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abstract: '[It is shown that the finite dimensional irreducible representaions of the quantum matrix algebra $ M_{ q,p}(2) $ ( the coordinate ring of $ GL_{q,p}(2) $) exist only when both q and p are roots of unity.  In this case th e space of states has either the topology of a torus or a cylinder which may be thought of as generalizations of cyclic representations.]{}' 
author:
- Vahid Karimipour
date:  
title: |
    **Representations\
    of The Quantum Matrix Algebra $ M_{q,p}(2) $**
---

= 0 cm = 0 cm = 16 cm = 22 cm

[*Department of Physics , Sharif University of Technology\
P.O.Box 11365-9161 Tehran, Iran\
Institute for studies in Theoretical Physics and Mathematics\
P.O.Box 19395-1795 Tehran, Iran\
Email.  Vahidka@irearn.bitnet*]{}. 

\
The representaion theory of quantized universal enveloping algebras \[ 1-5 \] has been extensively studied by many authors \[ 6-10 \] and many beautiful results have been obtained.  Among these are the cyclic representaions which occur when q is a root of unity.  However the representaion theory of the dual objects , that is the quantization of the algebra of functions on the group ( Quantum Matrix Algebras ) has not been systematically studied .  Only a few concrete representaions exist \[ 11,12 \] .  The first attempt toward such a goal has been reported in \[ 13 \], where the irreducible finite dimensional representations of $ M_q(2) $ were classified and it was shown that such irreducible representations exist only when q is a root of unity .  These representations were either cyclic or highest weight. 

Although the representation theory of a multiparametric quantum enveloping algebra is trivial ( since the extra parameters appear only in the coproducts in the form of a twisting \[ 14 \] ) , the corresponding task for the multiparametric quantum matrix group is far from straightforward , since the extra parameters now appear in the algebra itself.  As we will see this will lead to quite new features in the represen representaions of the quantum matrix algebra. 

In this paper we study the representations of $ M_{q,p} ( 2) $ ( the coordinate ring of $ GL_{q,p}(2) $) and classify its finite dimensional irreducible representations.  Our main reslults are the following:

i\) finite dimensional irreducible representations exist only when both q and p are roots of unity. 

ii\) the space of states has the topology of a torus or that of a c ylinder, depending on the value of some parameters which define the representation. 

iii\) when none of the parameters q or p is a root of unity , the states occupy the sites of an infinite rectangular lattice. 

iv\) when q is not a root of unity , but p is , the states occupy the sites of an infinitely long cylinder.\
[**II.  the quantum matrix algebra $ GL_{q,p}(2) $**]{}\
This algebra \[15\] is generated by the entries of a matrix   $ T=\left( \begin{array}{ll} a&b \\ c&d \end{array} \right) $\
and subject to the relations: $$R T_1 T_2 = T_2 T_1 R$$ where R is a two parametric solution of Yang Baxter equation:

$$R = \left( \begin{array}{llll} q &&& \\ & p && \\ & {
q-q^{-1}} & { p^{-1}}&\\ &&& {q} \end{array} \right)$$

The relations which follow from (1-2) are : $$ab = qp\ ba \hskip 2cm bd = { q\over p }\ db$$ $$ac = { q\over p}\  ca \hskip 2cm cd = { qp }\  dc$$ $$bc = {1\over  p^2} \ cb \hskip 2cm ad - da =  p (q-q^{-1}) \ bc$$ Compared with $ GL_q(2) $ this quantum group has two particular features which make its representation theory quite different.  The first is that the generators $ b $ and $ c$ no longer commute and hence one can not build the representation space from common eigenvectors of  $ b $ and $ c$ as in the work of \[ 13 \].  The second is that the quantum determinant $ D = ad - qp\  bc $ is not central, but satisfies the relations: $$D a = a D \hskip 2cm Db = p^2 bD$$ $$D d = d D \hskip 2cm Dc = p^{-2} cD$$ The following relations can also be obtained by repeated application of ( 3 )

$$a^n d - d a^n = qp\ ( 1- q^{-2n}) \ a^{n-1} bc$$ $$a d^n - d^ n a = { p\over q } ( q^{2n} - 1 )\
d^{n-1} bc$$ when $ q^n = p^n = 1 $ one sees from ( 3 ) and ( 5 ) that $ a^
n , b^n  , c^n  $ and $ d^n $ are central. 

In \[13\] both the commutativity of $ b $ and $ c $ and the centrality of the quantum determinant have been used to a large extent.  In our case when the above facts are no longer true , we must proceed in a different way. 

Lets denote the product of $ b $ and $ c $ by M ( $ M = bc $ ).  Then from (3) we find : $$M a = q^{-2} a M \hskip 2cm M b = p^2 b M$$ $$M c = p^{-2} c M \hskip 2cm M d = q^2 d M$$ Its clear from (4) that the operators M and D are commuting: $$M D = D M$$ We will use these commuting operators in the following sections to build up the irreps.  of $ M_{q,p}(2) $. 

\
Following \[13\] we call those $ M_{q,p}(2)$ modules in which one or more of the generators identically vanish, trivial modules .  In these cases the representation reduces to that of a simpler algebra.  Clearly the interesting representations are nontrivial ones to which we restrict ourselves in the rest of this paper.  The following lemma \[13\] establishes the condition for nontriviality of the representation.\
[**Lemma 1**]{}: Let V be a vector space .  Then an irreducible representation $ \rho : M_{q,p}(2)\longrightarrow End (V) $ is trivial if $ b $ or $ c
$ have zero eigenvalue in their spectrum.\
[**Proof**]{} : We follow a slightly different line of reasoning which is simpler than that of \[13\].  Without loss of generality , lets assume that  $ K_b
 \equiv Ker \ b \ne\  \{ 0\}\  $ , then since $ K_b $ is a subspace of V , we can choose a basis for it like :$ \{ e_1 , . . .   e_ m \} $ .  From eq. ( 3) we see that $ a K_ b \ , \ c K_b\  , $ and  $  d K_b\  $ are all subspaces of $ K_b $ .  Therefore the vectors $ \{ e_i\} $ transform among themselves under the action of $ a , b , c $ and $ d $ and hence $ K_b $ is an invariant subspace of V .  Since the representation is irreducible $ K_b = V $ .  Therefore $ b  K_b = b  V = 0 $ and the representation is trivial.  Hereafter we assume that $ K_b = K_c = \{ 0 \} $ .\
[**Lemma 2**]{}: A finite dimensional irreducible $M_{q,p}(2) $ module exists only when both $ q $ and $ p $ are roots of unity.\
[**Proof**]{}: Let $ v_0$ be a common eigenvector of $ M $ and $ D $ .  $$M v_0  = \mu v_ 0 \hskip 2cm D v_0  = \lambda v_0$$ Then from ( 3,4, and 6 ) one sees that the string of states $ v_n
 \equiv d^n v_0  $ satisfy: $$M v_n = q^{2n} \mu   v_n \hskip 2cm D v_n = \lambda
v_n$$ For the parameter q we adopt the reasoning of ref.\[13\].  To have finite dimensional representations one must have $ d^l v_0 = 0  $ for some $ l $ while all the vectors $ v_n $ for $ n<l $ are independent. Consider the string of states $ a^m u_0 $ where $ u_0 = v_{l-1} $.  Again one must have $ a^ {l'} u_0 = 0 $ but $ a^{ l'-1} u_0 \ne 0 $.  Then one will have $$0 = da^{l'}u_0 = {qp}( q^{-2l'}-1)\mu a^{l'-1}u_0$$ which means that q must be a root of unity.  On the other hand, suppose that $ p $ is not a root of unity.  then the string of states $ u_n = b^ n v_0 $ satisfy: $$M u_n = p^{2n}\mu\  u_n \hskip 2cm D u_n = p^{2n}\lambda u_n$$ If $ p $ not a root of unity all the above states will be independent and the representation can not be finite. 

Hereafter we set $ q^r = p^r = 1 $ ( note: $ q $ and $  p $ may be different roots of unity , i.e:  $  q^ {r_1} = p^{r_2 }  = 1 $ .  We set $ r $ to be the least common multiplier of $ r_1 $ and $ r_2 $ ). 

In this case $ a^r  , b^r , c^r  $ and $ d^ r $ are central and on V we set them equal to $ \eta_a , \eta_b ,
\eta _c $ and $ \eta _d $ respectively.  Clearly $ \eta _b $ and $
 \eta_c $ are both different from zero, otherwise $ K_b $ and $ K_c $ will have nonzero elements. 

We now classify the finite dimesnsional irreducible representations of $ M_{q,p}(2) $ which fall into three classes depending on the values of $ \eta_a $ and $ \eta_d $ .\
We denote the vector $ v_0 $ introduced in (8) by $ \vert 0 , 0 > $ and c onsider the lattice of states $W$ (fig.  1):

$$W = \{ \vert l , n > \equiv b^l d^n \vert 0 , 0 > \hskip 1cm
0\leq l , n \leq r-1 \}$$

These states are the common eigenvectors of $ M $ and $ D $ ( see 3-6).  $$M \vert l ,n > = p^{2l}q^{2n} \mu\vert l,n> \hskip
1cm D\vert l ,n > = p^{2l}\lambda \vert l , n >$$ We define the action of $ a $ and $ c $ on the state $ \vert 0 , 0
> $ as follows: $$a \vert 0 , 0 > = \alpha_0 \vert 0 , r-1 > \hskip 1
cm c\vert 0, 0 > = \gamma_0  \vert r-1 , 0 >$$

Then we have :

[**Theorem 3** ]{} : The following defines an irreducible representation of $ M_{q,p}(2) $.\
i) $ b\vert l , n> = \vert l+1 , n> \hskip 2cm b\vert r-1 , n> =
\eta _b\vert 0 , n > $\
ii)$ d\vert l , n> = ({p\over q })^l\vert l , n+1 > \hskip 2cm  d\vert
 l , r-1> = ( { p\over q})^l \eta_d \vert l , 0 >  $\
iii)$ c\vert l , n> = p^{2l}q^{2n}\eta_b \gamma_0 \vert l-1 , n >\hskip
 1cm  c\vert 0 , n > = q^{2n} \gamma_0 \vert r-1 , n >  $\
iv)$ a\vert l , n> = (qp)^l ( \alpha_0 \eta_ d + {p\over q } ( q^{2n}-
1)\gamma_0 \eta_b) \vert l , n-1 >$\
$ a\vert l , 0 > = ( qp)^l \alpha_0 \vert l , r-1 > $

[**Proof**]{}: i) and ii) are obvious .  we give an explicit verification of iii).  iv )is obtained by straightforward manipulations.  Acting with $ c $ on the state $ \vert l , n > $ and using the commutation relataions ( 3 ) we find : $$c \vert l,n > = c b^l d^n \vert 0, 0 > = p^{2l} ( qp )^n b^l d^
n c \vert 0 , 0 >$$ using ( 12 ) and the definition of states ( 10 ) we arrive at : $$c \vert l,n > = p^{2l} ( qp )^n b^l d^n \gamma_0  \vert r-1 , 0 >
= p^{2l} ( qp )^n \gamma_0 b^l d^n b^{r-1} \vert 0 , 0 >$$ againg using the commutation relations and the fact that $ b^r =
 \eta_b $ we finally find: $$c \vert l ,n > = p^{2l} ( qp )^n \gamma_0 (
{p\over q})^{n(r-1)}b^{l-1} d^n  \vert 0,0 > = p^{2l} q^{2n} \eta_b
\gamma_0 \vert l-1 , n >$$ The second part of iii) is proved similarly :

$$c\vert 0,n > = c  d^n \vert 0, 0 > =  ( qp )^n  d^n c \vert 0 ,0 >$$ $$= ( qp )^n d^n \gamma_0  \vert r-1 , 0 >
= ( qp )^n \gamma_0  d^n b^{r-1} \vert 0 , 0 >$$ $$= ( qp )^n \gamma_0 ( {p\over q})^{n(r-1)}b^{r-1} d^n \vert 0,0
 > = q^{2n} \gamma_0 \vert r-1 , n >$$ It is interesting to note that the space of states has the topology of a torus ( see fig.  2 where the actions of all the generators are shown graphically). 

The dimension of this representation is $ r^2 $.  To prove that it is the only irreducible representation in this case , we note that the dimension of V can not be greater than $ r^ 2 $ since otherwise the above lattice of states which is based on a single common eigenvector of $ M $ and $ D $ will provide an invariant subspace which contradicts the irreducibility of the representation.  The dimension of V can not be less that $ r^ 2 $ either since then one of the strings of states $ d^n \vert 0 , 0 > $ or $ b^n \vert 0 , 0 > $ must terminate for some value of $ n $ less than $ r $ .(  i.e:$ d^n \vert 0 , 0 > = 0 , n <
 r $) This then means that $$\eta_d\vert 0,0 > = d^ {r-n} ( d^n \vert 0, 0 >) = 0$$ which contradicts the assumption of $ \eta_d \ne 0 $.\
[**Remark** ]{}: The parameters $ \alpha_0 \ \  \gamma_0 \ \ \lambda $ an d $ \mu $ are not independent of $ \eta_a ,\eta_b ,\eta_c  $ and $ \eta_d $.  The following relations exist among them: $$\eta_c = \gamma_0^r \eta_b^{r-1} \hskip 3cm \mu =  \gamma
_0 \eta_b$$ $$\lambda = \alpha_0 \eta_d - {p\over q} \mu \hskip 2 cm
\eta_a = \alpha_0 \prod_{i=1}^{r-1} ( \lambda + { p\over q } \mu  q^{2i})$$ The proof of these formulas is given in the appendix.  We now turn to the second kind of representations.\
[**B. Cylindrical Representations $ (\eta_a = \eta_d = 0) $**]{}\
In this case the representation in theorem 3 is modified as follows:

$$d \vert l , r-1 > = 0 \ \ \ \ \ \ \ \ a \vert l , 0 >
= 0$$

$$a \vert l,n > = ( qp)^l( {p\over q} ( q^{2n} - 1 )
\gamma _0 \eta _b ) \vert l, n-1>$$

all the other relations remain intact.  This kind of representation may be called a cylindrical representation and may be thought of as a truncated form of the toroidal representation.(  fig.  3 )

C\) Where only one of the parameters ( say $ \eta_a $ ) is zero , the relations ( 15-16 ) are modified as follows: $$d \vert l , r-1 > = \eta_a \vert l ,0 > \ \ \ \ \ \
\ \ a \vert l , 0 > = 0$$ $$a \vert l,n > = ( qp)^l( {p\over q} ( q^{2n} - 1 )
 \gamma_0 \eta_b ) \vert l, n-1>$$ This then may be thought of as a semitoroidal representaion in which $ d $ traverses completely one of the cycles of the torus while $ a $ dose not. 

[**IV.  Infinite Dimensional Representations** ]{}\
In theorem 3 one can relax the conditions on the right hand side .  Then one can check easily that the left hand side equations define an infinite dimensional representation of $ M_{q,p}(2) $ on the two dimensional lattice : $ W = \{ \vert l,n > \ \ \ \ \ -\infty < l , n < \infty \} $ $$b\vert l , n> = \vert l+1 , n >$$ $$d\vert l , n> = ({p\over q })^l\vert l , n+1 >$$ $$c\vert l , n> = p^{2l}q^{2n}\mu \vert l-1 , n >$$ $$a\vert l , n> = (qp)^l ( \lambda + ({p\over q }\mu)  q^{2n}) \vert
l ,n-1>$$

[**Note**]{}: The states of this representation are not necessarily built up on a vaccum ( i.e $ \vert l ,n > \ne b^l d^n \vert 0 , 0 > $ ). 

If both $ q $ and $ p $ are roots of unity ( $ q^r = p^r = 1 $ ) then one can consistently identify the states as follows:

$$\vert l,n> \equiv \vert l+r , n > \ \ \ \ \ \ \ \ \vert l,n>
\equiv \vert l,n+r >$$ Representation ( 19 ) will induce a Toroidal representation on the equivalence class of these states. 

If only $ p $ is a root of unity , then one can consistently identify the states as follows :

$$\vert l+r , n > \equiv q^{nr} \vert l,n >$$ Eqs. ( 19) then induces a cylindrical representation on the the equivalence of these states.  By setting $ \lambda = -{ p\over q } \mu  $ one can also obtain a lowest weight module. 

[**Discussion** ]{}\
In the classical limit (q = p = 1),the coordinate ring of $ GL_{q,p}(2
) $ degenerates into a free abelian algebra whose irreducible finite dimensional representations are one dimensional where the generators $ a, b, c, $ and $ d $ are represented by by the pure numbers $ \eta_a , \eta_b , \eta_c , $ and $ \eta_d  $ respectively.  This limit is obtained from the representations in this paper by noting that when $ r = 1 $ , the lattice of states in fig.  1 has only one single state, namely $ \vert 0,0 > $ and from theorem I we have: $$a\vert 0, 0 > = \alpha_0 \vert 0, 0 >\ \ \ \ \ \ \ \ \  b\vert
0, 0 > = \eta _b \vert 0, 0 >$$ $$c\vert 0, 0 > = \gamma_0 \vert 0, 0 >\hskip 2cm    d\vert 0, 0 > =
\eta_d \vert 0, 0 >$$ From eqs.(14-15) we also obtain $ \gamma_0 = \eta_c $ and $ \alpha_0 =
\eta_a $ which proves the assertion. 

What appears to be very interesting about the representation theory of quantum matrix algebras compared with those of the quantized universal enveloping algebras is that in the latter case the classical limit is a lie algebra and one can use the decomposition of the root space of the lie algebra into the Cartan subalgebra and positive and negative root system for building up the represetation.  Recall that this decomposition remains essentially intact in the process of quantization .  This then leads to a paralellism between the represetation theory in the deformed and the undeformed case.  However for the case of quantum matrix algebras such a decomposition and the resulting paralellism does not exist. One expects that completely new feature s arise in their representation theory.(see for example \[16\] and \[17\]).\
\
[**Appendix: Proof of eqs. ( 13) and (14)** ]{}

Repeated use of theorem 3 gives the following: $$c^{r-1} \vert r-1,0> =(\eta_b \gamma_0)^{r-1} \vert 0 , 0 >$$ Acting by $ c $ on both sides we obtain $$\eta_c \vert r-1 , 0 > = (\eta_b \gamma_0)^{r-1}\gamma_0 \vert r-
1 , 0 >$$ where we have used (12) .  Comparison of both sides gives the first relation of (13).  To obtain the second relation of ( 13 ) we note that $$\mu \vert 0, 0
 >= M \vert 0 , 0> = bc \vert 0 , 0 > =
b ( \gamma_0 \vert r-1 , 0 > ) = \gamma_0 \eta_b \vert 0 , 0 >$$ To prove the first relation of (14) we act on the state $ \vert 0,0 > $ with $ D = ad - qp M $ and use theorem 3 .  Finally we note that: $$a^{r-1} \vert 0 , r-1 > =  \prod_{i=1}^{r-1} ( \lambda +({p\over q }
\mu) q^{2i})\vert 0 , 0 >$$

Acting on both sides with $ a $ and using (12) gives (14). 

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16. [ 16-]{} V. Karimipour; Representations of the Coordinate Ring of $ GL
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17. [ 17-]{} V. Karimipour; Representations of the Coordinate Ring of $
    GL_q(n) $, in preparation

FIGURE CAPTION

Fig.  3 : The Lattice of States W

Fig.  2 : Toroidal Representation

Fig.  3 : Cylindrical Representation
