Simple Modules Over The Quantum Affine Space
aa r X i v : . [ m a t h . R T ] J a n SIMPLE MODULES OVER THE QUANTUM AFFINE SPACE
SNEHASHIS MUKHERJEE AND SANU BERA Abstract.
The coordinate ring K Q [ x , · · · , x n ] of the quantum affine space isthe algebra generated over a field K by the variables x , · · · , x n satisfying therelations x i x j = q ij x j x i , ∀ ≤ i, j ≤ n . We construct simple K Q [ x , · · · , x n ]-modules in a more general setting where the entries q ij lie in a torsion subgroupof K ∗ and show analogous results hold as in [5]. Introduction
Let Q = ( q ij ) n × n be a multiplicatively anti-symmetric n × n matrix over a field K ,that is, q ii = 1 and q ij q ji = 1 for all 1 ≤ i, j ≤ n. The coordinate ring K Q [ x , · · · , x n ]of the quantum affine space is the algebra generated over a field K by the variables x , · · · , x n satisfying the relations(1.1) x i x j = q ij x j x i , ∀ ≤ i, j ≤ n. The coordinate ring K Q [ x , · · · , x n ] plays a fundamental role in non-commutativegeometry (see [2, 3]). Suitable localizations of the quantum affine space arise in therepresentation theory of torsion-free nilpotent groups. (see [1]).In [5], Kangju Min and Sei-Qwon Oh investigated the simple C Q [ x , · · · , x n ]-modules in the uniparameter case q ij = q, ∀ ≤ i < j ≤ n, where q is a primitive m -th root of unity and gave an explicit construction of thesimple modules. In particular, they established the following fact: Theorem 1.1 ([5], Theorem 5) . There is a surjective map Φ from C n onto the setof all simple C Q [ x , · · · , x n ] -modules in the case when q is a primitive m -th root ofunity such that dim C Φ( α ) = m [ p/ , where p is the number of non zero α i in α = ( α , · · · , α n ) and [ x ] denotes thegreatest integer ≤ x . Mathematics Subject Classification.
Key words and phrases.
Quantum affine space, Quantum torus, Simple modules.
In this article, we wish to generalize the above result for multiparameter caseassuming that K is an algebraically closed field and the group Λ generated by themultiparameters q ij is a torsion (and hence cyclic) subgroup of K ∗ . Let m be theorder and q be a generator of the group Λ . Now the multiparameters are of theform(1.2) q ij = q r ij , r ij ∈ Z m , ∀ ≤ i, j ≤ n. Throughout this paper a “module” means a right module. Given a simple module N over K Q [ x , · · · , x n ] we may assume that the action of the variables x , · · · , x n on N is non trivial. Otherwise, if the action of a variable, say, x j on N is trivial,then x j ∈ ann( N ) and N becomes a simple K Q [ x , · · · , x j − , x j +1 , · · · , x n ]-module.Let A Q be the localization of K Q [ x , · · · , x n ] with respect to the Ore set X generated by x , · · · , x n . Then A Q is the ring K Q [ x ± , · · · , x ± n ] generated by thevariables x , · · · , x n , together with their inverses, which satisfy the relations (1.1).In the literature the ring A Q is called a quantum torus of rank n . When n = 2, wewrite A q instead of A Q where xy = qyx .In [6], Karl-Hermann Neeb investigated the normal form of rational quantumtori, that is, assuming that the group Λ is torsion. If m = p l · · · p l k k be the primefactorization of m , then the set(1.3) P := { p j · · · p j k k | ≤ j i ≤ l i , i = 1 , · · · , k } ⊆ Z m is the multiplicatively closed set. In particular, the following result was shown: Theorem 1.2 ([6], Theorem III.4) . For a rational quantum torus A Q of rank n over K , there exists s ∈ N with s ≤ n and h | h | · · · | h s in P \ { } such that A Q ∼ = A q ⊗ A q h ⊗ A q h ⊗ · · · ⊗ A q hs ⊗ K [ Z n − s ] , s < n (1.4) or A Q ∼ = A q ⊗ A q h ⊗ A q h ⊗ · · · ⊗ A q hs − ⊗ A q zhs , s = n (1.5) for some z ∈ N with ord ( q zh s ) = ord ( q h s ) and for quantum tori A q hi (with h = 1 )of rank . IMPLE MODULES OVER THE QUANTUM AFFINE SPACE 3
Remark 1.1.
Since we are only interested in constructing simple A Q -modules uptoisomorphism, in view of Remark (4.1) it suffices to assume h s = zh s in (1.5). Thisallows us to work with the same normal form, viz A Q ∼ = A q ⊗ A q h ⊗ A q h ⊗ · · · ⊗ A q hs ⊗ K [ Z n − s ](1.6) in both cases of Theorem (1.2). Remark 1.2.
The explicit description of the set P in (1.3), it follows that every h i ∈ P \ { } divides m . We take g i := mh i . Notation 1.1.
We let X ± i and X ± i + s be the generators of A q hi with h = 1 , whichsatisfy the relation (1.7) X i X i + s = q h i X i + s X i . Also let the variables X s +1 , · · · , X n generate the group algebra K [ Z n − s ] . The following theorem generalizes the construction in [5] of simple modules forthe more general situation of relations (1.2).
Theorem A (Construction of Simple Modules) . For α = ( α , · · · , α n ) ∈ ( K ∗ ) n , let M ( α ) be the K -vector space with basis e ( a , · · · , a s ) , where ≤ a i ≤ g i − . Thenthere is an A Q -module structure on M ( α ) define as follows: e ( a , · · · , a s ) X i = α i e ( a , · · · , a i ∔ , · · · , a s ) , ∀ ≤ i ≤ se ( a , · · · , a s ) X i + s = α − i α i + s ( q h i ) a i − e ( a , · · · , a i ∔ ( − , · · · , a s ) , ∀ ≤ i ≤ se ( a , · · · , a s ) X s + j = α s + j e ( a , · · · , a s ) , ∀ ≤ j ≤ n − s where ∔ is addition in the additive group Z g i and h i ’s are as in the Theorem (1.2).Moreover, M ( α ) is a simple A Q -module of dimension s Q i =1 g i . The following results also hold.
Theorem B.
Each simple K Q [ x , · · · , x n ] -module has the form M ( α ) , for some α ∈ ( K ∗ ) n . Theorem C.
There is a surjective map Ψ from K n onto the set of all simple K Q [ x , · · · , x n ] -modules such that dim K Ψ( α ) = s Q i =1 g i , where g i and s are as definedabove. SNEHASHIS MUKHERJEE AND SANU BERA Preliminaries
We quote the following proposition from [5] which will use in the proof.
Proposition 2.1 ([5], Proposition 1) . Let R be an algebra over a field K , Z bea finitely generated subalgebra contained in the center of R and let R be finitelygenerated as Z -module. For a simple R -module N , the following hold.(i) R is Noetherian.(ii) N is finite dimensional vector space over K .(iii) ann R ( N ) is a maximal ideal of R .(iv) ann R ( N ) ∩ Z is a maximal ideal of Z . Corollary 2.1.
Every simple K Q [ x , · · · , x n ] -module is finite dimensional vectorspace over K . Proof.
Using Proposition (2.1). (cid:3)
Proposition 2.2.
Any simple K Q [ x , · · · , x n ] -module N with N x i = 0 for all i ,can be extended uniquely to a simple A Q -module. Moreover, the localization map N → N X − is an K Q [ x , · · · , x n ] -module isomorphism.Proof. Clearly N is X -torsion free. Since N is simple, N x = N for all x ∈ X .Thus N is X -divisible. Hence the assertion follows from Proposition (10.11) andCorollary (10.16) in [4]. (cid:3) Construction of a Simple Module
In this section we wish to construct simple modules over K Q [ x , · · · , x n ]. ByProposition (2.2), it suffices to do this for the quantum torus A Q of rank n . Wefollow the line of reasoning in [5] and our construction proceeds in the followingsteps.Step 1: (The representation space) For α = ( α , · · · , α n ) ∈ ( K ∗ ) n , let M ( α ) be the K -vector space with basis e ( a , · · · , a s ) for 0 ≤ a i ≤ g i − g i and s are as defined in Introduction section.Step 2: (Module structure) Using the normal form of quantum torus A Q in theRemark (1.1) along with the Notation (1.1), let us define the A Q -module IMPLE MODULES OVER THE QUANTUM AFFINE SPACE 5 structure on the K -space M ( α ) by the action of X i ’s on the basis vectors asfollows: e ( a , · · · , a s ) X i = α i e ( a , · · · , a i ∔ , · · · , a s ) , ∀ ≤ i ≤ se ( a , · · · , a s ) X i + s = α − i α i + s ( q h i ) a i − e ( a , · · · , a i ∔ ( − , · · · , a s ) , ∀ ≤ i ≤ se ( a , · · · , a s ) X s + j = α s + j e ( a , · · · , a s ) , ∀ ≤ j ≤ n − s where ∔ is addition in the additive group Z g i .Step 3: (Well-definedness) In order to establish the well-definedness of the aboverules, we need to check that for 1 ≤ i ≤ s and 0 ≤ a i ≤ g i − e ( a , · · · , a s ) X i X i + s = q h i e ( a , · · · , a s ) X i + s X i . (3.1) e ( a , · · · , a s ) X i X j = e ( a , · · · , a s ) X j X i , ∀ j = i + s. (3.2) e ( a , · · · , a s ) X i + s X j = e ( a , · · · , a s ) X j X i + s , ∀ j = i. (3.3) e ( a , · · · , a s ) X k X l = e ( a , · · · , a s ) X l X k , ∀ s + 1 ≤ k, l ≤ n. (3.4)For (3.1) we have the following calculation. e ( a , · · · , a s ) X i X i + s = α i e ( a , · · · , a i ∔ , · · · , a s ) X i + s = α i α − i α i + s ( q h i ) a i +1 − e ( a , · · · , a i ∔ ∔ ( − , · · · , a s )= q h i α − i α i + s ( q h i ) a i − α i e ( a , · · · , a i ∔ ( − ∔ , · · · , a s )= q h i α − i α i + s ( q h i ) a i − e ( a , · · · , a i ∔ ( − , · · · , a s ) X i = q h i e ( a , · · · , a s ) X i + s X i . The remaining (3.2)-(3.4) verification are similar. With this we have the following.
Theorem A.
The module M ( α ) is a simple A Q -module.Proof. Let P be a non-zero submodule of M ( α ). We claim that P contains a basisvector of the form e ( a , · · · , a s ). Indeed, any member p ∈ P is a finite K -linearcombination of such vectors. i.e., p := X λ k e ( a ( k )1 , · · · , a ( k ) s )for some λ k ∈ K . Suppose there exist two non-zero coefficients, say, λ u , λ v . We canchoose the smallest index r such that a ( u ) r = a ( v ) r . Now the vectors e ( a ( u )1 , · · · , a ( u ) s )and e ( a ( v )1 , · · · , a ( v ) s ) are eigenvectors of X r X r + s associated with the eigenvalues α r + s ( q h r ) a ( u ) r = µ u (say) and α r + s ( q h r ) a ( v ) r = µ v (say) respectively. We claim that SNEHASHIS MUKHERJEE AND SANU BERA µ u = µ v . Indeed, µ u = µ v = ⇒ q h r ( a ( u ) r − a ( v ) r ) = 1 = ⇒ g r | ( a ( u ) r − a ( v ) r ), whichis a contradiction. Now pX r X r + s − µ u p is an non zero element in P of smallerlength than p . Hence by induction it follows that every non zero submodule of M ( α ) contains a basis vector of the form e ( a , · · · , a s ). Thus M ( α ) is simple bythe actions of X i , ≤ i ≤ s . (cid:3) Main TheoremsTheorem B.
Let N be a simple K Q [ x , · · · , x n ] -module with N x i = 0 for all i .Then N is isomorphic to M ( α ) as A Q -module for some α ∈ ( K ∗ ) n .Proof. A given simple K Q [ x , · · · , x n ]-module N can be extended uniquely to asimple A Q -module by Proposition (2.2). Since each of the monomials X g i i , X i X i + s , X s +1 , X s +2 , · · · , X n , ∀ ≤ i ≤ s commutes and N is a finite dimensional vector space over K (see Corollary (2.1)),there is a common eigenvector v of X g i i , X i X i + s , X s +1 , X s +2 , · · · , X n , ∀ ≤ i ≤ s. Put vX g i i = ν i v, ∀ ≤ i ≤ svX i X i + s = α i + s v, ∀ ≤ i ≤ svX s + j = α s + j v, ∀ ≤ j ≤ n − s Let α i be a g i -th root of ν i for 1 ≤ i ≤ s . Since N x i = 0, all the α i ’s are nonzeroand so α ∈ ( K ∗ ) n . Also we can compute that for 0 ≤ i ≤ s and 0 ≤ a i ≤ g i − vX a s s · · · X a X i + s = α i + s ( q h i ) a i − vX a s s · · · X a i − i · · · X a , when a i > .ν − i α i + s ( q h i ) g i − vX a s s · · · X g i − i · · · X a , when a i = 0 . To have a homomorphism, we must have a map which takes X i X i + s -eigenvectorsof M ( α ) to X i X i + s -eigenvectors of N for the same eigenvalue. Define a lineartransformation φ : M ( α ) → N by φ ( e ( a , · · · , a s )) := α − a · · · α − a s s vX a s s · · · X a . IMPLE MODULES OVER THE QUANTUM AFFINE SPACE 7
To prove φ is an A Q -module homomorphism, it suffices to check that φ ( e ( a , · · · , a s ) X i ) = φ ( e ( a , · · · , a s )) X i , ∀ ≤ i ≤ s. (4.1) φ ( e ( a , · · · , a s ) X i + s ) = φ ( e ( a , · · · , a s )) X i + s , ∀ ≤ i ≤ s. (4.2) φ ( e ( a , · · · , a s ) X s + j ) = φ ( e ( a , · · · , a s )) X s + j , ∀ ≤ j ≤ n − s. (4.3)For (4.1) we have the following calculation. φ ( e ( a , · · · , a s ) X i ) = φ ( α i e ( a , · · · , a i ∔ , · · · , a s ))= α i φ ( e ( a , · · · , a i ∔ , · · · , a s ))= α i α − a · · · α − a i − i · · · α − a s s vX a s s · · · X a i +1 i · · · X a = α − a · · · α − a i i · · · α − a s s vX a s s · · · X a i +1 i · · · X a = α − a · · · α − a i i · · · α − a s s vX a s s · · · X a i i · · · X a X i = φ ( e ( a , · · · , a s )) X i For (4.2) we have the following calculation. If a i >
0, then φ ( e ( a , · · · , a s ) X i + s ) = φ ( α − i α i + s ( q h i ) a i − e ( a , · · · , a i ∔ ( − , · · · , a s ))= α − i α i + s ( q h i ) a i − φ ( e ( a , · · · , a i ∔ ( − , · · · , a s ))= α − i α i + s ( q h i ) a i − α − a · · · α − a i +1 i · · · α − a s s vX a s s · · · X a i +( − i · · · X a = α i + s ( q h i ) a i − α − a · · · α − a i i · · · α − a s s vX a s s · · · X a i +( − i · · · X a = φ ( e ( a , · · · , a s )) X i + s Similarly if a i = 0 the relation (4.2) also holds. For (4.3) we have the followingcalculation. φ ( e ( a , · · · , a s ) X s + j ) = φ ( α s + j e ( a , · · · , a s ))= α s + j φ ( e ( a , · · · , a s ))= α s + j α − a · · · α − a s s vX a s s · · · X a = α − a · · · α − a s s ( α s + j v ) X a s s · · · X a = α − a · · · α − a s s vX s + j X a s s · · · X a = α − a · · · α − a s s vX a s s · · · X a X s + j = φ ( e ( a , · · · , a s )) X s + j SNEHASHIS MUKHERJEE AND SANU BERA
Hence φ is an A Q -module homomorphism. Since M ( α ) and N are both simple A Q -module, by Schur’s lemma φ is an isomorphism. (cid:3) Theorem C.
There is a surjective map Ψ from K n onto the set of all simple K Q [ x , · · · , x n ] -modules such that dim K Ψ( α ) = s Q i =1 g i .Proof. Let N be a simple module over K Q [ x , · · · , x n ]. Let Z be the subalgebragenerated by x mi , i = 1 , · · · , n . Then Z is a finitely generated commutative K -subalgebra. Also ann( N ) ∩ Z is maximal ideal of Z by Proposition (2.1). Hencefor all i , x mi − λ i ∈ ann( N ) for some λ i ∈ K . If λ j = 0, then x j ∈ ann( N ) since x j is a normal element of K Q [ x , · · · , x n ] and ann( N ) is prime. Let us assume that λ i = 0 ∀ ≤ i ≤ p and λ i = 0 ∀ p < i ≤ n . So x i ∈ ann( N ) ∀ p < i ≤ n and K Q [ x , · · · , x n ] / h x p +1 , · · · , x n i is isomorphic to K Q [ x , · · · , x p ]. Therefore N becomes a simple K Q [ x , · · · , x p ]-module satisfying the hypothesis of Theorem B. Thus there exists α = ( α , · · · , α p ) ∈ ( K ∗ ) p such that N will be isomorphic to M ( α ) as A Q -module. (cid:3) Remark 4.1.
The map Ψ of Theorem C is not injective.Proof. Let α , β be two elements in ( K ∗ ) n such that α i = β i , i = 2 s and α s = q h s β s .Then define a map ψ : M ( α ) → M ( β ) by ψ ( e ( a , · · · , a s )) := e ( a , · · · , a s − , a s ∔ . It is easily verified that this map defines an A Q -module isomorphism. (cid:3) Remark 4.2.
All primitive ideals of K Q [ x , · · · , x n ] are annihilators of Ψ( α ) , α ∈ K n . Proof.
By definition, a primitive ideal in K Q [ x , · · · , x n ] is annihilator of a non zerosimple K Q [ x , · · · , x n ]-module. Hence the remark follows from Theorem C. (cid:3) Acknowledgement
We are very grateful to Dr. Ashish Gupta for setting up the problem, going throughthe paper and supervising it. We will also like to thank National Board of HigherMathematics, Department of Atomic Energy for funding our research.
IMPLE MODULES OVER THE QUANTUM AFFINE SPACE 9
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Snehashis Mukherjee And Sanu Bera School Of Mathematical Sciences,Ramakrishna Mission Vivekananda Educational and Research Institute (rkmveri),Belur Math, Howrah, Box: 711202, West Bengal, India.
E-mail address : [email protected] ; [email protected]; [email protected]