S. Duplij, "Higher braid groups and regular semigroups from polyadic binary correspondence"

arXiv:2102.04433v1
[math.GR]
8
Feb
2021
HIGHER BRAID GROUPS AND REGULAR SEMIGROUPS FROM
POLYADIC-BINARY CORRESPONDENCE
STEVEN DUPLIJ
Center for Information Technology (WWU IT), Universität Münster, Röntgenstrasse 7-13
D-48149 Münster, Germany
ABSTRACT. In this note we first consider a ternary matrix group related to the von Neumann regular
semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of
ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups
with their binary multiplication of components. We then generalize the construction to the higher arity
case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups
and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix
introduced by the author, which differ from any version of the well-known tetrahedron equation and
higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree
(in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are
connected only in the non-higher case.
CONTENTS
1. INTRODUCTION 2
2. PRELIMINARIES 2
3. TERNARY MATRIX GROUP CORRESPONDING TO THE REGULAR SEMIGROUP 3
4. POLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULAR
SEMIGROUP 4
5. TERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP 7
6. TERNARY MATRIX GENERATORS 8
7. GENERATED k-ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP 10
REFERENCES 17
E-mail address: douplii@uni-muenster.de; sduplij@gmail.com; https://ivv5hpp.uni-muenster.de/u/douplii.
Date: of start January 8, 2021. Date: of completion February 7, 2021.
Total: 27 references.
2010 Mathematics Subject Classification. 16T25, 17A42, 20B30, 20F36, 20M17, 20N15.
Key words and phrases. regular semigroup, braid group, generator, relation, presentation, Coxeter group, symmetric
group, polyadic matrix group, querelement, idempotence, finite order element.

1. INTRODUCTION
We begin by observing that the defining relations of the von Neumann regular semi-
groups (e.g. GRILLET [1995], HOWIE [1976], PETRICH [1984]) and the Artin braid group
KASSEL AND TURAEV [2008], KAUFFMAN [1991] correspond to such properties of ternary ma-
trices (over the same set) as idempotence and the orders of elements (period). We then generalize
the correspondence thus introduced to the polyadic case and thereby obtain higher degree (in our
definition) analogs of the former. The higher (degree) regular semigroups obtained in this way
have appeared previously in semisupermanifold theory DUPLIJ [2000], and higher regular cate-
gories in TQFT DUPLIJ AND MARCINEK [2002]. The representations of the higher braid relations
in vector spaces coincide with the higher braid equation and corresponding generalized R-matrix
obtained in DUPLIJ [2018b], as do the ordinary braid group and the Yang-Baxter equation TURAEV
[1988]. The proposed constructions use polyadic group methods and differ from the tetrahedron
equation ZAMOLODCHIKOV [1981] and n-simplex equations HIETARINTA [1997] connected with
the braid group representations LI AND HU [1995], HU [1997], also from higher braid groups of
MANIN AND SCHECHTMAN [1989]. Finally, we define higher degree (in our sense) versions of
the Coxeter group and the symmetric group and show that they are connected in the classical (i.e.
non-higher) case only.
2. PRELIMINARIES
There is a general observation NIKITIN [1984], that a block-matrix, forming a semisimple p2, kq-
ring (Artinian ring with binary addition and k-ary multiplication) has the shape
M pk ´ 1q ” Mppk´1qˆpk´1qq
“
¨
˚
˚
˚
˚
˚
˝
0 mpi1î2q
0 . . . 0
0 0 mpi2î3q
. . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... mpik´2îk´1q
mpik´1î1q
0 0 . . . 0
˛
‹
‹
‹
‹
‹
‚
. (2.1)
In other words, it is given by the cyclic shift pk ´ 1qˆpk ´ 1q matrix, in which identities are replaced
by blocks of suitable sizes and with arbitrary entries.
The set tM pk ´ 1qu is closed with respect to the product of k matrices, and we will therefore call
them k-ary matrices. They form a k-ary semigroup, and when the blocks are over an associative
binary ring, then total associativity follows from the associativity of matrix multiplication.
Our proposal is to use single arbitrary elements (from rings with associative multiplication) in
place of the blocks mpiˆjq
, supposing that the elements of the multiplicative part G of the rings form
binary (semi)groups having some special properties. Then we investigate the similar correspondence
between the (multiplicative) properties of the matrices Mpk´1q
, related to idempotence and order, and
the appearance of the relations in G leading to regular semigroups and braid groups, respectively. We
call this connection a polyadic matrix-binary (semi)group correspondence (or in short the polyadic-
binary correspondence).
– 2 –

PRELIMINARIES
In the lowest -arity case k “ 3, the ternary case, the 2 ˆ 2 matrices Mp2q
are anti-triangle. From
pM p2qq3
“ M p2q and pM p2qq3
„ E p2q (where E p2q is the ternary identity, see below), we obtain the
correspondences of the above conditions on M p2q with the ordinary regular semigroups and braid
groups, respectively. In this way we extend the polyadic-binary correspondence on -arities k ě 4 to
get the higher relations
pM pk ´ 1qqk
“
"
“ M pk ´ 1q corresponds to higher k-degree regular semigroups,
“ qE pk ´ 1q corresponds to higher k-degree braid groups,
(2.2)
where E pk ´ 1q is the k-ary identity (see below) and q is a fixed element of the braid group.
3. TERNARY MATRIX GROUP CORRESPONDING TO THE REGULAR SEMIGROUP
Let Gfree “ tG | µg
2u be a free semigroup with the underlying set G “ gpiq
(
and the binary
multiplication. The anti-diagonal matrices over Gfree
Mg
p2q “ Mp2ˆ2q
`
gp1q
, gp2q
˘
“
ˆ
0 gp1q
gp2q
0
˙
, gp1q
, gp2q
P Gfree (3.1)
form a ternary semigroup Mg
3 ” Mg
k“3 “ tMg
p2q | µg
3u, where Mg
p2q “ tMg
p2qu is the set of
ternary matrices (3.1) closed under the ternary multiplication
µg
3 rMg
1 p2q , Mg
2 p2q , Mg
3 p2qs “ Mg
1 p2q Mg
2 p2q Mg
3 p2q , @Mg
1 p2q , Mg
2 p2q , Mg
3 p2q P Mg
3 , (3.2)
being the ordinary matrix product. Recall that an element Mg
p2q P Mg
3 is idempotent, if
µg
3 rMg
p2q , Mg
p2q , Mg
p2qs “ Mg
p2q , (3.3)
which in the matrix form (3.2) leads to
pMg
p2qq3
“ Mg
p2q . (3.4)
We denote the set of idempotent ternary matrices by Mg
id p2q “ tMg
id p2qu.
Definition 3.1. A ternary matrix semigroup in which every element is idempotent (3.4) is called an
idempotent ternary semigroup.
Using (3.1) and (3.4) the idempotence expressed in components gives the regularity conditions
gp1q
gp2q
gp1q
“ gp1q
, (3.5)
gp2q
gp1q
gp2q
“ gp2q
, @gp1q
, gp2q
P Gfree. (3.6)
Definition 3.2. A binary semigroup Gfree in which any two elements are mutually regular (3.5)–(3.6)
is called a regular semigroup Greg.
Proposition 3.3. The set of idempotent ternary matrices (3.4) form a ternary semigroup Mg
3,id “
tMid p2q | µ3u, if Greg is abelian.
Proof. It follows from (3.5)–(3.6), that idempotence (and following from it regularity) is preserved
with respect to the ternary multiplication (3.2), only when any gp1q
, gp2q
P Gfree commute.
Definition 3.4. We say that the set of idempotent ternary matrices Mg
id p2q (3.4) is in ternary-binary
correspondence with the regular (binary) semigroup Greg and write this as
Mg
id p2q ≎ Greg (3.7)
This means that such property of the ternary matrices as their idempotence (3.4) leads to the
regularity conditions (3.5)–(3.6) in the correspondent binary group Gfree.
– 3 –

TERNARY MATRIX GROUP CORRESPONDING TO THE REGULAR SEMIGROUP
Remark 3.5. The correspondence (3.7) is not a homomorphism and not a bi-element mapping
BOROWIEC ET AL. [2006], and also not a heteromorphism in sense of DUPLIJ [2018a], because
we do not demand that the set of idempotent matrices Mg
id p2q form a ternary semigroup (which is
possible in commutative case of Gfree only, see Proposition 3.3).
4. POLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULAR SEMIGROUP
We next extend the ternary-binary correspondence (3.7) to the k-ary matrix case (2.1) and thereby
obtain higher k-regular binary semigroups1
.
Let us introduce the pk ´ 1q ˆ pk ´ 1q matrix over a binary group Gfree of the form (2.1)
Mg
pk ´ 1q ” Mppk´1qˆpk´1qq
`
gp1q
, gp2q
, . . . , gpk´1q
˘
“
¨
˚
˚
˚
˚
˚
˝
0 gp1q
0 . . . 0
0 0 gp1q
. . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... gpk´2q
gpk´1q
0 0 . . . 0
˛
‹
‹
‹
‹
‹
‚
,
(4.1)
where gpiq
P Gfree.
Definition 4.1. The set of k-ary matrices Mg
pk ´ 1q (4.1) over Gfree is a k-ary matrix semigroup
Mg
k “ tMg
pk ´ 1q | µg
ku, where the multiplication
µg
k rMg
1 pk ´ 1q , Mg
2 pk ´ 1q , . . . , Mg
k pk ´ 1qs
“ Mg
1 pk ´ 1q Mg
2 pk ´ 1q . . . Mg
k pk ´ 1q , Mg
i pk ´ 1q P Mg
k (4.2)
is the ordinary product of k matrices Mg
i pk ´ 1q ” Mppk´1qˆpk´1qq
´
g
p1q
i , g
p2q
i , . . . , g
pk´1q
i
¯
, see (4.1).
Recall that the polyadic power ℓ of an element M from a k-ary semigroup Mk is defined by (e.g.
POST [1940])
Mxℓyk “ pµkqℓ
»
—
–
ℓpk´1q`1
hkkkkikkkkj
M, . . . , M
fi
ffi
fl , (4.3)
such that ℓ coincides with the number of k-ary multiplications. In the binary case k “ 2 the polyadic
power is connected with the ordinary power p (number of elements in the product) as p “ ℓ ` 1, i.e.
Mxℓy2 “ Mℓ`1
“ Mp
. In the ternary case k “ 3 we have xℓy3 “ 2ℓ ` 1, and so the l.h.s. of (3.4) is
of polyadic power ℓ “ 1.
Definition 4.2. An element of a k-ary semigroup M P M3 is called idempotent, if its first polyadic
power coincides with itself
Mx1yk “ M, (4.4)
and xℓy-idempotent, if
Mxℓyk “ M, Mxℓ´1yk ‰ M. (4.5)
1
We use the following notation:
Round brackets: pkq is size of matrix k ˆ k, also the sequential number of a matrix element.
Square brackets: rks is number of multipliers in the regularity and braid conditions.
Angle brackets: xℓyk is the polyadic power (number of k-ary multiplications).
– 4 –

POLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULAR SEMIGROUP
Definition 4.3. A k-ary semigroup Mk is called idempotent (ℓ-idempotent), if each of its elements
M P Mk is idempotent (xℓy-idempotent).
Assertion 4.4. From Mx1yk “ M it follows that Mxℓyk “ M, but not vice-versa, therefore all x1y-
idempotent elements are xℓy-idempotent, but an xℓy-idempotent element need not be x1y-idempotent.
Therefore, the definition given in(4.5) makes sense.
Proposition 4.5. If a k-ary matrix Mg
pk ´ 1q P Mg
k is idempotent (4.4), then its elements satisfy
the pk ´ 1q relations
gp1q
gp2q
, . . . gpk´2q
gpk´1q
gp1q
“ gp1q
, (4.6)
gp2q
gp3q
, . . . gpk´1q
gp1q
gp2q
“ gp2q
, (4.7)
.
.
.
gpk´1q
gp1q
gp2q
, . . . gpk´2q
gpk´1q
“ gpk´1q
, @gp1q
, . . . , gpk´1q
P Gfree. (4.8)
Proof. This follows from (4.1), (4.2) and (4.4).
Definition 4.6. The relations (4.6)–(4.8) are called (higher) rks-regularity (or higher k-degree regu-
larity). The case k “ 3 is the standard regularity (r3s-regularity in our notation) (3.5)–(3.6).
Proposition 4.7. If a k-ary matrix Mg
pk ´ 1q P Mg
k is xℓy-idempotent (4.5), then its elements
satisfy the following pk ´ 1q relations
ℓ
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
`
gp1q
gp2q
. . . gpk´2q
gpk´1q
˘
. . .
`
gp1q
gp2q
, . . . gpk´2q
gpk´1q
˘
gp1q
“ gp1q
, (4.9)
ℓ
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
`
gp2q
gp3q
, . . . gpk´2q
gpk´1q
gp1q
˘
. . .
`
gp2q
gp3q
. . . gpk´2q
gpk´1q
gp1q
˘
gp2q
“ gp2q
, (4.10)
.
.
.
ℓ
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
`
gpk´1q
gp1q
gp2q
. . . gpk´3q
gpk´2q
˘
. . .
`
gpk´1q
gp1q
gp2q
. . . gpk´3q
gpk´2q
˘
gpk´1q
“ gpk´1q
, (4.11)
@gp1q
, . . . , gpk´1q
P Gfree.
Proof. This also follows from (4.1), (4.2) and (4.5).
Definition 4.8. The relations (4.6)–(4.8) are called (higher) rks-xℓy-regularity. The case k “ 3
(3.5)–(3.6) is the standard regularity (r3s-x1y-regularity in this notation).
Definition 4.9. A binary semigroup Gfree, in which any k ´ 1 elements are rks-regular (rks-xℓy-
regular), is called a higher rks-regular (rks-xℓy-regular) semigroup Greg rks (Gxℓy-reg rks).
Similarly to Assertion 4.4, it is seen that rks-xℓy-regularity (4.9)–(4.11) follows from rks-
regularity (4.6)–(4.8), but not the other way around, and therefore we have
Assertion 4.10. If a binary semigroup Greg rks is rks-regular, then it is rks-xℓy-regular as well, but
not vice-versa.
Proposition 4.11. The set of idempotent (xℓy-idempotent) k-ary matrices Mg
id pk ´ 1q form a k-ary
semigroup Mg
3,id “ tMg
id pk ´ 1q | µg
ku, if and only if Greg rks (Gℓ-reg rks) is abelian.
– 5 –

POLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULAR SEMIGROUP
Proof. It follows from (4.6)–(4.11) that the idempotence (xℓy-idempotence) and the following rks-
regularity (rks-xℓy-regularity) are preserved with respect the k-ary multiplication (4.2) only in the
case, when all gp1q
, . . . , gpk´1q
P Gfree mutually commute.
By analogy with (3.7), we have
Definition 4.12. We will say that the set of k-ary pk ´ 1q ˆ pk ´ 1q matrices Mg
id pk ´ 1q (4.1) over
the underlying set G is in polyadic-binary correspondence with the binary rks-regular semigroup
Greg rks and write this as
Mg
id pk ´ 1q ≎ Greg rks . (4.12)
Thus, using the idempotence condition for k-ary matrices in components (being simultaneously
elements of a binary semigroup Gfree) and the polyadic-binary correspondence (4.12) we obtain
the higher regularity conditions (4.6)–(4.11) generalizing the ordinary regularity (3.5)–(3.6), which
allows us to define the higher rks-regular binary semigroups Greg rks (Gxℓy-reg rks).
Example 4.13. The lowest nontrivial (k ě 3) case is k “ 4, where the 3 ˆ 3 matrices over Gfree are
of the shape
M p3q “ Mp3ˆ3q
“
¨
˝
0 a 0
0 0 b
c 0 0
˛
‚, a, b, c P Gfree, (4.13)
and they form the 4-ary matrix semigroup Mg
4 . The idempotence pM p3qqx1y4
“ pM p3qq3
“ M p3q
gives three r4s-regularity conditions
abca “ a, (4.14)
bcab “ b, (4.15)
cabc “ c. (4.16)
According to the polyadic-binary correspondence (4.12), the conditions (4.14)–(4.16) are r4s-
regularity relations for the binary semigroup Gfree, which defines to the higher r4s-regular binary
semigroup Greg r4s.
In the case ℓ “ 2, we have pM p3qqx2y4
“ pM p3qq7
“ M p3q, which gives three r4s-x2y-regularity
conditions (they are different from r7s-regularity)
abcabca “ a, (4.17)
bcabcab “ b, (4.18)
cabcabc “ c, (4.19)
and these define the higher r4s-x2y-regular binary semigroup Gx2y-reg r4s. Obviously, (4.17)–(4.19)
follow from (4.14)–(4.16), but not vice-versa.
The higher regularity conditions (4.14)–(4.16) obtained above from the idempotence of polyadic
matrices using the polyadic-binary correspondence, appeared first in DUPLIJ [1998] and were then
used for transition functions in the investigation of semisupermanifolds DUPLIJ [2000] and higher
regular categories in TQFT DUPLIJ AND MARCINEK [2001, 2002].
Now we turn to the second line of (2.2), and in the same way as above introduce higher degree
braid groups.
– 6 –

TERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP
5. TERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP
Recall the definition of the Artin braid group ARTIN [1947] in terms of generators and relations
KASSEL AND TURAEV [2008] (we follow the algebraic approach, see, e.g. MARKOV [1945]).
The Artin braid group Bn (with n strands and the identity e P Bn) has the presentation by n ´ 1
generators σ1, . . . , σn´1 satisfying n pn ´ 1q {2 relations
σiσi`1σi “ σi`1σiσi`1, 1 ď i ď n ´ 2, (5.1)
σiσj “ σjσi, |i ´ j| ě 2, (5.2)
where (5.1) are called the braid relations, and (5.2) are called far commutativity. A general element
of Bn is a word of the form
w “ σp1
i1
. . . σpr
ir
. . . σpm
im
, im “ 1, . . . , n, (5.3)
where pr P Z are (positive or negative) powers of the generators σir , r “ 1, . . . , m and m P N.
For instance, B3 is generated by σ1 and σ2 satisfying one relation σ1σ2σ1 “ σ2σ1σ2, and is
isomorphic to the trefoil knot group. The group B4 has 3 generators σ1, σ2, σ3 satisfying
σ1σ2σ1 “ σ2σ1σ2, (5.4)
σ2σ3σ2 “ σ3σ2σ3, (5.5)
σ1σ3 “ σ3σ1. (5.6)
The representation theory of Bn is well known and well established KASSEL AND TURAEV
[2008], KAUFFMAN [1991]. The connections with the Yang-Baxter equation were investigated,
e.g. in TURAEV [1988].
Now we build a ternary group of matrices over Bn having generators satisfying relations which
are connected with the braid relations (5.1)–(5.2). We then generalize our construction to a k-ary
matrix group, which gives us the possibility to “go back” and define some special higher analogs of
the Artin braid group.
Let us consider the set of anti-diagonal 2 ˆ 2 matrices over Bn
M p2q “ Mp2ˆ2q
`
bp1q
, bp2q
˘
“
ˆ
0 bp1q
bp2q
0
˙
, bp1q
, bp2q
P Bn. (5.7)
Definition 5.1. The set of matrices M p2q “ tM p2qu (5.7) over Bn form a ternary matrix semigroup
Mk“3 “ M3 “ tM p2q | µ3u, where k “ 3 is the -arity of the following multiplication
µ3 rM1 p2q , M2 p2q , M3 p2qs ” M1 p2q , M2 p2q , M3 p2q “ M p2q , (5.8)
b
p1q
1 b
p2q
2 b
p1q
3 “ bp1q
, (5.9)
b
p2q
1 b
p1q
2 b
p2q
3 “ bp2q
, b
p1q
i , b
p2q
i P Bn, Mi p2q “
˜
0 b
p1q
i
b
p2q
i 0
¸
(5.10)
and the associativity is governed by the associativity of both the ordinary matrix product in the r.h.s.
of (5.8) and Bn.
Proposition 5.2. Mp3q
is a ternary matrix group.
Proof. Each element of the ternary matrix semigroup M p2q P M3 is invertible (in the ternary sense)
and has a querelement M̄ p2q (a polyadic analog of the group inverse DÖRNTE [1929]) defined by
µ3
“
M p2q , M p2q , M̄ p2q
‰
“ µ3
“
M p2q , M̄ p2q , M p2q
‰
“ µ3
“
M̄ p2q , M p2q , M p2q
‰
“ M p2q .
(5.11)
– 7 –

TERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP
It follows from (5.8)–(5.10), that
M̄ p2q “ pM p2qq´1
“
˜
0
`
bp1q
˘´1
`
bp2q
˘´1
0
¸
, bp1q
, bp2q
P Bn, (5.12)
where
`
Mp2q
˘´1
denotes the ordinary matrix inverse (but not the binary group inverse which does
not exist in the k-ary case, k ě 3). Non-commutativity of µ3 is provided by (5.9)–(5.10).
The ternary matrix group M3 has the ternary identity
E p2q “
ˆ
0 e
e 0
˙
, e P Bn, (5.13)
where e is the identity of the binary group Bn, and
µ3 rM p2q , E p2q , E p2qs “ µ3 rE p2q , M p2q , E p2qs “ µ3 rE p2q , E p2q , M p2qs “ M p2q .
(5.14)
We observe that the ternary product µ3 in components is “naturally braided” (5.9)–(5.10). This
allows us to ask the question: which generators of the ternary group M3 can be constructed using
the Artin braid group generators σi P Bn and the relations (5.1)–(5.2)?
6. TERNARY MATRIX GENERATORS
Let us introduce pn ´ 1q2
ternary 2 ˆ 2 matrix generators
Σij p2q “ Σ
p2ˆ2q
ij pσi, σjq “
ˆ
0 σi
σj 0
˙
, (6.1)
where σi P Bn, i “ 1 . . . , n ´ 1 are generators of the Artin braid group. The querelement of Σij p2q
is defined by analogy with (5.12) as
Σ̄ij p2q “ pΣij p2qq´1
“
ˆ
0 σ´1
j
σ´1
i 0
˙
. (6.2)
Now we are in a position to present a ternary matrix group with multiplication µ3 in terms of
generators and relations in such a way that the braid group relations (5.1)–(5.2) will be reproduced.
Proposition 6.1. The relations for the matrix generators Σij p2q corresponding to the braid group
relations for σi (5.1)–(5.2) have the form
µ3 rΣi,j`1 p2q , Σi,j`1 p2q , Σi,j`1 p2qs “ µ3 rΣj`1,i p2q , Σj`1,i p2q , Σj`1,i p2qs
“ q
r3s
i E p2q , 1 ď i ď n ´ 2, (6.3)
µ3 rΣij p2q , Σij p2q , E p2qs “ µ3 rΣji p2q , Σji p2q , E p2qs , |i ´ j| ě 2, (6.4)
where q
r3s
i “ σiσi`1σi “ σi`1σiσi`1, and E p2q is the ternary identity (5.13).
Proof. Use µ3 as the triple matrix product (5.8)–(5.10) and the braid relations (5.1)–(5.2).
Definition 6.2. We say that the ternary matrix group Mgen-Σ
3 generated by the matrix generators
Σij p2q satisfying the relations (6.3)–(6.4) is in ternary-binary correspondence with the braid (binary)
group Bn, which is denoted as (cf. (3.7))
Mgen-Σ
3 ≎ Bn. (6.5)
Indeed, in components the relations (6.3) give (5.1), and (6.4) leads to (5.2).
– 8 –

TERNARY MATRIX GENERATORS
Remark 6.3. Note that the above construction is totally different from the bi-element representations
of ternary groups considered in BOROWIEC ET AL. [2006] (for k-ary groups see DUPLIJ [2018a]).
Definition 6.4. An element M p2q P M3 is of finite polyadic (ternary) order, if there exists a finite
ℓ such that
M p2qxℓy3
“ M p2q2ℓ`1
“ E p2q , (6.6)
where E p2q is the ternary matrix identity (5.13).
Definition 6.5. An element M p2q P M3 is of finite q-polyadic (q-ternary) order, if there exists a
finite ℓ such that
M p2qxℓy3
“ M p2q2ℓ`1
“ qE p2q , q P Bn. (6.7)
The relations (6.3) therefore say that the ternary matrix generators Σi,j`1 p2q are of finite q-ternary
order. Each element of Mgen-Σ
3 is a ternary matrix word (analogous to the binary word (5.3)),
being the ternary product of the polyadic powers (4.3) of the 2 ˆ 2 matrix generators Σ
p2q
ij and their
querelements Σ̄
p2q
ij (on choosing the first or second row)
W “
ˆ
Σi1j1 p2q
Σ̄i1j1 p2q
˙xℓ1y3
, . . . ,
ˆ
Σirjr p2q
Σ̄irjr p2q
˙xℓry3
, . . . ,
ˆ
Σimjm p2q
Σ̄imjm p2q
˙xℓmy3
“
ˆ
Σi1j1 p2q
Σ̄i1j1 p2q
˙2ℓ1`1
, . . . ,
ˆ
Σirjr p2q
Σ̄irjr p2q
˙2ℓr`1
, . . . ,
ˆ
Σimjm p2q
Σ̄imjm p2q
˙2ℓm`1
, (6.8)
where r “ 1, . . . , m, ir, jr “ 1, . . . , n (from Bn), ℓr, m P N. In the ternary case the total number
of multipliers in (6.8) should be compatible with (4.3), i.e. p2ℓ1 ` 1q ` . . . ` p2ℓr ` 1q ` . . . `
p2ℓm ` 1q “ 2ℓW ` 1, ℓW P N, and m is therefore odd. Thus, we have
Remark 6.6. The ternary words (6.8) in components give only a subset of the binary words (5.3), and
so Mgen-Σ
3 corresponds to Bn, but does not present it.
Example 6.7. For B3 we have only two ternary 2 ˆ 2 matrix generators
Σ12 p2q “
ˆ
0 σ1
σ2 0
˙
, Σ21 p2q “
ˆ
0 σ2
σ1 0
˙
, (6.9)
satisfying
pΣ12 p2qqx1y3
“ pΣ12 p2qq3
“ q
r3s
1 E p2q , (6.10)
pΣ21 p2qqx1y3
“ pΣ21 p2qq3
“ q
r3s
1 E p2q , (6.11)
where q
r3s
1 “ σ1σ2σ1 “ σ2σ1σ2, and both matrix relations (6.10)–(6.11) coincide in components.
Example 6.8. For B4, the ternary matrix group Mgen-Σ
3 is generated by more generators satisfying
the relations
pΣ12 p2qq3
“ q
p3q
1 E p2q , (6.12)
pΣ21 p2qq3
“ q
p3q
1 E p2q , (6.13)
pΣ23 p2qq3
“ q
p3q
2 E p2q , (6.14)
pΣ32 p2qq3
“ q
p3q
2 E p2q , (6.15)
Σ13 p2q Σ13 p2q E p2q “ Σ31 p2q Σ31 p2q E p2q , (6.16)
– 9 –

TERNARY MATRIX GENERATORS
where q
r3s
1 “ σ1σ2σ1 “ σ2σ1σ2 and q
r3s
2 “ σ2σ3σ2 “ σ3σ2σ3. The first two relations give the braid
relations (5.4)–(5.5), while the last relation corresponds to far commutativity (5.6).
7. GENERATED k-ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
The above construction of the ternary matrix group Mgen-Σ
3 corresponding to the braid group Bn
can be naturally extended to the k-ary case, which will allow us to “go in the opposite way” and build
so called higher degree analogs of Bn (in our sense: the number of factors in braid relations more
than 3). We denote such a braid-like group with n generators by Bn rks, where k is the number of
generator multipliers in the braid relations (as in the regularity relations (4.6)–(4.8)). Simultaneously
k is the -arity of the matrices (5.7), we therefore call Bn rks a higher k-degree analog of the braid
group Bn. In this notation the Artin braid group Bn is Bn r3s. Now we build Bn rks for any degree
k exploiting the “reverse” procedure, as for k “ 3 and Bn in SECTION 5. For that we need a k-
ary generalization of the matrices over Bn, which in the ternary case are the anti-diagonal matrices
M p2q (5.7), and the generator matrices Σij p2q (6.1). Then, using the k-ary analog of multiplication
(5.9)–(5.10) we will obtain the higher degree (than (5.1)) braid relations which generate the so called
higher k-degree braid group. In distinction to the higher degree regular semigroup construction
from SECTION 4, where the k-ary matrices form a semigroup for the Abelian group Gfree, using the
generator matrices, we construct a k-ary matrix semigroup (presented by generators and relations)
for any (even non-commutative) matrix entries. In this way the polyadic-binary correspondence will
connect k-ary matrix groups of finite order with higher binary braid groups (cf. idempotent k-ary
matrices and higher regular semigroups (4.12)).
Let us consider a free binary group Bfree and construct over it a k-ary matrix group along the
lines of NIKITIN [1984], similarly to the ternary matrix group M3 in (5.7)–(5.10).
Definition 7.1. A set M pk ´ 1q “ tM pk ´ 1qu of k-ary pk ´ 1q ˆ pk ´ 1q matrices
M pk ´ 1q “ Mppk´1qˆpk´1qq
`
bp1q
, bp2q
, . . . , bpk´1q
˘
“
¨
˚
˚
˚
˚
˚
˝
0 bp1q
0 . . . 0
0 0 bp1q
. . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... bpk´2q
bpk´1q
0 0 . . . 0
˛
‹
‹
‹
‹
‹
‚
, (7.1)
bpjq
P Bfree, j “ 1, . . . , k ´ 1,
form a k-ary matrix semigroup Mk “ tM pk ´ 1q | µku, where µk is the k-ary multiplication
µk rM1 pk ´ 1q , M2 pk ´ 1q , . . . , Mk pk ´ 1qs “ M1 pk ´ 1q M2 pk ´ 1q . . . Mk pk ´ 1q “ M pk ´ 1q ,
(7.2)
b
p1q
1 b
p2q
2 , . . . b
pk´1q
k´1 b
p1q
k “ bp1q
, (7.3)
b
p2q
1 b
p3q
2 , . . . b
p1q
k´1b
p2q
k “ bp2q
, (7.4)
.
.
.
b
pk´1q
1 b
p1q
2 , . . . b
pk´2q
k´1 b
pk´1q
k “ bpk´1q
, (7.5)
where the r.h.s. of (7.2) is the ordinary matrix multiplication of k-ary matrices (7.1) Mi pk ´ 1q “
Mppk´1qˆpk´1qq
´
b
p1q
i , b
p2q
i , . . . , b
pk´1q
i
¯
, i “ 1, . . . , k.
– 10 –

GENERATED k-ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
Proposition 7.2. Mk is a k-ary matrix group.
Proof. Because Bfree is a (binary) group with the identity e P Bfree, each element of the k-
ary matrix semigroup M pk ´ 1q P Mk is invertible (in the k-ary sense) and has a querelement
M̄ pk ´ 1q (see DÖRNTE [1929]) defined by (cf. (5.14))
µk
»
–
k´1
hkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkj
M pk ´ 1q , . . . , M pk ´ 1q, M̄ pk ´ 1q
ﬁ
ﬂ “ . . . “ M pk ´ 1q , (7.6)
where M̄ pk ´ 1q can be on any place, and so we have k conditions (cf. (5.11) for k “ 3).
The k-ary matrix group has the polyadic identity
E pk ´ 1q “ Eppk´1qˆpk´1qq
“
¨
˚
˚
˚
˚
˚
˝
0 e 0 . . . 0
0 0 e . . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... e
e 0 0 . . . 0
˛
‹
‹
‹
‹
‹
‚
, e P Bfree, (7.7)
satisfying
µk rM pk ´ 1q , E pk ´ 1q , . . . , E pk ´ 1qs “ . . . “ M pk ´ 1q , (7.8)
where M pk ´ 1q can be on any place, and so we have k conditions (cf. (5.14)).
Definition 7.3. An element of a k-ary group M pk ´ 1q P Mk has the polyadic order ℓ, if
pM pk ´ 1qqxℓyk
” pM pk ´ 1qqℓpk´1q`1
“ E pk ´ 1q , (7.9)
where E pk ´ 1q P Mk is the polyadic identity (7.7), for k “ 3 see (5.13).
Definition 7.4. An element (pk ´ 1q ˆ pk ´ 1q-matrix over Bfree) M pk ´ 1q P Mtku
is of finite
q-polyadic order, if there exists a finite ℓ such that
pM pk ´ 1qqxℓyk
” pM pk ´ 1qqℓpk´1q`1
“ qE pk ´ 1q , q P Bfree. (7.10)
Let us assume that the binary group Bfree is presented by generators and relations (cf. the Artin
braid group (5.1)–(5.2)), i.e. it is generated by n ´ 1 generators σi, i “ 1, . . . , n ´ 1. An element of
Bgen-σ
n ” Bfree pe, σiq is the word of the form (5.3). To find the relations between σi we construct
the corresponding k-ary matrix generators analogous to the ternary ones (6.1). Then using a k-ary
version of the relations (6.3)–(6.4) for the matrix generators, as the finite order conditions (7.10),
we will obtain the corresponding higher degree braid relations for the binary generators σi, and can
therefore present a higher degree braid group Bn rks in the form of generators and relations.
Using n ´ 1 generators σi of Bgen-σ
n we build pn ´ 1qk
polyadic (or k-ary) pk ´ 1q ˆ pk ´ 1q-
matrix generators having k ´ 1 indices i1, . . . , ik´1 “ 1, . . . , n ´ 1, as follows
Σi1,...,ik´1
pk ´ 1q ” Σ
ppk´1qˆpk´1qq
i1,...,ik´1
`
σi1 , . . . , σik´1
˘
“
¨
˚
˚
˚
˚
˚
˝
0 σi1 0 . . . 0
0 0 σi2 . . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... σik´2
σik´1
0 0 . . . 0
˛
‹
‹
‹
‹
‹
‚
. (7.11)
For the matrix generator Σi1,...,ik´1
pk ´ 1q (7.11) its querelement Σ̄i1,...,ik´1
pk ´ 1q is defined by
(7.6).
– 11 –

We now build a k-ary matrix analog of the braid relations (5.1), (6.3) and of far commutativity
(5.2), (6.4). Using (7.11) we obtain pk ´ 1q conditions that the matrix generators are of finite polyadic
order (analog of (6.3))
µk rΣi,i`1,...,i`k´2 pk ´ 1q , Σi,i`1,...,i`k´2 pk ´ 1q , . . . , Σi,i`1,...,i`k´2 pk ´ 1qs (7.12)
“ µk rΣi`1,i`2,...,i`k´2,i pk ´ 1q , Σi`1,i`2,...,i`k´2,i pk ´ 1q , . . . , Σi`1,i`2,...,i`k´2,i pk ´ 1qs (7.13)
.
.
.
µk rΣi`k´2,i,i`1,...,i`k´3 pk ´ 1q , Σi`k´2,i,i`1,...,i`k´3 pk ´ 1q , . . . , Σi`k´2,i,i`1,...,i`k´3 pk ´ 1qs
(7.14)
“ q
rks
i E pk ´ 1q , 1 ď i ď n ´ k ` 1, (7.15)
where E pk ´ 1q are polyadic identities (7.7) and q
rks
i P Bgen-σ
n .
We propose a k-ary version of the far commutativity relation (6.4) in the following form
µk
»
–
k´1
hkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkj
Σi1,...,ik´1
pk ´ 1q , . . . , Σi1,...,ik´1
pk ´ 1q ,E pk ´ 1q
fi
fl “ . . . (7.16)
“ µk
»
–
k´1
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
Στpi1q,τpi2q,...,τpik´1q pk ´ 1q , . . . , Στpi1q,τpi2q,...,τpik´1q pk ´ 1q ,E pk ´ 1q
fi
fl ,
(7.17)
if all |ip ´ is| ě k ´ 1, p, s “ 1, . . . , k ´ 1, (7.18)
where τ is an element the permutation symmetry group τ P Sk´1.
In matrix form we can define
Definition 7.5. A k-ary (generated) matrix group Mgen-Σ
k is presented by the pk ´ 1q ˆ pk ´ 1q
matrix generators Σi1,...,ik´1
pk ´ 1q (7.11) and the relations (we use (7.2))
pΣi,i`1,...,i`k´2 pk ´ 1qqk
(7.19)
“ pΣi`1,i`2,...,i`k´2,i pk ´ 1qqk
(7.20)
.
.
.
pΣi`k´2,i,i`1,...,i`k´3 pk ´ 1qqk
(7.21)
“ q
rks
i E pk ´ 1q , 1 ď i ď n ´ k ` 1,
and
´
Σ
pk´1q
i1,i2,...,ik´1
¯k´1
E pk ´ 1q “
´
Σ
pk´1q
τpi1q,τpi2q,...,τpik´1q
¯k´1
E pk ´ 1q , (7.22)
if all |ip ´ is| ě k ´ 1, p, s “ 1, . . . , k ´ 1,
where τ P Sk´1 and q
rks
i P Bgen-σ
n .
Each element of Mgen-Σ
k is a k-ary matrix word (analogous to the binary word (5.3)) being the k-
ary product of the polyadic powers (4.3) of the matrix generators Σi1,...,ik´1
pk ´ 1q and their querele-
ments Σ̄i1,...,ik´1
pk ´ 1q as in (6.8).
– 12 –

Similarly to the ternary case k “ 3 (SECTION 5) we now develop the k-ary “reverse” procedure
and build from Bgen-σ
n the higher k-degree braid group Bn rks using (7.11). Because the presen-
tation of Mgen-Σ
k by generators and relations has already been given in (7.19)–(7.22), we need to
expand them into components and postulate that these new relations between the (binary) generators
σi present a new higher degree analog of the braid group. This gives
Definition 7.6. A higher k-degree braid (binary) group Bn rks is presented by pn ´ 1q generators
σi ” σ
rks
i (and the identity e) satisfying the following relations
‚ pk ´ 1q higher braid relations
k
σiσi`1 . . . σi`k´3σi`k´2σi (7.23)
“ σi`1σi`2 . . . σi`k´2σiσi`1 (7.24)
.
.
.
“ σi`k´2σiσi`1σi`2 . . . σiσi`1σi`k´2 ” q
rks
i , q
rks
i P Bn rks , (7.25)
i P Ibraid “ t1, . . . , n ´ k ` 1u , (7.26)
‚ pk ´ 1q-ary far commutativity
k´1
σi1 σi2 . . . σik´3
σik´2
σik´1
(7.27)
.
.
.
“ στpi1qστpi2q . . . στpik´3qστpik´2qστpik´1q, (7.28)
if all |ip ´ is| ě k ´ 1, p, s “ 1, . . . , k ´ 1, (7.29)
Ifar “ tn ´ k, . . . , n ´ 1u , (7.30)
where τ is an element of the permutation symmetry group τ P Sk´1.
A general element of the higher k-degree braid group Bn rks is a word of the form
w “ σp1
i1
. . . σpr
ir
. . . σpm
im
, im “ 1, . . . , n, (7.31)
where pr P Z are (positive or negative) powers of the generators σir , r “ 1, . . . , m and m P N.
Remark 7.7. The ternary case k “ 3 coincides with the Artin braid group Br3s
n “ Bn (5.1)–(5.2).
Remark 7.8. The representation of the higher k-degree braid relations in Bn rks in the tensor product
of vector spaces (similarly to Bn and the Yang-Baxter equation TURAEV [1988]) can be obtained
using the n1
-ary braid equation introduced in DUPLIJ [2018b] (Proposition 7.2 and next there).
Definition 7.9. We say that the k-ary matrix group Mgen-Σ
k generated by the matrix generators
Σi1,i2,...,ik´1
pk ´ 1q satisfying the relations (7.19)–(7.22) is in polyadic-binary correspondence with
the higher k-degree braid group Bn rks, which is denoted as (cf. (6.5))
Mgen-Σ
k ≎ Bn rks . (7.32)
Example 7.10. Let k “ 4, then the 4-ary matrix group Mgen-Σ
4 is generated by the matrix generators
Σi1,i2,i3 p3q satisfying (7.19)–(7.22)
‚ 4-ary relations of q-polyadic order (6.6)
pΣi,i`1,i`2 p3qq4
“ pΣi`2,i,i`1 p3qq4
“ pΣi`1,i`2,i p3qq4
“ q
r3s
i E p3q , 1 ď i ď n ´ 3, (7.33)
– 13 –

‚ far commutativity
pΣi1,i2,i3 p3qq3
E p3q “ pΣi3,i1,i2 p3qq3
E p3q
“ pΣi1,i3,i2 p3qq3
E p3q , (7.34)
|i1 ´ i2| ě 3, |i1 ´ i3| ě 3, |i2 ´ i3| ě 3.
Let σi ” σ
r4s
i P Bn r4s, i “ 1, . . . , n ´ 1, then we use the 4-ary 3 ˆ 3 matrix presentation for the
generators (cf. Example 4.13)
Σi1,i2,i3 p3q ” Σp3ˆ3q
pσi1 , σi2 , σi3 q “
¨
˝
0 σi1 0
0 0 σi2
σi3 0 0
˛
‚, i1, i2, i3 “ 1, . . . , n ´ 1. (7.35)
The querelement Σ̄i1,i2,i3 p3q satisfying
pΣi1,i2,i3 p3qq3
Σ̄i1,i2,i3 p3q “ Σi1,i2,i3 p3q , (7.36)
has the form
Σ̄i1,i2,i3 p3q “
¨
˝
0 σ´1
i3
σ´1
i2
0
0 0 σ´1
i1
σ´1
i3
σ´1
i2
σ´1
i1
0 0
˛
‚. (7.37)
Expanding (7.33)–(7.34) in components, we obtain the relations for the higher 4-degree braid
group Bn r4s as follows
‚ higher 4-degree braid relations
σiσi`1σi`2σi “ σi`1σi`2σiσi`1 “ σi`2σiσi`1σi`2 ” q
r4s
i , 1 ď i ď n ´ 3, (7.38)
‚ ternary far (total) commutativity
σi1 σi2 σi3 “ σi2 σi3 σi1 “ σi3 σi1 σi2 “ σi1 σi3 σi2 “ σi2 σi1 σi3 “ σi3 σi2 σi1 , (7.39)
|i1 ´ i2| ě 3, |i1 ´ i3| ě 3, |i2 ´ i3| ě 3. (7.40)
In the higher 4-degree braid group the minimum number of generators is 4, which follows from
(7.38). In this case we have a braid relation for i “ 1 only and no far commutativity relations,
because of (7.40). Then
Example 7.11. The higher 4-degree braid group B4 r4s is generated by 3 generators σ1, σ2, σ3,
which satisfy only the braid relation
σ1σ2σ3σ1 “ σ2σ3σ1σ2 “ σ3σ1σ2σ3. (7.41)
If n ď 7, then there will no far commutativity relations at all, which follows from (7.40), and so
the first higher 4-degree braid group containing far commutativity should have n “ 8 elements.
Example 7.12. The higher 4-degree braid group B8 r4s is generated by 7 generators σ1, . . . , σ7,
which satisfy the braid relations with i “ 1, . . . , 5
σ1σ2σ3σ1 “ σ2σ3σ1σ2 “ σ3σ1σ2σ3, (7.42)
σ2σ3σ4σ2 “ σ3σ4σ2σ3 “ σ4σ2σ3σ4, (7.43)
σ3σ4σ5σ3 “ σ4σ5σ3σ4 “ σ5σ3σ4σ5, (7.44)
σ4σ5σ6σ4 “ σ5σ6σ4σ5 “ σ6σ4σ5σ6, (7.45)
σ5σ6σ7σ5 “ σ6σ7σ5σ6 “ σ7σ5σ6σ7, (7.46)
– 14 –

together with the ternary far commutativity relation
σ1σ4σ7 “ σ4σ7σ1 “ σ7σ1σ4 “ σ1σ7σ4 “ σ4σ1σ7 “ σ7σ4σ1. (7.47)
Remark 7.13. In polyadic group theory there are several possible modifications of the commutativ-
ity property, but nevertheless we assume here the total commutativity relations in the k-ary matrix
generators and the corresponding far commutativity relations in the higher degree braid groups.
If Bn rks Ñ Z is the abelianization defined by σ˘
i Ñ ˘1, then σp
i “ e, if and only if p “ 0, and
σi are of infinite order. Moreover, we can prove (as in the ordinary case k “ 3 DYER [1980])
Theorem 7.14. The higher k-degree braid group Bn rks is torsion-free.
Recall (see, e.g. KASSEL AND TURAEV [2008]) that there exists a surjective homomorphism of
the braid group onto the finite symmetry group Bn Ñ Sn by σi Ñ si “ pi, i ` 1q P Sn. The
generators si satisfy (5.1)–(5.2) together with the finite order demand
sisi`1si “ si`1sisi`1, 1 ď i ď n ´ 2, (7.48)
sisj “ sjsi, |i ´ j| ě 2, (7.49)
s2
i “ e, i “ 1, . . . , n ´ 1, (7.50)
which is called the Coxeter presentation of the symmetry group Sn. Indeed, multiplying both sides
of (7.48) from the right successively by si`1, si, and si`1, using (7.50), we obtain psisi`1q3
“ 1, and
(7.49) on si and sj, we get psisjq2
“ 1. Therefore, a Coxeter group BRIESKORN AND SAITO [1972]
corresponding (7.48)–(7.50) is presented by the same generators si and the relations
psisi`1q3
“ 1, 1 ď i ď n ´ 2, (7.51)
psisjq2
“ 1, |i ´ j| ě 2, (7.52)
s2
i “ e, i “ 1, . . . , n ´ 1. (7.53)
A general Coxeter group Wn “ Wn pe, riq is presented by n generators ri and the relations
BJÖRNER AND BRENTI [2005]
prirjqmij
“ e, mij “

1, i “ j,
ě 2, i ‰ j.
(7.54)
By analogy with (7.48)–(7.50), we make the following
Definition 7.15. A higher analog of Sn, the k-degree symmetry group Sn rks “ Srks
n pe, siq, is
presented by generators si, i “ 1, . . . , n ´ 1 satisfying (7.23)–(7.28) together with the additional
condition of finite pk ´ 1q-order s
pk´1q
i “ e, i “ 1, . . . , n.
Example 7.16. The lowest higher degree case is S4 r4s which is presented by three generators s1,
s2, s3 satisfying (see (7.41))
s1s2s3s1 “ s2s3s1s2 “ s3s1s2s3, (7.55)
s3
1 “ s3
2 “ s3
3 “ e. (7.56)
In a similar way we define a higher degree analog of the Coxeter group (7.54).
– 15 –

Definition 7.17. A higher k-degree Coxeter group W
n rks “ Wrks
n pe, riq is presented by n genera-
tors ri obeying the relations
`
ri1 , ri2 , . . . , rik´1
˘mi1i2
,...,ik´1 “ e, (7.57)
mi1i2
,...,ik´1
“

1, i1 “ i2 “ . . . “ ik´1,
ě k ´ 1, |ip ´ is| ě k ´ 1, p, s “ 1, . . . , k ´ 1.
(7.58)
It follows from (7.58) that all generators are of pk ´ 1q order rk´1
i “ e. A higher k-degree Coxeter
matrix is a hypermatrix M
rk´1s
n,Cox
¨
˝
k´1
hkkkkkkkkikkkkkkkkj
n ˆ n ˆ . . . ˆ n
˛
‚having 1 on the main diagonal and other entries
mi1i2
,...,ik´1
.
Example 7.18. In the lowest higher degree case k “ 4 and all mi1i2
,...,ik´1
“ 3 we have (instead of
commutativity in the ordinary case k “ 3)
prirjq2
“ r2
j r2
i , (7.59)
rirjri “ r2
j r2
i r2
j . (7.60)
Example 7.19. A higher 4-degree analog of (7.51)–(7.53) is given by
priri`1ri`2q4
“ 1, 1 ď i ď n ´ 3, (7.61)
pri1 ri2 ri3 q3
“ 1, |i1 ´ i2| ě 3, |i1 ´ i3| ě 3, |i2 ´ i3| ě 3, (7.62)
r3
i “ e, i “ 1, . . . , n ´ 1. (7.63)
It follows from (7.62), that
pri1 ri2 ri3 q2
“ r2
i3
r2
i2
r2
i1
, (7.64)
which cannot be reduced to total commutativity (7.39). From the first relation (7.61) we obtain
riri`1ri`2ri “ r2
i`2r2
i`1, (7.65)
which differs from the higher 4-degree braid relations (7.38).
Example 7.20. In the simplest case the higher 4-degree Coxeter group W
4 r4s has 3 generator r1, r2,
r3 satisfying
pr1r2r3q4
“ r3
1 “ r3
2 “ r3
3 “ e. (7.66)
Example 7.21. The minimal case, when the conditions (7.62) appear is W
8 r4s
r1r2r3r1 “ r2
3r2
2, (7.67)
r2r3r4r2 “ r2
4r2
3, (7.68)
r3r4r5r3 “ r2
5r2
4, (7.69)
r4r5r6r4 “ r2
6r2
5, (7.70)
r5r6r7r5 “ r2
7r2
6, (7.71)
and an analog of commutativity
pr1r4r7q2
“ r2
7r2
4r2
1. (7.72)
Thus, we arrive at
Theorem 7.22. The higher k-degree Coxeter group can present the k-degree symmetry group in the
lowest case only, if and only if k “ 3.
– 16 –

As a further development, it would be interesting to consider the higher degree (in our sense)
groups constructed here from a geometric viewpoint (e.g., BIRMAN [1976], KAUFFMAN [1991]).
Acknowledgement. The author is grateful to Mike Hewitt, Thomas Nordahl, Vladimir Tkach and
Raimund Vogl for the numerous fruitful discussions and valuable support.
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