Classification of Uq(sl2)-module algebra structures on the quantum plane

Journal of Mathematical Physics, Analysis, Geometry
2010, v. 6, No. 4, pp. 1–25
Classiﬁcation of Uq(sl2)-Module Algebra Structures
on the Quantum Plane
S. Duplij
Theory Group, Nuclear Physics Laboratory, V.N. Karazin Kharkiv National University,
4 Svoboda Sq., Kharkiv, 61077, Ukraine
E-mail:sduplij@gmail.com
S. Sinel’shchikov
Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:sinelshchikov@ilt.kharkov.ua
Received , 2010
A complete list of Uq(sl2)-module algebra structures on the quantum
plane is produced and the (uncountable family of) isomorphism classes of
these structures are described. The composition series of representations in
question are computed. The classical limits of the Uq(sl2)-module algebra
structures are discussed.
Key words: quantum universal enveloping algebra, Hopf algebra, Verma
module, representation, composition series, weight.
Mathematics Subject Classiﬁcation 2000: 33A15, 33B15, 33D05.
1. Introduction
The quantum plane [11] is known to be a starting point in studying modules
over quantum universal enveloping algebras [3]. The structures existing on the
quantum plane are widely used as a background to produce associated structures
for more sophisticated quantum algebras [5, 4, 10]. There is one distinguished
structure of Uq(sl2)-module algebra on the quantum plane which was widely
considered before (see, e.g., [8]). In addition, one could certainly mention the
structure h(v) = ε(h)v, where h ∈ Uq(sl2), ε is the counit, v is a polynomial on
the quantum plane. Normally it is disregarded because of its triviality.
Nevertheless, it turns out that there exist more (in fact, an uncountable family
of nonisomorphic) Uq(sl2)-module algebra structures which are nontrivial and can
be used in further development of the quantum group theory.
c S. Duplij and S. Sinel’shchikov, 2010

S. Duplij and S. Sinel’shchikov
In this paper we suggest a complete description and classification of Uq(sl2)-
module algebra structures existing on the quantum plane. Specifically, in Sec-
tion 3 we use a general form of the automorphism of quantum plane to render the
notion of weight for Uq(sl2)-actions considered here. In Section 4 we present our
classification in terms of a pair of symbolic matrices, which relies upon consider-
ing the low dimensional (0-th and 1-st) homogeneous components of an action.
In Section 5 we describe the composition series for the above structures viewed
as representations in vector spaces.
2. Preliminaries
Let H be a Hopf algebra whose comultiplication is ∆, counit is ε, and antipode
is S [1]. Also let A be a unital algebra with unit 1. We will also use the Sweedler
notation ∆ (h) = i hi ⊗ hi [13].
Definition 2.1. By a structure of H-module algebra on A we mean a homo-
morphism π : H → EndC A such that:
(i) π(h)(ab) = i π(hi)(a) · π(hi )(b) for all h ∈ H, a, b ∈ A;
(ii) π(h)(1) = ε(h)1 for all h ∈ H.
The structures π1, π2 are said to be isomorphic if there exists an automorphism
Ψ of the algebra A such that Ψπ1(h)Ψ−1 = π2(h) for all h ∈ H.
Throughout the paper we assume that q ∈ C {0} is not a root of the unit
(qn = 1 for all non-zero integers n). Consider the quantum plane which is a unital
algebra Cq[x, y] with two generators x, y and a single relation
yx = qxy. (2.1)
The quantum universal enveloping algebra Uq (sl2) is a unital associative al-
gebra determined by its (Chevalley) generators k, k−1, e, f, and the relations
k−1
k = 1, kk−1
= 1, (2.2)
ke = q2
ek, (2.3)
kf = q−2
fk, (2.4)
ef − fe =
k − k−1
q − q−1
. (2.5)
The standard Hopf algebra structure on Uq(sl2) is determined by
∆(k) = k ⊗ k, (2.6)
∆(e) = 1 ⊗ e + e ⊗ k, (2.7)
∆(f) = f ⊗ 1 + k−1
⊗ f, (2.8)
S(k) = k−1
, S(e) = −ek−1
, S(f) = −kf,
ε(k) = 1, ε(e) = ε(f) = 0.
2 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4

Classiﬁcation of Uq(sl2)-Module Algebra Structures on the Quantum Plane
3. Automorphisms of the Quantum Plane
Denote by Cq[x, y]i the i-th homogeneous component of Cq[x, y], which is a
linear span of the monomials xmyn with m + n = i. Also, given a polynomial
p ∈ Cq[x, y], denote by (p)i the i-th homogeneous component of p, that is the
projection of p onto Cq[x, y]i parallel to the direct sum of all other homogeneous
components of Cq[x, y].
We rely upon a result by J. Alev and M. Chamarie which gives, in particular,
a description of automorphisms of the algebra Cq[x, y] [2, Prop. 1.4.4(i)]. In fact,
their claim is much more general, so in the special case we need here we present
a quite elementary proof for the reader’s convenience.
Proposition 3.1. Let Ψ be an automorphism of Cq[x, y], then there exist
nonzero constants α, β such that
Ψ : x → αx, y → βy. (3.1)
First note that an automorphism as in (3.1) is well deﬁned on the entire
algebra, because the ideal of relations generated by (2.1) is Ψ-invariant. We split
the proof into a series of lemmas.
Lemma 3.2. One has (Ψ(x))0 = (Ψ(y))0 = 0.
P r o o f. We start with proving (Ψ(x))0 = 0. Suppose the contrary, that is
(Ψ(x))0 = 0. As Ψ(y) = 0, we choose the lowest i with (Ψ(y))i = 0. Apply Ψ to
the relation yx = qxy and then project it to the i-th homogeneous component of
Cq[x, y] (parallel to the direct sum of all other homogeneous components) to get
(Ψ(y)Ψ(x))i = q(Ψ(x)Ψ(y))i. Clearly, (Ψ(y)Ψ(x))i is the lowest homogeneous
component of Ψ(y)Ψ(x), and (Ψ(y)Ψ(x))i = (Ψ(y))i(Ψ(x))0. In a similar way
q(Ψ(x)Ψ(y))i = q(Ψ(x))0(Ψ(y))i. Because (Ψ(x))0 is a constant, it commutes
with (Ψ(y))i, then (Ψ(y))i(Ψ(x))0 = q(Ψ(y))i(Ψ(x))0, and since (Ψ(x))0 = 0,
we also have (Ψ(y))i = q(Ψ(y))i. Recall that q = 1, hence (Ψ(y))i = 0 which
contradicts to our choice of i. Thus our claim is proved. The proof of another
claim goes in a similar way.
Lemma 3.3. One has (Ψ(x))1 = 0, (Ψ(y))1 = 0.
P r o o f. Let us prove that (Ψ(x))1 = 0. Suppose the contrary, which by
virtue of Lemma 3.2 means that Ψ(x) = i aixmi yni with mi + ni > 1. The
subsequent application of the inverse automorphism gives Ψ−1(Ψ(x)) which is
certainly x. On the other hand,
Ψ−1
(Ψ(x)) =
i
ai(Ψ−1
(x))mi
(Ψ−1
(y))ni
.
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 4 3

By Lemma 3.2 every nonzero monomial in Ψ−1(x) and Ψ−1(y) has degree at least
one, which implies that Ψ−1(Ψ(x)) is a sum of monomials of degree at least 2.
In particular, Ψ−1(Ψ(x)) can not be x. This contradiction proves the claim. The
rest of the statements can be proved in a similar way.
Lemma 3.4. There exist nonzero constants α, β, γ, δ such that (Ψ(x))1 = αx,
(Ψ(y))1 = βy.
P r o o f. Let us apply Ψ to (2.1), then project it to Cq[x, y]2 to get
(Ψ(y)Ψ(x))2 = q(Ψ(x)Ψ(y))2. It follows from Lemmas 3.2, 3.3 that (Ψ(y)Ψ(x))2 =
(Ψ(y))1(Ψ(x))1 and (Ψ(x)Ψ(y))2 = (Ψ(x))1(Ψ(y))1. Let (Ψ(x))1 = αx + µy and
(Ψ(y))1 = βy + νx, which leads to (βy + νx)(αx + µy) = q(αx + µy)(βy + νx).
This, together with (2.1) and Lemma 3.3, implies that µ = ν = 0, α = 0, and
β = 0.
Denote by C[x] and C[y] the linear spans of {xn| n ≥ 0} and {yn| n ≥ 0},
respectively. Obviously, one has the direct sum decompositions
Cq[x, y] = C[x] ⊕ yCq[x, y] = C[y] ⊕ xCq[x, y].
Given any polynomial P ∈ Cq[x, y], let (P)x be its projection to C[x] parallel
to yCq[x, y], and in a similar way deﬁne (P)y. Obviously, C[x] and C[y] are
commutative subalgebras.
Lemma 3.5. One has (Ψ(x))y = (Ψ(y))x = 0.
P r o o f. First we prove that (Ψ(x))y = 0. Project yx = qxy to C[y] to
obtain (Ψ(y))y(Ψ(x))y = q(Ψ(x))y(Ψ(y))y. On the other hand, (Ψ(y))y(Ψ(x))y =
(Ψ(x))y(Ψ(y))y, so that (1 − q)(Ψ(x))y(Ψ(y))y = 0. Since q = 1, we deduce that
(Ψ(x))y(Ψ(y))y = 0. It follows from Lemma 3.4 that (Ψ(y))y = 0, and since
Cq[x, y] is a domain [7], we ﬁnally obtain (Ψ(x))y = 0. The proof of another
claim goes in a similar way.
P r o o f of Proposition 3.1. It follows from Lemma 3.5 that Ψ(x) = xP
for some P ∈ Cq[x, y]. An application of Ψ−1 gives x = Ψ−1(x)Ψ−1(P). Since
deg x = 1, one should have either deg Ψ−1(x) = 0 or deg Ψ−1(P) = 0. Lemma 3.2
implies that deg Ψ−1(x) = 0, hence deg Ψ−1(P) = 0, that is Ψ−1(P) is a nonzero
constant, and so P = ΨΨ−1(P) is the same constant (we denote it by α).
The second claim can be proved in a similar way.

4. The Structures of Uq(sl2)-Module Algebra on the Quantum
Plane
We describe here the Uq (sl2)-module algebra structures on Cq[x, y] and then
classify them up to isomorphism.
For the sake of brevity, given a Uq(sl2)-module algebra structure on Cq[x, y],
we can associate a 2 × 3 matrix with entries from Cq[x, y]
M
def
=
k
e
f
· x, y =
k(x) k(y)
e(x) e(y)
f(x) f(y)
, (4.1)
where k, e, f are the generators of Uq(sl2) and x, y are the generators of Cq[x, y].
We call M a full action matrix. Conversely, suppose we have a matrix M with
entries from Cq[x, y] as in (4.1). To derive the associated Uq(sl2)-module algebra
structure on Cq[x, y] we set (using the Sweedler notation)
(ab)u
def
= a(bu), a, b ∈ Uq(sl2), u ∈ Cq[x, y], (4.2)
a(uv)
def
= Σi(aiu) · (ai v), a ∈ Uq(sl2), u, v ∈ Cq[x, y], (4.3)
which determines a well-defined action of Uq(sl2) on Cq[x, y] iff the following
properties hold. Firstly, an application (defined by (4.2)) of an element from
the relation ideal of Uq(sl2) (2.2)–(2.5) to any u ∈ Cq[x, y] should produce zero.
Secondly, a result of application (defined by (4.3)) of any a ∈ Uq(sl2) to an
element of the relation ideal of Cq[x, y] (2.1) vanishes. These conditions are to
be verified in the specific cases considered below.
Note that, given a Uq (sl2)-module algebra structure on the quantum plane,
the action of the generator k determines an automorphism of Cq[x, y], which is a
consequence of invertibility of k and ∆ (k) = k ⊗ k. In particular, it follows from
(3.1) that k is determined completely by its action Ψ on the generators presented
by a 1 × 2-matrix Mk as follows
Mk
def
= k (x) , k (y) = αx, βy (4.4)
for some α, β ∈ C {0}(which is certainly a minor of M (4.1)). Therefore every
monomial xnym ∈ Cq[x, y] is an eigenvector for k, and the associated eigenvalue
αnβm will be referred to as a weight of this monomial, which will be written as
wt (xnym) = αnβm.
We will also need another minor of M as follows
Mef
def
=
e(x) e(y)
f(x) f(y)
, (4.5)
and we call Mk and Mef an action k-matrix and an action ef-matrix, respectively.

It follows from (2.3)–(2.4) that each entry of M is a weight vector, in particu-
lar, all the nonzero monomials which constitute a specific entry should be of the
same weight. Specifically, by some abuse of notation we can write
wt(M)
def
=


wt(k(x)) wt(k(y))
wt(e(x)) wt(e(y))
wt(f(x)) wt(f(y))




wt(x) wt(y)
q2wt(x) q2wt(y)
q−2wt(x) q−2wt(y)

 =


α β
q2α q2β
q−2α q−2β

 ,
where the relation between the two matrices A = (aij) and B = (bij) is defined
as follows:
Notation. A B if for every pair of indices i, j such that both aij and bij
are nonzero, one has aij = bij, e.g.,
1 0
0 2
1 3
0 0
.
As an immediate consequence, we also have
Proposition 4.1. Suppose that α/β is not a root of the unit. Then every
homogeneous component (e(x))n, (e(y))n, (f(x))n, (f(y))n, n ≥ 0, if nonzero,
reduces to a monomial.
P r o o f. Under our assumptions on α, β, the weights of the monomials
xiyn−i, 0 ≤ i ≤ n, of degree n are pairwise different. Since e(x), e(y), f(x), f(y)
are weight vectors, our claim follows.
Our basic observation is that the Uq(sl2)-actions in question are actually de-
termined to a large extent by the projections of M to the lower homogeneous
components of Cq[x, y].
Next, we denote by (M)i the i-th homogeneous component of M, whose
elements are just the i-th homogeneous components of the corresponding entries
of M. Thus every matrix element of M, if nonzero, admits a well-defined weight.
Let us introduce the constants a0, b0, c0, d0 ∈ C such that zero degree compo-
nent of the full action matrix is
(M)0 =


0 0
a0 b0
c0 d0


0
. (4.6)
Here we keep the subscript 0 to the matrix in the r.h.s. to emphasize the origin of
this matrix as the 0-th homogeneous component of M. Note that the weights of
nonzero projections of (weight) entries of M should have the same weight. Hence
wt ((M)0)


0 0
q2α q2β
q−2α q−2β


0
. (4.7)

On the other hand, as all the entries of (M)0 are constants (4.6), one also deduces
wt ((M)0)


0 0
1 1
1 1


0
, (4.8)
where the relation is understood as a set of elementwise equalities iff they are
applicable, that is, when the corresponding entry of the projected matrix (M)0
is nonzero. Therefore, it is not possible to have all nonzero entries in the 0-th
homogeneous component of M simultaneously.
The classification of Uq(sl2)-module algebra structures on the quantum plane
we are about to suggest will be done in terms of a pair of symbolic matrices
derived from the minor Mef only. Now we use (Mef)i to construct a symbolic
matrix Mef
i
whose entries are symbols 0 or as follows: a nonzero entry of
(Mef)i is replaced by , while a zero entry is replaced by the symbol 0.
In the case of 0-th components the specific elementwise relations involved in
(4.7) imply that each column of Mef
0
should contain at least one 0, and so
that Mef
0
can be either of the following 9 matrices:
0 0
0 0 0
,
0
0 0 0
,
0
0 0 0
,
0 0
0 0
,
0 0
0 0
, (4.9)
0 0 0
,
0 0
0
,
0
0 0
,
0
0 0
.
An application of e and f to (2.1) by using (4.4) gives
ye(x) − qβe(x)y = qxe(y) − αe(y)x, (4.10)
f(x)y − q−1
β−1
yf(x) = q−1
f(y)x − α−1
xf(y). (4.11)
After projecting (4.10)–(4.11) to Cq[x, y]1 we obtain
a0(1 − qβ)y = b0(q − α)x,
d0 1 − qα−1
x = c0 q − β−1
y,
which certainly implies
a0(1 − qβ) = b0(q − α) = d0 1 − qα−1
= c0 q − β−1
= 0.

This determines the weight constants α and β as follows:
a0 = 0 =⇒ β = q−1
, (4.12)
b0 = 0 =⇒ α = q, (4.13)
c0 = 0 =⇒ β = q−1
, (4.14)
d0 = 0 =⇒ α = q. (4.15)
This deduction compared to (4.7), (4.8) implies that the symbolic matrices
from (4.9) containing two ’s should be excluded. Also, using (4.7) and (4.12)–
(4.15) we conclude that the position of in the remaining symbolic matrices
completely determines the associated weight constants by
0
0 0 0
=⇒ α = q−2
, β = q−1
, (4.16)
0
0 0 0
=⇒ α = q, β = q−2
, (4.17)
0 0
0 0
=⇒ α = q2
, β = q−1
, (4.18)
0 0
0 0
=⇒ α = q, β = q2
. (4.19)
As for the matrix
0 0
0 0 0
, it does not determine the weight constants at all.
Next, for the 1-st homogeneous component, one has wt(e(x)) = q2wt(x) =
wt(x) (because q2 = 1), which implies (e(x))1 = a1y, and in a similar way we
have
(Mef)1 =
a1y b1x
c1y d1x 1
with a1, b1, c1, d1 ∈ C. This allows us to introduce a symbolic matrix Mef
1
as
above. Using the relations between the weights similar to (4.7), we obtain
wt((Mef)1)
q2α q2β
q−2α q−2β 1
β α
β α 1
, (4.20)
here is implicit for a set of the elementwise equalities applicable iﬀ the respec-
tive entry of the projected matrix (M)1 is nonvanishing.
This means that every row and every column of Mef
1
may contain at least
one 0. Now project (4.10)–(4.11) to Cq[x, y]2 to obtain
a1(1 − qβ)y2
= b1(q − α)x2
,
d1 1 − qα−1
x2
= c1 q − β−1
y2
,

whence a1(1 − qβ) = b1(q − α) = d1 1 − qα−1 = c1 q − β−1 = 0. As a
consequence we have
a1 = 0 =⇒ β = q−1
, (4.21)
b1 = 0 =⇒ α = q, (4.22)
c1 = 0 =⇒ β = q−1
, (4.23)
d1 = 0 =⇒ α = q. (4.24)
A comparison of (4.20) with (4.21)–(4.24) allows one to discard the symbolic
matrix
0
0 1
from the list of symbolic matrices with at least one 0 at every
row or column. As for other symbolic matrices with the above property, we get
0
0 0 1
=⇒ α = q−3
, β = q−1
, (4.25)
0
0 0 1
=⇒ α = q, β = q−1
, (4.26)
0 0
0 1
=⇒ α = q, β = q−1
, (4.27)
0 0
0 1
=⇒ α = q, β = q3
, (4.28)
0
0 1
=⇒ α = q, β = q−1
. (4.29)
The matrix
0 0
0 0 1
does not determine the weight constants in the way
described above.
In view of the above observations we see that in most cases a pair of symbolic
matrices corresponding to 0-th and 1-st homogeneous components determines
completely the weight constants of the conjectured associated actions. It will
be clear from the subsequent arguments that the higher homogeneous compo-
nents are redundant within the presented classiﬁcation. Therefore, we intro-
duce the table of families of Uq(sl2)-module algebra structures, each family is
labelled by two symbolic matrices Mef
0
, Mef
1
, and we call such a family
a Mef
0
; Mef
1
-series. Note that the series labelled with pairs of nonzero
symbolic matrices at both positions are empty, because each of the matrices de-
termines a pair of speciﬁc weight constants α and β (4.16)–(4.19) which fails to
coincide to any pair of such constants associated to the set of nonzero symbolic

matrices at the second position (4.25)–(4.29). Also, the series with zero symbolic
matrix at the first position and symbolic matrices containing only one at the
second position are empty.
For instance, show that
0 0
0 0 0
;
0
0 0 1
-series is empty. If we sup-
pose the contrary, then it follows from (2.5) that within this series we have
e(f(x)) − f(e(x)) = −(1 + q2
+ q−2
)x.
We claim that the projection of the l.h.s. to Cq[x, y]1 is zero. Start with observing
that, if the first symbolic matrix consists of 0’s only, one cannot reduce a degree
of any monomial by applying e or f. On the other hand, within this series f(x)
is a sum of the monomials whose degree is at least 2. Therefore, the term e(f(x))
has zero projection to Cq[x, y]1. Similarly, f(e(x)) has also zero projection to
Cq[x, y]1. The contradiction we get proves our claim.
In a similar way, one can prove that all other series with zero symbolic matrix
at the first position and symbolic matrices containing only one at the second
position are empty.
In the framework of our classification we obtained 24 “empty” Mef
0
; Mef
1
-
series. Next turn to “nonempty” series. We start with the simplest case in which
the action ef-matrix is zero, while the full action matrix is
M =
αx βy
0 0
0 0
.
Theorem 4.2. The
0 0
0 0 0
;
0 0
0 0 1
-series consists of 4 Uq(sl2)-
module algebra structures on the quantum plane given by
k(x) = ±x, k(y) = ±y, (4.30)
e(x) = e(y) = f(x) = f(y) = 0, (4.31)
which are pairwise nonisomorphic.
P r o o f. It is evident that (4.30)–(4.31) determine a well-defined Uq(sl2)-
action consistent with the multiplication in Uq(sl2) and in the quantum plane, as
well as with comultiplication in Uq(sl2). Prove that there are no other Uq(sl2)-
actions here. Note that an application of the l.h.s. of (2.5) to x or y has zero
projection to Cq[x, y]1, because in this series e and f send any monomial to a sum
of the monomials of higher degree. Therefore, k − k−1 (x) = k − k−1 (y) = 0,

and hence α−α−1 = β −β−1 = 0, which leads to α, β ∈ {1, −1}. To prove (4.31),
note that wt(e(x)) = q2wt(x) = ±q2 = ±1. On the other hand, the weight of
any nonzero weight vector in this series is ±1. This and similar arguments which
involve e, f, x, y imply (4.31).
To see that the Uq(sl2)-module algebra structures are pairwise non-isomorphic,
observe that all the automorphisms of the quantum plane commute with the ac-
tion of k (see Sect. 3).
The action we reproduce in the next theorem is well known [9, 12], and here
is the place for it in our classification.
Theorem 4.3. The
0 0
0 0 0
;
0
0 1
-series consists of a one-para-
meter (τ ∈ C {0}) family of Uq(sl2)-module algebra structures on the quantum
plane
k(x) = qx, k(y) = q−1
y, (4.32)
e(x) = 0, e(y) = τx, (4.33)
f(x) = τ−1
y, f(y) = 0. (4.34)
All these structures are isomorphic, in particular, to the action as above with
τ = 1.
The full action matrix related to (4.32)–(4.34) is
M =
qx q−1y
0 x
y 0
.
P r o o f. It is easy to check that (4.32)–(4.34) are compatible to all the
relations in Uq(sl2) and Cq[x, y], hence determine a well-defined Uq(sl2)-module
algebra structure on the quantum plane [12].
Prove that the
0 0
0 0 0
;
0
0 1
-series contains no other actions ex-
cept those given by (4.32)–(4.34). Let us first prove that the matrix elements of
Mef (4.5) contain no terms of degree higher than one, i.e. (Mef)n = 0 for n ≥ 2.
A general form for e(x) and e(y) here is
e(x) =
m+n≥2
¯ρmnxm
yn
, e(y) = τex +
m+n≥2
¯σmnxm
yn
, (4.35)
where τe, ¯ρmn, ¯σmn ∈ C, τe = 0. Note that in this series
wt (Mef) =
q3 q
q−1 q−3 .

In particular, wt(e(x)) = q3 and wt(e(y)) = q, which reduces the general
form (4.35) to a sum of terms with each one having the same fixed weight
e(x) =
m≥0
ρmxm+3
ym
, (4.36)
e(y) = τex +
m≥0
σmxm+2
ym+1
. (4.37)
Substitute (4.36)–(4.37) to (4.10) and then project it to the one-dimensional
subspace Cxm+3ym+1 (for every m ≥ 0) to obtain
ρm
σm
= −q
1 − qm+1
1 − qm+3
.
In a similar way, the relations wt(f(x)) = q−1 and wt(f(y)) = q−3 imply that
f(x) = τf y +
n≥0
ρnxn+1
yn+2
, (4.38)
f(y) =
n≥0
σnxn
yn+3
, (4.39)
where τf ∈ C {0}. An application of (4.38)–(4.39) and (4.11) with subsequent
projection to Cxn+1yn+3 (for every n ≥ 0) allows one to get
ρn
σn
= −q−1 1 − qn+3
1 − qn+1
.
Thus we have
Mef =
0 τex
τf y 0
+
n≥0
−µnq(1 − qn+1)xn+3yn µn(1 − qn+3)xn+2yn+1
νn(1 − qn+3)xn+1yn+2 −νnq(1 − qn+1)xnyn+3 ,
where µn, νn ∈ C. We intend to prove that the second matrix in this sum is
zero. Assume the contrary. In the case there exist both nonzero µn’s and νn’s,
and since the sums here are finite, for the first row choose the largest index ne
with µne = 0 and for second row, the largest index nf with νnf
= 0. Then using
(2.7)–(2.8), we deduce that the highest degree of the monomials in (ef −fe)(x) is
2ne + 2nf + 5. This monomial appears to be unique, and its precise computation
gives µne νnf
qnenf −1(1 − qn2+nf +4)(1 − q2ne+2nf +6)xne+nf +3yne+nf +2. Therefore,
(ef −fe)(x) has a nonzero projection onto the one dimensional subspace spanned
by the monomial xne+nf +3yne+nf +2, the latter being of degree higher than 1. This
contradicts to (2.5) whose r.h.s. applied to x has degree 1.

In the case when all νn’s are zero and some µn’s are nonvanishing we have
that the highest degree monomial of (ef − fe)(x) is of the form
τf µne
(1 − qne+3)(1 − q2ne+4)
qne+1(1 − q2)
xne+2
yne+1
,
which is nonzero under our assumptions on q. This again produces the same con-
tradiction as above. In the opposite case when all µn’s are zero and some νn’s are
nonvanishing, a similar computation works, which also leads to a contradiction.
Therefore, all µn’s and νn’s are zero.
Finally, an application of (2.5) to x yields τeτf = 1 so that τe = τ and
τf = τ−1 for some τ ∈ C {0}.
We claim that all the actions corresponding to nonzero τ are isomorphic to the
speciﬁc action with τ = 1. The desired isomorphism is given by the automorphism
Φτ : x → x, y → τy. In particular, Φτ eτ Φ−1
τ (y) = τ−1Φτ (τx) = x = e1 (y),
where eτ (y) denotes the action from (4.33) with an arbitrary τ = 0.
Now we consider the actions whose symbolic matrix Mef
0
contains one .
Seemingly, the corresponding actions described below never appeared in the
literature before, so we present a more detailed computations.
Theorem 4.4. The
0
0 0 0
;
0 0
0 0 1
meter (b0 ∈ C {0}) family of Uq(sl2)-module algebra structures on the quantum
plane
k(x) = qx, k(y) = q−2
y, (4.40)
e(x) = 0, e(y) = b0, (4.41)
f(x) = b−1
0 xy, f(y) = −qb−1
0 y2
. (4.42)
All these structures are isomorphic, in particular to the action as above with
b0 = 1.
The full action matrix of an action within this isomorphism class is of the
form
M =
qx q−2y
0 1
xy −qy2
.
P r o o f. First we demonstrate that an extension of (4.40)–(4.42) to the
entire action of Uq(sl2) on Cq[x, y] passes through all the relations. It is clear

that (4.40) is compatible with the relation kk−1
= k−1k = 1. Then we apply the
relations (2.3)–(2.5) to the quantum plane generators
(ke − q2
ek)(x) = k(0) − q3
e(x) = 0,
(ke − q2
ek)(y) = k(b0) − e(y) = b0 − b0 = 0,
(kf − q−2
fk)(x) = k b−1
0 xy − q−1
f(x)
= b−1
0 q−1
xy − q−1
b−1
0 xy = 0,
(kf − q−2
fk)(y) = k −qb−1
0 y2
− q−4
f(y)
= −qb−1
0 q−4
y2
+ q−4
qb−1
0 y2
= 0,
ef − fe −
k − k−1
q − q−1
(x) = e b−1
0 xy − f(0) − x = b−1
0 e(xy) − x
= b−1
0 xe(y) + b−1
0 e(x)k(y) − x = 0,
ef − fe −
k − k−1
q − q−1
(y) = −qb−1
0 e y2
− f(b0) −
q−2 − q2
q − q−1
y
= −qb−1
0 e y2
+ q + q−1
y
= −qb−1
0 ye(y) − qb−1
0 e(y)k(y) + q + q−1
y
= −qy − q−1
y + q + q−1
y = 0.
Now apply the generators of U2 (sl2) to (2.1) and get
k (yx − qxy) = q−2
y · qx − qqx · q−2
y = 0,
e (yx − qxy) = ye (x) + e (y) k (x) − qxe (y) − qe (x) k (y)
= 0 + b0qx − qxb0 − 0 = 0,
f (yx − qxy) = f (y) x + k−1
(y) f (x) − qf (x) y − qk−1
(x) f (y)
= −qb−1
0 y2
x + q2
yb−1
0 xy − qb−1
0 xy · y + qq−1
x · qb−1
0 y2
= −q3
b−1
0 xy2
+ q3
b−1
0 xy2
− qb−1
0 xy2
+ qb−1
0 xy2
= 0.
Next prove that
0
0 0 0
;
0 0
0 0 1
-series contains no actions except
(4.40)–(4.42). Show that the matrix elements of Mef (4.5) have no terms of degree
higher than two, viz. (Mef)n = 0 for n ≥ 3. Now a general form for e(x), e(y),
f(x), f(y) is
e(x) =
m+n≥0
¯ρmnxm
yn
, e(y) =
m+n≥0
¯σmnxm
yn
, (4.43)
f(x) =
m+n≥0
¯ρmnxm
yn
, f(y) =
m+n≥0
¯σmnxm
yn
(4.44)

where ¯ρmn, ¯σmn, ¯ρmn, ¯σmn ∈ C. Within this series one has the matrix of weights
wt(Mef) =
q3 1
q−1 q−4 .
In view of this, the general form (4.43)–(4.44) should be a sum of terms of
the same weight
e(x) =
m≥0
ρmx2m+3
ym
, (4.45)
e(y) = b +
m≥0
σmx2m+2
ym+1
, (4.46)
f(x) = b xy +
n≥0
ρnx2n+3
yn+2
, (4.47)
f(y) = b y2
+
n≥0
σnx2n+2
yn+3
. (4.48)
Now we combine (4.45)–(4.46), (4.47)–(4.48)) with (4.10), (4.11), respectively,
then project the resulting relation to the one-dimensional subspace Cx2m+3ym+2
(resp. Cx2n+3yn+3) (for every m ≥ 0, resp. n ≥ 0) to obtain
ρm
σm
= −q2 1 − qm+1
1 − q2m+4
,
ρn
σn
= −q−1 1 − qn+3
1 − q2n+4
.
Thus we get
Mef =
0 b
b xy b y2
+
n≥0
µnq2(1 − qn+1)x2n+3yn −µn(1 − q2n+4)x2n+2yn+1
−νn(1 − qn+3)x2n+3yn+2 νnq(1 − q2n+4)x2n+2yn+3 , (4.49)
where µn, νn ∈ C. To prove that the second matrix vanishes, assume the contrary.
First consider the case when there exist both nonzero µn’s and νn’s. As the sums
here are ﬁnite, for the ﬁrst row choose the largest index ne with µne = 0 and for
the second row, the largest index nf with νnf
= 0. After applying (2.7)–(2.8) one
concludes that the highest degree of monomials in (ef − fe)(x) is 3ne + 3nf + 7.
This monomial is unique, and its computation gives
µne νnf
q2nenf +2ne
(1 − qne+nf +4
)(1 − q2ne+2nf +6
)x2ne+2nf +5
yne+nf +2
. (4.50)

Under our assumptions on q, since ne ≥ 0, nf ≥ 0, µne νnf
= 0, it becomes
clear that (4.50) is a nonzero monomial of degree higher than 1. This breaks
(2.5) whose r.h.s. applied to x has degree 1. An application of (2.5) to x and y
together with (4.49) leads to (up to terms of degree higher than 1)
ef − fe −
k − k−1
q − q−1
(x) = 0 = b b x − x,
ef − fe −
k − k−1
q − q−1
(y) = 0 = b b (1 + q−2
)y + q + q−1
y,
which yields
b = b0, b = b−1
0 , b = −qb−1
0
for some b0 = 0.
A similar, but simpler computation also shows that in the case when all νn’s
are zero and some µn’s are nonzero we have the highest degree monomial of
(ef − fe)(x) of the form
b−1
0 µne
(1 − qne+3)(q2ne+4 − 1)
1 − q2
x2ne+3
yne+1
.
This monomial is nonzero due to our assumption on q, which gives the same
contradiction as above. The opposite case, when all µn’s are zero and some
νn’s are nonvanishing, can be treated similarly and also leads to a contradiction.
Therefore, all µn’s and νn’s are zero. This gives the desired relations (4.40)–(4.42).
Finally we show that the actions (4.40)–(4.42) with nonzero b0 are isomorphic
to the speciﬁc action with b0 = 1. The desired isomorphism is as follows Φb0 :
x → x, y → b0y. In fact,
Φb0 eb0 Φ−1
b0
(y) = Φb0 eb0 b−1
0 y = b−1
0 Φb0 (b0) = Φb0 (1) = 1 = e1(y),
Φb0 fb0 Φ−1
b0
(x) = Φb0 fb0 (x) = b−1
0 Φb0 (xy) = b−1
0 b0xy = xy = f1(x),
Φb0 fb0 Φ−1
b0
(y) = Φb0 fb0 b−1
0 y = b−1
0 Φb0 −qb−1
0 y2
= −qb−2
0 b2
0y2
=
= −qy2
= f1(y).
The theorem is proved.
Theorem 4.5. The
0 0
0 0
;
0 0
0 0 1
meter (c0 ∈ C {0}) family of Uq(sl2)-module algebra structures on the quantum
plane
k(x) = q2
x, k(y) = q−1
y, (4.51)
e(x) = −qc−1
0 x2
, e(y) = c−1
0 xy, (4.52)
f(x) = c0, f(y) = 0. (4.53)

All these structures are isomorphic, in particular to the action as above with
c0 = 1.
The full action matrix for this isomorphism class (with c0 = 1) is
M =
q2x q−1y
−qx2 xy
1 0
.
P r o o f. Quite literally repeats that of the previous theorem.
Theorem 4.6. The
0
0 0 0
;
0 0
0 0 1
-series consists of a three-para-
meter (a0 ∈ C {0}, s, t ∈ C) family of Uq(sl2)-actions on the quantum plane
k(x) = q−2
x, k(y) = q−1
y, (4.54)
e(x) = a0, e(y) = 0, (4.55)
f(x) = −qa−1
0 x2
+ ty4
, f(y) = −qa−1
0 xy + sy3
. (4.56)
The generic domain {(a0, s, t)| s = 0, t = 0} with respect to the parameters
splits into uncountably many disjoint subsets {(a0, s, t)|s = 0, t = 0, ϕ = const},
where ϕ =
t
a0s2
. Each of those subsets corresponds to an isomorphism class of
Uq(sl2)-module algebra structures. Additionally, there exist three more isomor-
phism classes corresponding to the subsets
{(a0, s, t)| s = 0, t = 0}, {(a0, s, t)|s = 0, t = 0}, {(a0, s, t)| s = 0, t = 0}.
P r o o f. A routine veriﬁcation demonstrates that (4.54)–(4.56) pass through
all the relations as before, hence admit an extension to a well-deﬁned series of
Uq(sl2)-actions on the quantum plane.
Now check that
0
0 0 0
;
0 0
0 0 1
-series contains no other actions
except (4.54)–(4.56). First consider the polynomial e(x). Since its weight is
q2wt(x) = 1, and the weight of any monomial other than constant is a negative
degree of q (within the series under consideration), hence not 1, one gets e(x) =
a0. In a similar way, the only possibility for e(y) is zero, because if not, wt(e(y)) =
q2wt(y) = q, which is impossible in view of the above observations.
Turn to f (x) and observe that wt (f (x)) = q−4. It is easy to see that all the
monomials with this weight are x2, xy2, y4, that is f (x) = ux2 + vxy2 + wy4. In
a similar way wt (f (y)) = q−3 and so f (y) = zxy + sy3. A substitution to (2.5)
yields 1 + q−2 ua0 = − q + q−1 , v = 0, za0q−1 = −1. Note that (4.11) gives
no new relations for u, v, z and provides no restriction on w and s at all. This
leads to (4.56).

To distinguish the isomorphism classes of the structures within this series, we
use Theorem 3.1 in writing down the general form of an automorphism of Cq[x, y]
as Φθ,ω : x → θx, y → ωy. Certainly, this commutes with the action of k. For
other generators we get
Φθ,ωea0,s,tΦ−1
θ,ω (x) = Φθ,ωea0,s,t θ−1
x = θ−1
a0,
Φθ,ωea0,s,tΦ−1
θ,ω (y) = Φθ,ωea0,s,t ω−1
y = ω−1
Φθ,ωea0,s,t(y) = 0,
Φθ,ωfa0,s,tΦ−1
θ,ω (x) = Φθ,ωfa0,s,t θ−1
x = θ−1
Φθ,ω −qa−1
0 x2
+ ty4
= −qa−1
0 θx2
+ θ−1
tω4
y4
,
Φθ,ωfa0,s,tΦ−1
θ,ω (y) = Φθ,ωfa0,s,t ω−1
y = ω−1
Φθ,ω −qa−1
0 xy + sy3
= −qθa−1
0 xy + sω2
y3
.
That is, the automorphism Φθ,ω transforms the parameters of actions (4.55)–
(4.56) as follows:
a0 → θ−1
a0, s → ω2
s, t → θ−1
ω4
t.
In particular, this means that within the domain {s = 0, t = 0} one obtains an
invariant ϕ =
t
a0s2
of the isomorphism class. Obviously, the complement to this
domain further splits into three distinct subsets {s = 0, t = 0}, {s = 0, t = 0},
{s = 0, t = 0} corresponding to the isomorphism classes listed in the formulation,
and our result follows.
Note that up to isomorphism of Uq(sl2)-module algebra structure, the full
action matrix corresponding to (4.54)–(4.56) is of the form
M =
q−2x q−1y
1 0
−qx2 + ty4 −qxy + sy3
.
Theorem 4.7. The
0 0
0 0
;
0 0
0 0 1
-series consists of three-parameter
(d0 ∈ C {0}, s, t ∈ C) family of Uq(sl2)-actions on the quantum plane
k(x) = qx, k(y) = q2
y, (4.57)
e(x) = −qd−1
0 xy + sx3
, e(y) = −qd−1
0 y2
+ tx4
, (4.58)
f(x) = 0, f(y) = d0. (4.59)

Here we have the domain {(d0, s, t)| s = 0, t = 0} which splits into the
disjoint subsets {(d0, s, t)| s = 0, t = 0, ϕ = const} with ϕ =
t
d0s2
. This
uncountable family of subsets is in one-to-one correspondence to the isomor-
phism classes of Uq(sl2)-module algebra structures. Aside of those, one also has
three more isomorphism classes labelled by the subsets {(d0, s, t)| s = 0, t = 0},
{(d0, s, t)| s = 0, t = 0}, {(d0, s, t)| s = 0, t = 0}.
P r o o f. Is the same as that of the previous theorem.
Here, also up to isomorphism of Uq(sl2)-module algebra structures, the full
action matrix is
M =
qx q2y
−qxy + sx3 −qy2 + tx4
0 1
.
R e m a r k 4.8. There could be no isomorphisms between the Uq (sl2)-module
algebra structures on Cq[x, y] picked from different series. This is because every
automorphism of the quantum plane commutes with the action of k, hence, the
restrictions of isomorphic actions to k are always the same. On the other hand,
the actions of k in different series are different.
R e m a r k 4.9. The list of Uq (sl2)-module algebra structures on Cq[x, y]
presented in the theorems of this section is complete. This is because the assump-
tions of those theorems exhaust all admissible forms for the components (Mef)0,
(Mef)1 of the action ef-matrix.
R e m a r k 4.10. In all series of Uq(sl2)-module algebra structures listed in
Theorems 4.2–4.7, except the series
0 0
0 0 0
;
0 0
0 0 1
, the weight con-
stants α and β satisfy the assumptions of Proposition 4.1. So the claim of this
proposition is well visible in a rather simple structure of nonzero homogeneous
components of e(x), e(y), f(x), f(y), which everywhere reduce to monomials.
5. Composition Series
Let us view the Uq (sl2)-module algebra structures on Cq[x, y] listed in the
theorems of the previous section merely as representations of Uq (sl2) in the vector
space Cq[x, y]. Our immediate intention is to describe the composition series for
these representations.
Proposition 5.1. The representations corresponding to
0 0
0 0 0
;
0 0
0 0 1
-
series described in (4.30)–(4.31) split into the direct sum Cq[x, y] = ⊕∞
m=0 ⊕∞
n=0

Cxmyn of (irreducible) one-dimensional subrepresentations. These subrepresen-
tations may belong to two isomorphism classes, depending on the weight of a
specific monomial xmyn which can be ±1 (see Th. 4.2).
P r o o f. Since e and f are represented by zero operators and the monomials
xmyn are eigenvectors for k, then every direct summand is Uq(sl2)-invariant.
Now turn to nontrivial Uq (sl2)-module algebra structures and start with the
well-known case [8, 12].
0 0
0 0 0
;
0
0 1
-
series described in (4.32)–(4.34) split into the direct sum Cq[x, y] = ⊕∞
n=0Cq[x, y]n
of irreducible finite-dimensional subrepresentations, where Cq[x, y]n is the n-th
homogeneous component (introduced in Sect. 3) with dim Cq[x, y]n = n + 1 and
the isomorphism class of this subrepresentation is V1,n [8, Ch. VI].
P r o o f. Is that of Theorem VII.3.3 (b) from [8].
In the subsequent observations we encounter a split picture which does not
reduce to a collection of purely finite-dimensional sub- or quotient modules. We
recall the definition of the Verma modules in our specific case of Uq (sl2).
Definition 5.3. A Verma module V(λ) (λ ∈ C {0}) is a vector space with
a basis {vi, i ≥ 0}, where the Uq(sl2) action is given by
kvi = λq−2i
vi, k−1
vi = λ−1
q2i
vi,
ev0 = 0, evi+1 =
λq−i − λ−1qi
q − q−1
vi, fvi =
qi+1 − q−i−1
q − q−1
vi+1.
Note that the Verma module V (λ) is generated by the highest weight vector
v0 whose weight is λ (for details see, e.g., [8]).
0
0 0 0
;
0 0
0 0 1
-
series described in (2.2)–(4.42) split into the direct sum of subrepresentations
Cq[x, y] = ⊕∞
n=0Vn, where Vn = xnC[y]. Each Vn admits a composition series of
the form 0 ⊂ Jn ⊂ Vn. The simple submodule Jn of dimension n+1 is the linear
span of xn, xny, . . . , xnyn−1, xnyn, whose isomorphism class is V1,n and Jn is not
a direct summand in the category of Uq(sl2)-modules (there exist no submodule
W such that Vn = Jn ⊕ W). The quotient module Vn Jn = Zn is isomorphic to
the (simple) Verma module V q−n−2 .
P r o o f. Due to the isomorphism statement of Theorem 4.4, it suffices to
set the parameter of the series b0 = 1 in (2.2)–(4.42). An application of e and f

to the basis elements of Cq[x, y] gives
e(xn
yp
) = q1−p qp − q−p
q − q−1
xn
yp−1
= 0, ∀p > 0, (5.1)
e(xn
) = 0, (5.2)
f(xn
yp
) = q−n q2n − q2p
q − q−1
xn
yp+1
, ∀p ≥ 0, (5.3)
which already implies that each Vn is Uq(sl2)-invariant. Also Jn is a submodule
of Vn generated by the highest weight vector xn, as the sequence of weight vectors
f(xnyp) terminates because f(xnyn) = 0. The highest weight of Jn is qn, hence
by Theorem VI.3.5 of [8], the submodule Jn is simple and its isomorphism class
is V1,n.
Now assume the contrary to our claim, that is Vn = Jn ⊕W for some submod-
ule W of Vn, and Vn xnyn+1 = u + w, u ∈ Jn, w ∈ W is the associated decom-
position. In view of (5.1)–(5.2), an application of en+1 gives A(q)xn = en+1(w)
for some nonzero constant A(q), because en+1|Jn = 0. This is a contradiction,
because Jn ∩ W = {0}, thus there exist no submodule W as above.
The quotient module Zn is spanned by its basis vectors zn+1,zn+2, . . . which
are the projections of xnyn+1, xnyn+2, . . . respectively, to Vn Jn. It follows from
(5.1), that zn+1 is the highest weight vector whose weight is q−n−2, and it gen-
erates Zn by (5.3). Now the universality property of the Verma modules (see,
e.g., [8, Prop. VI.3.7]) implies that there exists a surjective morphism of modules
Π : V q−n−2 → Zn. It follows from Proposition 2.5 of [7] that ker Π = 0, hence
Π is an isomorphism.
The next series, unlike the previous one, involves the lowest weight Verma
modules. In all other respects the proof of the following proposition is the same
(we also set here d0 = 1).
0 0
0 0
;
0 0
0 0 1
-
Cq[x, y] = ⊕∞
n=0Vn, where Vn = C[x]yn. Each Vn admits a composition series of
the form 0 ⊂ Jn ⊂ Vn. The simple submodule Jn of dimension n+1 is the linear
span of yn, xyn, . . . , xn−1yn, xnyn. This is a ﬁnite-dimensional Uq(sl2)-module
whose lowest weight vector is yn with weight q−n, and its isomorphism class is
V1,n. Now the submodule Jn is not a direct summand in the category of Uq(sl2)-
modules (there exists no submodule W such that Vn = Jn ⊕ W). The quotient
module Vn Jn = Zn is isomorphic to the (simple) Verma module with lowest
weight qn+2.
Now turn to considering the three parameter series as in Theorems 4.6, 4.7.
Despite we have now three parameters, the entire series has the same split picture.

0
0 0 0
;
0 0
0 0 1
-
Cq[x, y] = ⊕∞
n=0Vn, where Vn is a submodule generated by its highest weight vector
yn. Each Vn with n ≥ 1 is isomorphic to a simple highest weight Verma module
V (q−n). The submodule V0 admits a composition series of the form 0 ⊂ J0 ⊂ V0,
where J0 = C1. The submodule J0 is not a direct summand in the category of
Uq(sl2)-modules (there exists no submodule W such that V0 = J0 ⊕ W). The
quotient module V0 J0 is isomorphic to the (simple) Verma module V q−2 .
P r o o f. First, let us consider the special case of (4.55), (4.56) in which
s = t = 0 and a0 = 1. Then Vn = C[x]yn are Uq(sl2)-invariant, and we calculate
e(xp
yn
) = q−n−p+1 qp − q−p
q − q−1
xp−1
yn
= 0, ∀p > 0,
e(yn
) = 0,
f(xp
yn
) = qn+p qp+n − q−p−n
q − q−1
xp+1
yn
, ∀p ≥ 0. (5.4)
Note that f(xpyn) = 0 only when p = n = 0. Therefore Vn admits a generating
highest weight vector yn whose weight is q−n. As in the proof of Proposition 5.4
we deduce that each Vn with n ≥ 1 is isomorphic to the (highest weight simple)
Verma module V (q−n). In the case n = 0, it is clear that V0 contains an obvious
submodule C1 which is not a direct summand by an argument in the proof of
Proposition 5.4.
Turn to the general case when the three parameters are unrestricted. The
formulas (4.54)–(4.56) imply the existence of a descending sequence of submod-
ules
. . . ⊂ Fn+1 ⊂ Fn ⊂ Fn−1 ⊂ . . . ⊂ F2 ⊂ F1 ⊂ F0 = Cq[x, y],
where Fn = ∪∞
k=nC[x]yk, because operators of the action, being applied to a
monomial, can only increase its degree in y. Note that the quotient module
Fn Fn+1 with unrestricted parameters is isomorphic to the module C[x]yn ∼=
V (q−n), just as in the case s = t = 0.
Now we claim that Fn+1 is a direct summand in Fn, namely Fn = Vn ⊕Fn+1,
n ≥ 0, with Vn = Uq (sl2) yn for n ≥ 1 and V0 = Uq (sl2) x.
First consider the case n ≥ 1. By virtue of (4.54)–(4.56), yn is a generating
highest weight vector of the submodule Vn = Uq(sl2)yn, whose weight is q−n.
Another application of the argument in the proof of Proposition 5.4 establishes
an isomorphism Vn
∼= V (q−n); in particular, Vn is a simple module by Proposition
2.5 of [7]. Hence Vn ∩Fn+1 can not be a proper submodule of Vn. Since Vn is not
contained in Fn+1 (as yn /∈ Fn+1), the latter intersection is zero, and the sum

Vn + Fn+1 is direct. On the other hand, a comparison of (4.56) and (5.4) allows
one to deduce that Vn + Fn+1 contains all the monomials xpym, m ≥ n, p ≥ 0.
This already proves Fn = Vn ⊕ Fn+1.
Turn to the case n = 0. The composition series 0 ⊂ C1 ⊂ V0 = Uq(sl2)x
is treated in the same way as that for V0 in Proposition 5.4; in particular, the
quotient module V0/C1 is isomorphic to the simple Verma module V q−2 . Let
π : V0 → V0/C1 be the natural projection map. Obviously, F1 does not contain
C1, hence the restriction of π to V0 ∩ F1 is one-to-one. Thus, to prove that
the latter intersection is zero, it suffices to verify that π(V0 ∩ F1) is zero. As
the module V0/C1 is simple, the only alternative to π(V0 ∩ F1) = {0} could
be π(V0 ∩ F1) = V0/C1. Under the latter assumption, there should exist some
element of V0 ∩ F1, which is certainly of the form Py for some P ∈ Cq[x, y], and
such that π(x) = π(Py). This relation is equivalent to x − Py = γ for some
constant γ, which is impossible, because the monomials that form Py, together
with x and 1, are linearly independent. The contradiction we get this way proves
that V0 ∩ F1 = {0}, hence the sum V0 + F1 is direct. On the other hand, a
comparison of (4.56) and (5.4) allows one to deduce that V0 + F1 contains all
the monomials xpym, with m, p ≥ 0. Thus the relation Fn = Vn ⊕ Fn+1 is now
proved for all n ≥ 0. This, together with ∩∞
i=0Fi = {0}, implies that
Cq[x, y] = (⊕∞
n=1Uq(sl2)yn
) ⊕ Uq(sl2)x,
which was to be proved.
In a similar way we obtain the following
0 0
0 0
;
0 0
0 0 1
-
Cq[x, y] = ⊕∞
n=0Vn, where Vn is a submodule generated by its lowest weight vector
xn. Each Vn with n ≥ 1 is isomorphic to a simple lowest weight Verma module
whose lowest weight is qn. The submodule V0 admits a composition series of the
form 0 ⊂ J0 ⊂ V0, where J0 = C1. The submodule J0 is not a direct sum-
mand in the category of Uq(sl2)-modules (there exists no submodule W such that
V0 = J0 ⊕ W). The quotient module V0 J0 is isomorphic to the (simple) lowest
weight Verma module whose lowest weight is q2.
The associated classical limit actions of the Lie algebra sl2 (here it is the Lie
algebra generated by e, f, h subject to the relations [h, e] = 2e, [h, f] = −2f,
[e, f] = h) on C[x, y] by differentiations is derived from the quantum action via
substituting k = qh with subsequent formal passage to the limit as q → 1.
In this way we present all quantum and classical actions in Table 1. It should
be noted that there exist more sl2-actions on C[x, y] by differentiations (see, e.g.,

[6]) than one can see in Table 1. It follows from our results that the rest of the
classical actions admit no quantum counterparts. On the other hand, among the
quantum actions listed in the ﬁrst row of Table 1, the only one to which the above
classical limit procedure is applicable, is the action with k(x) = x, k(y) = y.
The rest three actions of this series admit no classical limit in the above sense.
Table 1.
Symbolic matrices Uq(sl2) − symmetries
Classical limit
sl2 − actions
by diﬀerentiations
0 0
0 0 0
;
0 0
0 0 1
k(x) = ±x, k(y) = ±y,
e(x) = e(y) = 0,
f(x) = f(y) = 0,
h(x) = 0, h(y) = 0,
e(x) = e(y) = 0,
f(x) = f(y) = 0,
0
0 0 0
;
0 0
0 0 1
k(x) = qx,
k(y) = q−2
y,
e(x) = 0, e(y) = b0,
f(x) = b−1
0 xy,
f(y) = −qb−1
0 y2
h(x) = x,
h(y) = −2y,
e(x) = 0, e(y) = b0,
f(x) = b−1
0 xy,
f(y) = −b−1
0 y2
0 0
0 0
;
0 0
0 0 1
k(x) = q2
x,
k(y) = q−1
y,
e(x) = −qc−1
0 x2
,
e(y) = c−1
0 xy,
f(x) = c0, f(y) = 0,
h(x) = 2x,
h(y) = −y,
e(x) = −c−1
0 x2
,
e(y) = c−1
0 xy,
f(x) = c0, f(y) = 0.
0
0 0 0
;
0 0
0 0 1
k(x) = q−2
x,
k(y) = q−1
y,
e(x) = a0, e(y) = 0,
f(x) = −qa−1
0 x2
+ ty4
,
f(y) = −qa−1
0 xy + sy3
.
h(x) = −2x,
h(y) = −y,
e(x) = a0, e(y) = 0,
f(x) = −a−1
0 x2
+ ty4
,
f(y) = −a−1
0 xy + sy3
.
0 0
0 0
;
0 0
0 0 1
k(x) = qx, k(y) = q2
y,
e(x) = −qd−1
0 xy + sx3
,
e(y) = −qd−1
0 y2
+ tx4
,
f(x) = 0, f(y) = d0,
h(x) = x, h(y) = 2y,
e(x) = −d−1
0 xy + sx3
,
e(y) = −d−1
0 y2
+ tx4
,
f(x) = 0, f(y) = d0,
0 0
0 0 0
;
0
0 1
k(x) = qx,
k(y) = q−1
y,
e(x) = 0, e(y) = τx,
f(x) = τ−1
y, f(y) = 0,
h(x) = x,
h(y) = −y,
e(x) = 0, e(y) = τx,
f(x) = τ−1
y, f(y) = 0.

Acknowledgements. One of the authors (S.D.) is thankful to Yu. Be-
spalov, J. Cuntz, B. Dragovich, J. Fuchs, A. Gavrilik, H. Grosse, D. Gurevich,
J. Lukierski, M. Pavlov, H. Steinacker, Z. Rakić, W. Werner, and S. Worono-
wicz for fruitful discussions. Also he is grateful to the Alexander von Humboldt
Foundation for valuable support as well as to J. Cuntz for kind hospitality at the
Mathematisches Institut, Universität Münster, where this paper was finalized.
Both authors would like to express their gratitude to D. Shklyarov who attracted
their attention to the fact that the results of this work were actually valid for q
not being a root of unit, rather than for 0 < q < 1. We are also grateful to the
referee who pointed out some inconsistencies in a previous version of the paper.
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