFLIS LAWYERS ON THE “PACIFICATION” IN TRANS-CAUCASIA, 1905-1906 THE EXCERPTS HERE PRINTED ARE NOT OF EXCEPTIONAL CASES, BUT ARE APPALLINGLY REPRESENTATIVE OF THE ENTIRE TEXT. The Village Sos, April 4, 1905. (1) Parish Priest Ter-Akop Bagdasaryan: We learned that a special detachment of Cossacks, under the command of Colonel Vevern, was coming; that the detachment was going from village to village, instructing the Tartars as well as the Armenians to live peacefully, threatening to punish severely all those that will disturb the peace. We were glad of this, and when we learned that the detachment was approaching our village, we at once set out to prepare bread, meat, forage, and also a lodging for the detachment. On the 11th of March, at about 2 o’clock, we noticed the detachment from afar. I called together the prominent people of the village, donned my vestments, took a cross and a Bible, bread and salt, and we started out to greet the detachment. In front of the Cossacks walked many Armenians from various villages, leading the Cossacks’ horses. These Armenians, on noticing the women in our village, were astonished, and they said: “What does this mean? Have they lost their reason? Why have they left their women in the village? The Cossacks violate the women everywhere.” When our women learned of this, they began to run from the village. Justice of the Peace Yermolayev rode first. He said to us in the language of the Tartars: “Go back, you are not worthy to receive us.” After that the same Yermolayev had a conversation with the commander of the detachment, and then turned to me and to our representative people and said: “Your bread and salt cannot be accepted. There will be a different
- 38. settlement with you.” We returned to the village in a painful frame of mind. As soon as the Cossacks entered the village—there were several hundred of them—a signal was sounded. The Cossacks dismounted and rushed after the women; they caught them in the ravines, on the roads, in the forests. Terrible cries were heard on all sides. The Cossacks violated the women, tore off their headgear, their ornaments, and other valuables which they had taken along with them as they hastened from the houses. All this was witnessed by the officers, the district chief, and the justice of the peace, but they did not stop them. Among the women that were violated in the outskirts of the village was a girl of 16-17 years of age, Kola Arutyunyanz. As there were some women that did not succeed in running away in time, I asked all those that remained to come to my house and I said: “As long as I am alive I will defend your honor, and if they kill me, then you shall also die.” Some twenty women gathered in my house, but there were still some women that remained in their houses. Some of these were old, and they thought that they would not be attacked on that account; others did not have time enough to take their children along; still others had sick children. When it became dark the Cossacks began to break into the houses, to plunder, beat and violate the women that were in the houses. Cries of men and women for help came from everywhere. The authorities heard the sobs of the unfortunates, they saw and knew what indecencies were being perpetrated, but they did not check them. It was about 12 o’clock at midnight I was called out of the house. I asked what I was wanted for. I was told that the Cossacks had beaten Ovanes Airetetyan Krikoryanz, that Ovanes was dying, and that they wanted me to come and give him the communion. I went to Ovanes’s house and found him unconscious. The mother of Ovanes, the old woman Nubara, related the following: “When the Cossacks began to break into the houses Ovanes went down to guard the yard, and told me to lock myself in the house and watch it. Suddenly the dogs began to bark. The Cossacks had entered the yard. Ovanes (he was a reservist of low rank) began to implore the Cossacks, half in Russian, half in Tartarian, to spare his life. At that time a powerful blow resounded and right after it Ovanes cried out: ‘Oh, I am dying!’ For a short time a faint rattling was heard, and then all became quiet. A few minutes later the Cossacks turned to the doors of our house and started to break in; at last the doors gave way and the Cossacks came in; there was no light in the house and they did not see that I was an old woman. Despite all entreaties they
- 39. threw me down and violated me, one after another.” After the assault the old woman, almost 70 years old, did not come to herself for half an hour. Having heard Nubara’s statement and finding it impossible to give the communion to Ovanes, as he was in a state of unconsciousness, I returned to my house. In the morning I was notified that Ovanes died. Then I went to the superior officer of the district, Freilich. Yermolayev was also there. In answer to my information he said: ‘Well, what of it? If he died, bury him.’ After I had left, Freilich and Yermolayev went to the commander of the detachment and told him what I had said about Ovanes. He sent two soldiers to investigate. These reported to the commander that Ovanes was alive. Then the commander ordered me to appear before him, and told me that I gave him a false report. Yermolayev, who was present, began to assail me, saying that it was I who had organized the attack upon the Tartars, and that I and my daughter led the attack upon Kadjakh, and that I was in general a dangerous man. I remarked to Yermolayev that his accusations were unjustified, that my daughter had been studying in the Moscow Gymnasium, that she had been in Caucasia for two years and that she had been in Siberia since September, visiting at her brother’s. The commander of the detachment ordered my arrest for the “false” report. The detachment stayed in our village until 2 o’clock of the next day and before leaving heaped the most painful indecencies upon the population. The Cossacks dishonored another girl who was suffering from paralysis, Nubata Musayanz, 12 years old. Her grandfather, Musa, a man of about 70, took his grandchild into his arms and was about to carry her away from the Cossacks, but they threw the old man down and beat him mercilessly, and trampled him with their boots; he is very sick now and the doctors say that unless he undergoes a serious operation he will die soon. The paralyzed little girl, Nubata, was dishonored by the Cossacks in front of the old man. The Village Sos, April 5, 1905. (1) Kola Arutyunyanz, 18 years old: “I ran together with Saarnaza Arutyunyanz. Three Cossacks overtook us and violated us. I was a virgin. The assault was committed upon us after a hard struggle. After the first three Cossacks, three others came, and they also violated us.” (2 and 3) Saarnaza Arutyunyanz and Tuti Kasparyanz corroborated the above given testimony, adding that the Cossacks robbed them of several
- 40. valuable things which they managed to take along with them. Tuti showed the skirt that was torn while she was dishonored. Saarnaza is 40 years old and Tuti—50. The Cossacks tore from the sufferers their silver head- ornaments. (4) Nubara Krikoryanz, 70-75 years old, mother of Ovanes Krikoryanz. She corroborated all the testimony given by the priest, and added the following: “I was violated by five Cossacks. It was dark in the room. The Cossacks, entering the room, lit a match, which was soon extinguished. Seeing that I was a woman, the Cossacks seized me and violated me, one after another. It was at midnight. The Cossacks plundered our house. The wife of Ovanes was hiding in the mountains with others, and only thanks to this circumstance she escaped disgrace.” APPENDIX B THE REPLY TO THE CROWN SPEECH BY THE FIRST DUMA, 1906[24] Your Majesty: In a speech addressed to the representatives of the people it pleased your Majesty to announce your resolution to keep unchanged the decree by which the people were assembled to carry out legislative functions in coöperation with their monarch. The State Duma sees in this solemn promise of the monarch to the people a lasting pledge for the strengthening and the further development of legislative procedure in strict conformity with constitutional principles. The State Duma, on its side, will direct all its efforts toward perfecting the principles of national representation and will present for your Majesty’s confirmation a law for national representation, based, in accordance with the manifest will of the people, upon principles of universal suffrage. Your Majesty’s summons to us to coöperate in a work which shall be useful to the country finds an echo in the hearts of all the members of the State Duma. The State Duma, made up of representatives of all classes and all races inhabiting Russia, is united in a warm desire to regenerate Russia and to create within her a new order, based upon the peaceful coöperation of all classes and races, upon the firm foundation of civic liberty. But the State Duma deems it its duty to declare that while present conditions exist, such reformation is impossible.
- 41. The country recognizes that the ulcer in our present régime is in the arbitrary power of officials who stand between the Czar and the people, and seized with a common impulse, the country has loudly declared that reformation is possible only upon the basis of freedom of action and the participation by the nation itself in the exercise of the legislative power and the control of the executive. In the manifesto of October 17, 1905, your Majesty was pleased to announce from the summit of the throne a firm determination to employ these very principles as the foundation for Russia’s future, and the entire nation hailed these good tidings with a universal cry of joy. Yet the very first days of freedom were darkened by the heavy affliction into which the country was thrown by those who would bar the path leading to the Czar; those who by trampling down the very fundamental principles of the imperial manifesto of October 17, 1905, overwhelmed the land with the disgrace of organized massacres, military reprisals, and imprisonments without trial. The impression of these recent administrative acts has been felt so keenly by the people that no pacification of the country is possible until the people are assured that henceforth arbitrary acts of officials shall cease, nor be longer shielded by the name of your Majesty; until all the ministers shall be held responsible to the representatives of the people, and that the administration in every step of state service shall be reformed accordingly. Sire: The idea of completely freeing the monarch from responsibility can be implanted in the minds of the nation only by making the ministers responsible to the people. Only a ministry fully trusted by the majority of the Duma can establish confidence in the government; and only in the presence of such confidence is the peaceful and regular work of the State Duma possible. But above all it is most needful to free Russia from the operation of exceptional laws for so-called “special and extraordinary protection,” and “martial law,” under cover of which the arbitrary authority of irresponsible officials has grown up and still continues to grow. Side by side with the establishment of the principle of responsibility of the administration to the representatives of the people, it is indispensable, for the successful work of the Duma, that there should be implanted, and definitely adopted, the fundamental principle of popular representation based on the coöperation of the monarch with the people, as the only source
- 42. of legislative power. Therefore all barriers between the imperial power and the people must be removed. No branch of legislative power should ever be closed to the inspection of the representative of the people, in coöperation with the monarch. The State Duma considers it its duty to state to your Majesty, in the name of the people, that the whole nation, with true inspiration and energy, with genuine faith in the near prosperity of the country, will only then fulfil its work of reformation, when the Council of State, which stands between it and the throne, shall cease to be made up, even in part, of members who have been appointed instead of being elected; when the law of collecting taxes shall be subject to the will of the representatives of the people; and when there shall be no possibility, by any special enactment, of limiting the legislative jurisdiction of the representatives of the people. The State Duma also considers it inconsistent with the vital interests of the people that any bill imposing taxes, when once passed by the Duma, should be subject to amendment on the part of any body which is not representative of the mass of taxpayers. In the domain of its future legislative activity, the State Duma, performing the duty definitely imposed upon it by the people, deems it necessary to provide the country, without delay, with a strict law providing for the inviolability of the person, freedom of conscience, liberty of speech, freedom of the press, freedom of association, convinced that without the strict observance of these principles, the foundation of which was laid in the manifesto of October 17, 1905, no social reform can be realized. The Duma also considers it necessary to secure for all citizens the right of petition to the people’s representatives. The State Duma has further the inflexible conviction that neither liberty nor order can be made firm and secure except on the broad foundation of equality before the law of all citizens without exception. Therefore the State Duma will establish a law for the perfect equality before the law of all citizens, abolishing all limitations dependent upon estate, nationality, religion, and sex. The Duma, however, while striving to free the country from the binding fetters of administrative guardianship and leaving the limitation of the liberty of the citizen to the independent judicial authorities, still deems the application of capital punishment, even in accordance with a legal sentence, as inadmissible. A death sentence should never be pronounced. The Duma holds that it has the right to proclaim, as the unanimous desire of the people, that a day should come when a law forever abolishing capital punishment here shall be
- 43. established. In anticipation of that law the country to-day is looking to your Majesty for a suspension of all death sentences. The investigation of the needs of the rural population and the undertaking of legislative measures to meet those wants will be considered among the first problems of the State Duma. The most numerous part of the population, the hard-working peasants, impatiently await the satisfaction of their acute want of land and the first Russian State Duma would be recreant to its duty were it to fail to establish a law to meet this primary want by resorting to the use of lands belonging to the state, the crown, the royal family, and monastic and church lands; also private landed property on the principle of the law of eminent domain. The Duma also deems it necessary to create laws giving equality to the peasantry, removing the present degrading limitations which separate them from the rest of the people. The Duma considers the needs of working people as pressing and that there should be legislative measures taken for the protection of hired labor. The first step in that direction ought to be to give freedom to the hired laborer in all branches of work, freedom to organize, freedom to act, and to secure his material and spiritual welfare. The Duma will also deem it its duty to employ all its forces in raising the standard of intelligence, and above all it will occupy itself in framing laws for free and general education. Along with the aforementioned measures the Duma will pay special attention to the just distribution of the burden of taxation, unjustly imposed at present upon the poorer classes of inhabitants; and to the reasonable expenditure of the means of the state. Not less vital in legislative work will be a fundamental reform of local government and of self-government, extending the latter to all the inhabitants upon the principles of universal suffrage. Bearing in mind the heavy burden imposed upon the people by your Majesty’s army and navy, the Duma will secure principles of right and justice in those branches of the service. Finally, the Duma deems it necessary to point out as one of the problems pressing for solution the long-crying demands of the different nationalities. Russia is an empire inhabited by many different races and nationalities. Their spiritual union is possible only by meeting the needs of each one of
- 44. them, and by preserving and developing their national characteristics. The Duma will try to satisfy those reasonable wants. Your Majesty: On the threshold of our work stands one question which agitates the soul of the whole nation; and which agitates us, the chosen and elected of the people, and which deprives us of the possibility of undisturbedly proceeding toward the first part of our legislative activity. The first word uttered by the State Duma met with cries of sympathy from the whole Duma. It was the word amnesty. The country thirsts for amnesty, to be extended to all those whose offenses were the result of either religious or political convictions; and all persons implicated in the agrarian movement. These are demands of the national conscience which cannot be overlooked; the fulfilment of which cannot be longer delayed. Sire, the Duma expects of you full political amnesty as the first pledge of mutual understanding and mutual agreement between the Czar and his people. APPENDIX C A RUSSIAN AUTHORITY ON THE POLICE PARTICIPATION OF POGROMS M. LOPUCHIN’S LETTER TO M. STOLYPIN Herewith we give the translation in full of the letter of M. Alexis Lopuchin, formerly Director of the Police Department of Russia. This is made from a German translation of the original Russian, and is vouched for, as to its correctness, by the author of the letter. Honored Sir: I deemed it my duty to bring to your attention through my letter of the 26th of May the fact that I gave to the editor of the journal “Retsch” the copy of the report of the chief of the special division of the police department to the minister of the interior, concerning the organization of the pogrom against the Jews in Alexandrovsk (government of Yekaterinoslaff), and touching the participation therein of the authorities of the police department. I did this in the firm conviction that it was only through the imperial Duma, when well informed by the public press, we could hope, once for all, to destroy the great danger menacing the State because of the systematic preparation by government officials of Jewish and other pogroms. I informed you of my action lest some subordinate of
- 45. Your Excellency might be held responsible for having furnished that journal with the report. I deemed it unnecessary in my communication to impart to you the facts detailed in the report of Markaroff, and with which I was familiar; I refrained from doing so because it was furthest from my thoughts that it could be possible that Your Excellency would conceal the truth that was revealed by the investigation called forth at the request of the Duma, in connection with the report of Markaroff. But yet must I be convinced from the newspaper reports of the Duma session of June 21st, that in your answer to the inquiry of the Duma, the material that was put into your hands for the proper preparation thereof, the real facts in the case, were substantially set aside. I, therefore, conceived it to be my bounden duty to impart to you, in this communication, facts that are well known to me. In January of this year several persons informed me that there were indications of the preparation in different sections of Russia of a Jewish pogrom, and they appealed for my help to prevent such misfortune. Investigations that were made established the truth of their statements, and satisfied me of the participation by public officials in the preparations for a pogrom. They brought me on the trail of a printing- office in the police department. On January 20, Count Witte, the president of the Council of Ministers, invited me to his office and asked me to give him my views on the Jewish question, and as to the reason for the participation of the Jewish proletariat in the revolutionary movement. After I had clearly presented to him my main point of view on the question, I told him that, aside from the judicial aspect of the question, there was another of great importance, namely, anti-Semitism, that not only existed because of the long-continued period in which the Jews were without rights, but because, as well, of the direct provocations against them on the part of persons in public authority. As a special indication of such provocation, I pointed to the incident of the printing-office in the police department, of whose output, however, I had no sufficient evidence in my hands, and Count Witte assigned to me, as an officer of the Minister of the Interior, the duty of making a close investigation into the matter. I proved the following conclusively: After the manifesto of the 17th October, 1905, thanks to the disturbances that broke out in many places after this act of the government, evidence of a reaction appeared in circumscribed sections of society. Ratschkowski, chief of the political division of the police department, an officer assigned to special duty by the Minister of the Interior, undertook to maintain and strengthen this reaction by the
- 46. issuing of effective proclamations. They were printed by an officer of the gendarmerie, in the building of the gendarmerie in St. Petersburg, upon a printing-press that was taken from revolutionaries when a house search was made. I had in my hand one of these proclamations; it was addressed to the working people, bore the signature “Group of Russian Factory Workers of St. Petersburg,” and sought to destroy the faith of working- men in their radical leaders by maintaining that these leaders had misappropriated funds that had been collected for the political campaign. This proclamation was not the only one that was printed in the headquarters of the gendarmerie; but at the time of investigation I could not get others because they had all been distributed. As the printing-press that served the purpose of the revolutionaries failed to satisfy the present needs, a complete one was purchased at the expense of the police department that was capable of printing one thousand per hour. This was set up in the secret service section of the police department. Captain Comisaroff was given its supervision, and two compositors were employed upon the work. On this machine there were printed in December, 1905, and in January, 1906, not one but a vast number of proclamations, all composed variously, but all of the same general tenor. In all these proclamations, alongside of a condemnation of the revolutionary movement, the information was offered that non-believers, mainly the Jews, were responsible therefor, and their purpose was to provoke an uprising against these people. I had in my hands three proclamations that were printed in the printing-office of the police department. As I positively proved, they were not the only ones; the fourth one was just set up at that moment (February 3). It contained the most ridiculous complaints against the Jews, and urged that they be boycotted in the Duma elections. But of the printed proclamations that I had in my hand one appears especially as law violating; the author, addressing himself to the soldiers, calls upon the army for a campaign against “the Poles, Armenians, and Jews.” Thousands of copies were printed, of every proclamation. Of the proclamation addressed to the soldiers, 5,000 copies were sent to Vilna by the officers on special duty to Mr. Schkott, the governor-general, for distribution in that city. Schkott distributed a portion of them himself in the evening in the streets of the city, and gave the rest of them to the chief of police of Vilna, who, on January 28, telegraphed to the police department that in view of the great success that attended the distribution of the proclamation addressed to the soldiers, to send him a new lot. Several thousand copies more were printed and sent on to the Vilna chief of police. The same proclamation was sent in thousands of copies to
- 47. Kursk, being taken by Surgeon Michailoff, assigned there to duty, who, at the request of M. Ratschkowski, was appointed secret agent of the police department. Michailoff also telegraphed (February 1 or 2) for a new lot of these proclamations in view of their great success among the soldiers. Aside from these, the appeals printed by the police department were distributed in St. Petersburg through M. Dubrovin, and the League of the Russian People, over which he presided; in Moscow through the publisher of the “Viedomosti,” Gringmut, who was given a large number of these appeals in December, 1905, by Ratschkowski personally. The provocative appeals of the police department were also distributed in other states, by the police and gendarmerie. All that is narrated above I imparted in January of this year to Count Witte, president of the committee of ministers, and I gave him specimens of all the proclamations above referred to (for that reason I have none at hand for present use). Count Witte at once called before him Captain Comisaroff, who acknowledged the truth of all this information. To me, also, he confirmed all these statements without exception. At the same time he declared that he acted under orders of Herr Ratschkowski; that he then presented the text of the proclamation to Wuitsch, the director of the police department, and did not at any time put them in type until the director stated in writing that he had read the proclamation. Express orders were issued by Secretary of State Witte that the printing-office of the police department should be wiped out of existence. However Captain Comisaroff merely took apart the printing- press as a precaution against the printing of further proclamations, by order of Ratschkowski, in spite of Witte’s orders; and to make that altogether impossible the press was taken from the police department to the residence of Captain Comisaroff. Aside from this and altogether without regard thereto, Your Excellency was confidentially informed that the proclamations which called for the extermination of the Jews in the city of Alexandrovsk (Yekaterinoslaff government) were circulated even after all the uprisings ceased, even after December 27, 1905; I consider it my duty to attach herewith a specimen of a proclamation that was distributed in the city of Alexandrovsk February 7 and 8, and that called for the extermination of the Jews on the 9th of February, the anniversary of the breaking out of the war with Japan. Your Excellency was confidentially informed that the officer for special duty, Ratschkowski, remained at the head of the political division of the police department until the end of April; that although this office was wiped out by the highest authority, he remained at the head of the entire secret and protective police; that the right was given him to
- 48. supervise, so far as he deemed it necessary, the course of all political occurrences and trials that affected the police department, and he was further authorized to utilize the social organizations in the interest of the government. * * * * * * * * Permit me, sir, to regard it as my moral duty, aside from imparting to you this information, to convey to you, as a former director of the police department, the reasons, incomprehensible at a first glance, why it is not only impossible for the central government to suppress the pogrom politics of the local authorities when the organization of a pogrom originates with them, but not even to be well informed as to the organization of the pogrom itself. One of these reasons is the freedom from punishment of the officers of the government who are responsible for the pogroms—no proof need be given of this. But there are other reasons of a general character; at the time I was director of the police department a pogrom occurred; that of Kishineff. The foreign and our own illegal press that then had the privilege to speak out on our internal conditions as well as several circles of society, put upon the police department the responsibility for the organization of this pogrom. There was no responsibility that could be attached to the police department; yet the charge was not groundless in so far as they started out with the supposition that the police department and the ministry of the interior were possessed of all possible power. In spite of the closest investigation as to the participation of officers of the government in the organization of the Kishineff pogrom, it was impossible for me; as director of the police department, to absolutely prove the fact, and yet there could be no doubt whatever of their participation. And what is especially characteristic, the secret working of the pogrom organization became clear to me only after I ceased to hold an official position in the ministry of the interior. And in such a position does every official of the central government find himself if he yields no sympathy to pogrom politics. That is to be accounted for by the fact that the minister of the interior and the central political organization are altogether powerless—the police and the gendarmerie are not in his hands, but precisely the reverse: he is in the hands of the superiors of these officials. The fact is that, through the organization of the secret political police, because of the exceptional law providing for extraordinary military protection, and I the long continuance of that condition in the country, the whole power has been transferred from above to below. Aside from the continued causes that have been uncovered, the weakness of the governmental authority, there are existing at present
- 49. other causes. I met no one among the political or general police officials who was not absolutely and thoroughly convinced that in reality there were two governments in existence, each of which drove its own politics to the other, one embodied in the person of Secretary of State Witte, the other in the person of Trepoff, who, according to general conviction, brought to the Czar reports of the condition of affairs in the empire, different than those that Count Witte brought to him, and in this wise developed a different political position. This point of view finds its foundation in the fact that General Trepoff, after his appointment as commander of the palace, succeeded in having special funds put at his command for the engagement of a separate force of secret agents, and he, therefore, became possessed of tools in hand that should only be in control of the minister of the interior. This point of view finds further foundation in the fact that General Trepoff, even after he gave up the post he held in the ministry of the interior, in October, 1905, succeeded without the knowledge of the minister of the interior, in getting out of the police department all the documents, except those of no moment, for the purpose of looking through them; not only current documents, but those of no present use— even though all these had nothing whatever to do with the commander of the palace. As to what purpose General Trepoff had in mind with reference to the secret funds, and the documents of the police department, in what direction he was inclined to utilize his position in regard to these, there exists, Your Excellency, in the mind of the undersigned, a firm conviction—rightly or wrongly—that General Trepoff sought to influence the politics of the government. This conviction, indeed, is as firm as the conviction that General Trepoff sympathized with the pogroms politics. And whatever power the ministry may set to work in opposition to pogroms, they will be valueless so long as the local police are convinced of the lack of power of the ministry and the possession of power of other authorities. APPENDIX D REPORT OF CAPTAIN PIETUCHOW, OF THE GENDARMERIE ADMINISTRATION OF THE STATE OF SIEDLCE, TO THE ASSISTANT GOVERNOR-GENERAL AT WARSAW. The provisional governor-general of the government of Siedlce, Major- general Engelke, by virtue of order No. 12, of August 10, this year, named
- 50. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookultra.com