Cuspidal modules for the derivation Lie algebra over a rational quantum torus
aa r X i v : . [ m a t h . R T ] A p r Cuspidal modules for the derivation Lie algebraover a rational quantum torus
Chengkang Xu ∗ School of Mathematical Science, Shangrao Normal University, Shangrao, China
Abstract:
Let C Q denote a rational quantum torus with d variables, and Z be the centre of C Q . In this paper we give a explicit description of the structure of the cuspidal modules for thederivation Lie algebra D over C Q , with an extra associative Z -action. Keywords : derivation Lie algebra; cuspidal module; rational quantum torus; module of tensorfields.
In the past few decades, the representation theory of the derivation algebra over a torus hasattracted many mathematicians and physicists. Let d > A = C [ x ± , · · · , x ± d ] the commuting torus in variables x , · · · , x d , and by W d thederivation algebra over A . Irreducible modules with finite dimensional weight spaces over W d were classified in [BF1]. They are modules of tensor fields constructed independentlyby Shen[S] and Larsson[Lar], and modules of highest weight type given in [BB]. Forthe derivation algebra D over a rational quantum torus C Q , although the modules oftensor fields were constructed in [LT] and [LiuZ2] and the modules of highest weight typewere given in [Xu], we are still far from the complete classification of irreducible modules * Corresponding author. The author is supported by the National Natural Science Foundation ofChina(No. 11626157, 11801375). E-mail: [email protected]. D . However, some partial classifications areknown to us. For example, irreducible D -modules with certain C Q -action were classifiedin [RBS]. In [LiuZ2], Liu and Zhao classified irreducible cuspidal D -modules with an extraassociative action of the centre Z of C Q . These modules were constructed using a finitedimensional irreducible gl d -module V and a finite dimensional Z d /R -graded irreducible gl N -module W . Here N is a positive number closely related to the structure of C Q , and R is a subgroup of Z d corresponding to the center Z . To achieve their classification resultin [LiuZ2], Liu and Zhao used heavy computations(totally five pages of them). Moreover,the appearance of the algebra gl N in the proof seems a little farfetched, and it is not clearwhat is the relation between the gl d -module V and the Z d /R -graded gl N -module W .In this paper we study D -modules(not necessarily irreducible) with an extra associative Z -action. We will simply call such a module a ZD -module. Our main result is thatcuspidal ZD -modules with support lying in one coset α + Z d , α ∈ C d , are in a one-to-one correspondence with finite dimensional Γ-graded modules over a subalgebra G ofDer( AC ), where AC = C [ x , · · · , x d ] ⊕ C Q / Z . The algebra G has a quotient isomorphic to gl d ⊕ gl N . We will use some results about the solenoidal Lie algebras over A and C Q , whichwere given in [BF2] and [Xu2] separately. As a corollary, we reprove the classification ofirreducible cuspidal ZD -modules, in a more conceptional way. From our proof one cansee clearly that the algebras gl d , gl N and their modules appear naturally. We mentionthat Liu and Zhao classified irreducible cuspidal D -modules with no other restriction in[LiuZ] using the classification result in [LiuZ2] and a method introduced in [BF1].The present paper is arranged as follows. In Section 2 we recall some results aboutthe quantum torus C Q , the algebras D , the solenoidal Lie algebra W µ over A and thesolenoidal Lie algebra g µ over C Q . Section 3 is devoted to finite dimensional G -modules.In Section 4 we prove our main theorem about the cuspidal ZD -modules and classifyirreducible cuspidal ZD -modules once again.Throughout this paper, C , Z , N , Z + refer to the set of complex numbers, integers,nonnegative integers and positive integers respectively. For a Lie algebra G , we denoteby U ( G ) the universal enveloping algebra of G . Fix 1 < d ∈ Z + and a standard basis ǫ , · · · , ǫ d for the space C d . Denote by ( · | · ) the inner product on C d .2 Notations and Preliminaries
For a Lie algebra, a weight module is called a cuspidal module if all weight spaces areuniformly bounded. The support of a weight module is defined to be the set of allweights.For m = ( m , · · · , m d ) T ∈ Z d we denote by x m the monomial x m · · · x m d d in A . Thenthe derivation Lie algebra W d over A has a basis { x m x i ∂∂x i | m ∈ Z d , ≤ i ≤ d } subjectto the Lie bracket [ x m x i ∂∂x i , x n x j ∂∂x j ] = x m + n ( n i x j ∂∂x j − m j x i ∂∂x i ) . Let µ = ( µ , · · · , µ d ) T ∈ C d be generic, which means that µ , · · · , µ d are linearly in-dependent over the field of rational numbers. The solenoidal Lie algebra W µ over A isthe subalgebra of W d spanned by { x m P di =1 µ i x i ∂∂x i | m ∈ Z d } . Many others call W µ (centerless) higher rank Virasoro algebra.Let Q = ( q ij ) be a d × d complex matrix with all q ij being roots of unity and satisfying q ii = 1 , q ij q ji = 1 for all 1 ≤ i = j ≤ d. The rational quantum torus relative to Q is theunital associative algebra C Q = C [ t ± , · · · , t ± d ] with multiplication t i t j = q ij t j t i for all 1 ≤ i = j ≤ d. For m = ( m , · · · , m d ) T ∈ Z d we denote t m = t m · · · t m d d . For m , n ∈ Z d , set σ ( m , n ) = Y ≤ i
In this section we study cuspidal ZD -modules with support lying in one coset α + Z d forsome α ∈ C d . Fix M = L s ∈ Z d M α + s such a ZD -module, where M α + s is the weight spacewith weight α + s . Clearly we have t r M α + s ⊆ M α + r + s . Since M is a free Z -module, we7ay write M ∼ = M s U s ⊗ Z , where U s = M α + s for s ∈ Γ . Set U = L s ∈ Γ U s , which is a Γ-graded space.We should introduce some operators in U ( D ⋉ Z ) that act on U . Set D ( u , m ) = t − m ∂ ( u , m ) for u ∈ C d and m ∈ R , acts on each U s , hence on U . For r ∈ Γ , n ∈ R ,define the operator L ( n , r ) to be the restriction of t − n t n + r mapping U s to U r + s for each s ∈ Γ. The commutators among these operators are[ D ( u , m ) , D ( v , n )] = ( u | n )( D ( v , m + n ) − D ( v , n )) − ( v | m )( D ( u , m + n ) − D ( u , m ));[ D ( u , m ) , L ( n , s )] = ( u | n + s ) L ( m + n , s ) − ( u | n ) L ( n , s );[ L ( m , r ) , L ( n , s )] = ( σ ( r , s ) − σ ( s , r )) L ( m + n , r + s ) , (4.1)where u , v ∈ C d , m , n ∈ R, r , s ∈ Γ . Since for v ∈ U s , n ∈ R , we have ∂ ( u , m )( t n v ) = ( u | n ) t m + n v + t m + n D ( u , m ) v ; t m + r ( t n v ) = t n ( t m + r v ) = t m + n ( L ( m , r ) v ) , (4.2)the operators D ( u , m ) , L ( m , r ) completely determine the D -action on M . Furthermore,the operators D ( u , m ) may be restricted to U s , and L ( n , r ) may be considered as anoperator from U s to U r + s for each s ∈ Γ. In other words, D ( u , m ) are of degree and L ( n , r ) are of degree r in End( U ).Let µ , · · · , µ d be a basis of C d with all µ i being generic. Recall the solenoidalLie algebra g µ over C Q , the solenoidal Lie algebra W µ over the commuting torus A = C [ x ± , · · · , x ± d ]. Then the algebra D may be decomposed into direct sum of subalgebras D = g µ ⊕ W µ ⊕ · · · ⊕ W µ d . Here W µ i are solenoidal Lie algebras over the commuting torus Z . By results from [BF1]and [Xu2], we know that the operators D ( µ i , m ) , ≤ i ≤ d and L ( m , r ) , r ∈ Γ , havepolynomial dependence on m ∈ R with coefficients in End( U ). Moreover, the constantterm of D ( µ i , m ) on each U s is ( u | α + s ) Id . Hence D ( u , m ) are polynomials on m , sincethey are linear combinations of D ( µ i , m ). 8et D ( u , m ) = X p ∈ N d m p p ! f u ( p ); L ( m , r ) = X p ∈ N d m p p ! g r ( p ) , where f u ( p ) are operators in End( U ) of degree and g r ( p ) ∈ End( U ) are operators ofdegree r .Now by expanding the equations in (4.1) and comparing coefficients at both sides, weget the following commutators[ f u ( p ) , f v ( l )] = d P i =1 u i l i f v ( p + l − ǫ i ) − d P i =1 v i p i f u ( p + l − ǫ i ) , if p , l = ;0 , if p = or l = , [ f u ( p ) , g s ( l )] = ( u | s ) g s ( l ) , if p = ;( u | s ) g s ( p + l ) + d P i =1 u i l i g s ( p + l − ǫ i ) , if p = , [ g r ( p ) , g s ( l )] = ( σ ( r , s ) − σ ( s , r )) g r + s ( p + l ) , if r + s / ∈ R ;0 , if r + s ∈ R, (4.3)where u , v ∈ C d , p , l ∈ N d and r , s ∈ Γ . Recall the Lie bracket of the algebra G fromequation (3.1) and we see from equations in (4.3) that the operators { f u ( p ) , g s ( l ) | u ∈ C d , s ∈ Γ , p ∈ N d \{ } , l ∈ N d } yield a finite dimensional Γ-graded representation for G on U . Denote by ρ this represen-tation map. Then combining equation (4.2) we obtain Theorem 4.1.
There exists an equivalence between the category of finite dimensional Γ -graded G -modules and the category of cuspidal ZD -modules with support lying in somecoset α + Z d . This equivalence functor associates to a finite dimensional Γ -graded G -module U = L s ∈ Γ U s an ZD -module M = M s ∈ Z d U s ⊗ t s ith the D -action ∂ ( u , m )( v s ⊗ t s + n ) = ( u | α + n + s )Id + X p ∈ N d \{ } m p p ! ρ ( f u ( p )) v s ⊗ t s + n + m ; t m + r ( v s ⊗ t s + n ) = X p ∈ N d m p p ! ρ ( x p t r ) v s ⊗ t s + n + m + r , (4.4) where m , n ∈ R, r ∈ Γ \ R, s ∈ Γ and v s ∈ U s . As a consequence we classify irreducible cuspidal ZD -modules, which was first donein [LiuZ2]. First we recall modules of tensor fields over the algebra D . Let α ∈ C d andlet V be a finite dimensional gl d -module, W = L s ∈ Γ W s a finite dimensional Γ-graded gl N -module. On the tensor space N s ∈ Z d ( V ⊗ W s ⊗ t s ) there is a D -module structuredefined by ∂ ( u , m )( v ⊗ w ⊗ t s ) = ( u | α + s ) v + ( d X i,j =1 m i u j E ij ) v ! ⊗ w ⊗ t m + s ; t r ( v ⊗ w ⊗ t s ) = v ⊗ ( X r w ) ⊗ t r + s , for u ∈ C d , m ∈ R, r / ∈ R, s ∈ Z d , v ∈ V and w ∈ W s . We denote this module by V α ( V, W ), called a module of tensor fields for D . Clearly, V α ( V, W ) becomes a ZD -module provided the Z -action t n ( v ⊗ w ⊗ t s ) = v ⊗ w ⊗ t n + s for n ∈ R. Moreover, V α ( V, W ) is irreducible as a ZD -module as long as V, W are irreducible.
Theorem 4.2.