BGG category for the quantum Schrödinger algebra
aa r X i v : . [ m a t h . R T ] A p r BGG CATEGORY FOR THE QUANTUM SCHR ¨ODINGER ALGEBRA
GENQIANG LIU, YANG LI
Abstract.
In this paper, we study the BGG category O for the quantum Schr¨odingeralgebra U q ( s ), where q is a nonzero complex number which is not a root of unity. If thecentral charge ˙ z = 0, using the module B ˙ z over the quantum Weyl algebra H q , we show thatthere is an equivalence between the full subcategory O [ ˙ z ] consisting of modules with thecentral charge ˙ z and the BGG category O ( sl ) for the quantum group U q ( sl ). In the casethat ˙ z = 0, we study the subcategory A consisting of finite dimensional U q ( s )-modules oftype 1 with zero action of Z . Motivated by the ideas in [DLMZ, Mak], we directly constructan equivalent functor from A to the category of finite dimensional representations of aninfinite quiver with some quadratic relations. As a corollary, we show that the category offinite dimensional U q ( s )-modules is wild. Keywords:
BGG category, highest weight module, quiver, wild.1.
Introduction
In this paper, we denote by Z , Z + , N , C and C ∗ the sets of all integers, nonnegative integers,positive integers, complex numbers, and nonzero complex numbers, respectively. Let q be anonzero complex number which is not a root of unity. For n ∈ Z , denote [ n ] = q n − q − n q − q − .The BGG category O for complex semisimple Lie algebras was introduced by Joseph Bern-stein, Israel Gelfand and Sergei Gelfand in the early 1970s, see [BGG], which includes allhighest weight modules such as Verma modules and finite dimensional simple modules. Thiscategory is influential in many areas of representation theory. About the knowledge of O ,one can see the recent monograph [Hu] for details.The Schr¨odinger Lie algebra s is the semidirect product of sl and the three-dimensionalHeisenberg Lie algebra. This algebra can describe symmetries of the free particle Schr¨odingerequation, see [DDM1, Pe]. The representation theory of the Schr¨odinger algebra has beenstudied by many authors. A classification of the simple highest weight representations of theSchr¨odinger algebra were given in [DDM1]. All simple weight modules with finite dimen-sional weight spaces were classified in [D], see also [LMZ]. All simple weight modules of theSchr¨odinger algebra were classified in [BL2, BL3]. The BGG category O of s was studied in[DLMZ].In 1996, in order to research the q -deformed heat equations, a q -deformation of the universalenveloping algebra of the Schr¨odinger Lie algebra was introduced in [DDM2]. This algebrais called the quantum Schr¨odinger algebra. The quantum Schr¨odinger algebra U q ( s ) over C is generated by the elements E , F , K , K − , X , Y , Z with defining relations:(1.1) [ E, F ] = K − K − q − q − , KXK − = qX,KEK − = q E, KF K − = q − F,KY K − = q − Y, qY X − XY = Z,EX = qXE, EY = X + q − Y E,F X = Y K − + XF, F Y = Y F, where Z is central in U q ( s ). This definition is somewhat different from that of [DDM2] inform. This algebra is a kind of quantized symplectic oscillator algebra of rank one, see [GK].The paper [GK] gave the PBW Theorem and showed that the category O is a highest weightcategory. In the present paper, we will give specific characterizations for each block of O for U q ( s ) using some quivers.The subalgebra generated by E, F, K, K − is the quantum group U q ( sl ). The subalgebragenerated by X, Y, Z is called the quantum Weyl algebra H q , see [B]. All simple weightmodules over U q ( s ) with zero action of Z were classified in [BL1].The paper is organized as follows. In Section 2 we recall some basic facts about the category O for U q ( s ) . For a ˙ z ∈ C , we denote by O [ ˙ z ] the full subcategory of O consisting of allmodules which are annihilated by some power of the maximal ideal h Z − ˙ z i of C [ Z ]. InSection 3, in case of ˙ z = 0, using the modules B ˙ z over the quantum Weyl algebra H q , weshow that the functor − ⊗ ˜ B ˙ z gives an equivalence between O ( sl ) and O [ ˙ z ] , where O ( sl ) isthe BGG category of U q ( sl ), see Theorem 8. A Weight U q ( s )-module M is of type 1 if theSupp( M ) ⊂ q Z . In Section 4, we study the category A of finite dimensional U q ( s )-modulesof type 1 with zero action of Z . It is shown that there is an equivalence between A andthe category of finite dimensional representations of an infinite quiver with some quadraticrelations, see Theorem 17. In [DLMZ], the grading technique was used in the study of finitedimensional modules for the Schr¨odinger Lie algebra.2. Basic properties of the category O The definition of Category O . Let U q ( n + ) be the subalgebra of U q ( s ) generated bythe elements E, X and let U q ( n − ) be the subalgebra generated by F, Y . Moreover let U q ( h )be the subalgebra generated by the elements K, K − , Z . We write ⊗ for ⊗ C .Then we have the following triangular decomposition:(2.1) U q ( s ) = U q ( n − ) ⊗ U q ( h ) ⊗ U q ( n + ) . A U q ( s )-module V is called a weight module if K acts diagonally on V , i.e., V = ⊕ λ ∈ C ∗ V λ , where V λ = { v ∈ V | Kv = λv } . For a weight module V , letsupp( V ) = { λ ∈ C ∗ | V λ = 0 } . Next, we introduce the category O for U q ( s ). GG CATEGORY FOR THE QUANTUM SCHR ¨ODINGER ALGEBRA 3
Definition 1.
A left module M over U q ( s ) is said to belong to category O if(1) M is finitely generated over U q ( s ) ;(2) M is a weight module;(3) The action of U q ( n + ) on M is locally finite, i.e., dim U q ( n + ) v < ∞ for any v ∈ M . For a weight module V , a weight vector v λ ∈ V λ is called a highest weight vector if Ev λ = Xv λ = 0. A module M is called a highest weight module of highest weight λ if there exists ahighest weight vector v λ in M which generates M . For λ ∈ C ∗ and ˙ z ∈ C , let ∆( λ, ˙ z ) be theVerma module generated by v λ , where Kv λ = λv λ , Zv λ = ˙ zv λ . Then { Y k F l v λ | k, l ∈ Z + } isa basis of ∆( λ, ˙ z ). Let R ( λ, ˙ z ) be the largest proper submodule of ∆( λ, ˙ z ). Hence L ( λ, ˙ z ) =∆( λ, ˙ z ) /R ( λ, ˙ z ) is the unique simple quotient module of ∆( λ, z ).2.2. Basic properties of O . By the similar arguments as those in [Hu], we can see thatevery module M in O has the following standard properties. Lemma 2.
The category O is closed with respect to taking submodules, quotient modulesand finite direct sums. That is, the category O is an abelian category. Lemma 3.
Let M be any module in O . ( ) The module M has a finite filtration M ⊂ M ⊂ · · · ⊂ M n = M such that each subquotient M j /M j − for j n is a highest weight module. ( ) Each weight space of M is finite dimensional. ( ) Any simple module in O is isomorphic to some L ( λ, ˙ z ) , for λ, ˙ z ∈ C . Blocks of nonzero central charge
In this section, we assume that ˙ z = 0. We denote by O [ ˙ z ] the full subcategory of O consistingof all modules which are annihilated by some power of the maximal ideal h Z − ˙ z i of C [ Z ].Let O ( sl ) denote the BGG category for U q ( sl ). We will show that there is an equivalencebetween O ( sl ) and O [ ˙ z ] . Firstly, we found that the structure of Verma modules over U q ( s )is similar as that of Verma modules over U q ( sl ).3.1. The tensor product realizations of highest modules.
In this subsection, we willgive tensor product realizations of Verma modules using Verma modules over U q ( sl ) and H q . This construction is crucial to the study of the category O [ ˙ z ] .For a nonzero ˙ z ∈ C , let B ˙ z := H q / ( H q ( Z − ˙ z ) + H q X ) GENQIANG LIU, YANG LI which is a simple H q -module. Denote the image of Y i in B ˙ z by v i for i ∈ Z + . We can seethat(3.1) Xv i = − ˙ z ( q i − q − v i − , i ∈ Z + . Define the action of U q ( sl ) on B ˙ z by(3.2) Kv i = q − − i v i ,F v i = q ˙ z ( q + 1) v i +2 ,Ev i = − ˙ z ( q i − q i − − q i − ( q − q − v i − . Then we can check that ( EF − F E ) v i = K − K − q − q − v i ,EY v i = Xv i + q − Y Ev i ,F Xv i = Y K − v i + XF v i . Thus the action (3.2) indeed makes B ˙ z to be a module over U q ( s ). We denote this U q ( s )-module by e B ˙ z . In fact, e B ˙ z ∼ = L ( q − , ˙ z ).We can make a U q ( sl )-module N to be a U q ( s )-module be defining H q N = 0. We denotethe resulting U q ( s )-module by e N . Lemma 4.
The following map (3.3) ∆ : U q ( s ) → U q ( s ) ⊗ U q ( s ) defined by ∆( E ) = 1 ⊗ E + E ⊗ K, ∆( F ) = K − ⊗ F + F ⊗ K ) = K ⊗ K, ∆( K − ) = K − ⊗ K − , ∆( X ) = 1 ⊗ X, ∆( Y ) = 1 ⊗ Y. ∆( Z ) = 1 ⊗ Z can define an algebra homomorphism. Remark:
There is no algebra homomorphism ǫ : U q ( s ) → C such that the following diagramcommutes: U q ( s ) id (cid:15) (cid:15) ∆ / / U q ( s ) ⊗ U q ( s ) ⊗ ǫ (cid:15) (cid:15) U q ( s ) can / / U q ( s ) ⊗ C So we can not define a bialgebra structure on U q ( s ) from ∆.Via the map ∆, the space e N ⊗ e B ˙ z can be defined as a U q ( s )-module for any U q ( sl )-module N . More precisely, for u ∈ U q ( s ), if ∆( u ) = Σ i u i ⊗ w i , then the action of U q ( s ) on e N ⊗ e B ˙ z GG CATEGORY FOR THE QUANTUM SCHR ¨ODINGER ALGEBRA 5 is defined by u ( n ⊗ b ) = Σ i u i n ⊗ w i b, n ∈ N, b ∈ e B ˙ z . Let ∆ sl ( λ ) be the Verma module over U q ( sl ) with the highest weight λ whose unique simplequotient module is L sl ( λ ). It is well known that the module ∆ sl ( λ ) is reducible if and onlyif λ ∈ q Z + , see [J]. For each d ∈ Z + , we have non-split short exact sequences0 → ∆ sl ( q − d − ) → ∆ sl ( q d ) → L sl ( q d ) → , and 0 → ∆ sl ( − q − d − ) → ∆ sl ( − q d ) → L sl ( − q d ) → . The structure of ∆( λ, ˙ z ) was determined in [DDM2]. The following proposition give a con-structive proof. Proposition 5. [CCL]
The following results hold.(1) If ˙ z = 0 , then ∆( λ, ˙ z ) ∼ = e ∆ sl ( λq ) ⊗ e B ˙ z . Therefore the Verma module ∆( λ, ˙ z ) isreducible if and only if λ ∈ q Z + − . Moreover, L ( λ, z ) ∼ = e L sl ( λq ) ⊗ e B ˙ z .(2) For each d ∈ Z + , we have a non-split short exact sequences → ∆( q − d − , ˙ z ) → ∆( q d − , ˙ z ) → L ( q d − , ˙ z ) → , and → ∆( − q − d − , ˙ z ) → ∆( − q d − , ˙ z ) → L ( − q d − , ˙ z ) → . Equivalence between O ( sl ) and O [ ˙ z ] .Lemma 6. If ˙ z = 0 , then any module in O [ ˙ z ] has finite composition length.Proof. According to (1) in Lemma 3, any nonzero module M in O [ ˙ z ] has a finite filtrationwith sub-quotients given by highest weight modules. Hence it suffices to treat the case that M is the Verma module ∆( λ, ˙ z ). By Proposition 5, ∆( λ, ˙ z ) has finite composition length if˙ z = 0. (cid:3) Proposition 7.
Suppose that V is a module in O [ ˙ z ] with nonzero central charge ˙ z . Then V ∼ = e N ⊗ e B ˙ z for some U q ( sl ) -module N .Proof. By Lemma 6, V has a finite composition length l. We will proceed the proof byinduction on l . Firstly, we consider the case l = 1. The fact that V ∈ O forces that V isa simple highest weight U q ( s )-module. Then V is a simple quotient module of some Vermamodule ∆( λ, z ) . Note that ∆( λ, z ) ∼ = e ∆ sl ( λq ) ⊗ e B ˙ z . Thus V ∼ = e L sl ( λq ) ⊗ e B ˙ z .Next, we consider the general case. Let M be a maximal submodule of V . By the inductionhypothesis, we see that M ∼ = f N ⊗ e B ˙ z , V /M ∼ = f N ⊗ e B ˙ z , where N , N are U q ( sl )-modules. GENQIANG LIU, YANG LI
As vector spaces, we can assume that V = N ⊗ B ˙ z , where N is a vector space such that N ⊆ N and N/N ∼ = N . Moreover, u ( w ⊗ v ) = w ⊗ uv, K ( w ⊗ v ) = ( Kw ) ⊗ ( Kv ), for u ∈ H q , w ∈ N, v ∈ B ˙ z .For w ∈ N , we can find w i ∈ N, i k such that E ( w ⊗ v ) = P ki =0 w i ⊗ v i , where v i isthe image of Y i in B ˙ z . From X k E = q − k EX k , Xv = 0, we have X k E ( w ⊗ v ) = w k ⊗ X k v k = q − k ( w ⊗ X k v ) = 0 . By (3.1), when k > X k v k = 0 . So we must have that k = 0. Denote w E = q w . Then E ( w ⊗ v ) = w E ⊗ Kv .From Ev i = − ˙ z ( q i − q i − − q i − ( q − q − v i − and(3.4) EY i = q − i Y i E + [ i ] Y i − X − ( q i − q i − − q i − ( q − q − ZY i − , we obtain that(3.5) E ( w ⊗ v i ) = EY i ( w ⊗ v )= (cid:16) q − i Y i E + [ i ] Y i − X − ( q i − q i − − q i − ( q − q − ZY i − (cid:17) ( w ⊗ v )= w E ⊗ Kv i + w ⊗ Ev i . By the action of F on N/N ⊗ e B ˙ z , there exist w F ∈ N, w ′ i ∈ N , 1 i k satisfying F ( w ⊗ v ) = w F ⊗ v + K − w ⊗ F v + k X i =1 w ′ i ⊗ v i , where P ki =1 w ′ i ⊗ v i ∈ N ⊗ e B ˙ z .From 0 = F ( w ⊗ Xv ) = F X ( w ⊗ v )= XF ( w ⊗ v ) + Y K − ( w ⊗ v )= K − w ⊗ XF v + k X i =1 w ′ i ⊗ Xv i + q ( K − w ) ⊗ v = − q ( K − w ) ⊗ v − ˙ z ( q i − q − k X i =1 w ′ i ⊗ v i − + q ( K − w ) ⊗ v = − ˙ z ( q i − q − k X i =1 w ′ i ⊗ v i − , we have w ′ i = 0, for any 1 i k. Then F ( w ⊗ v ) = w F ⊗ v + K − w ⊗ F v . GG CATEGORY FOR THE QUANTUM SCHR ¨ODINGER ALGEBRA 7
Consequently, from
Y F = F Y and Y i v = v i , we have(3.6) F ( w ⊗ v i ) = w F ⊗ v i + K − w ⊗ F v i , i ∈ Z + . We can define the action of U q ( sl ) on N as follows: E · w = w E , F · w = w F , w ∈ N. From (3.5) and (3.6), we can see that V ∼ = e N ⊗ e B ˙ z . The proof is complete. (cid:3) By Lemma 7, we have the following category equivalence.
Theorem 8. If ˙ z = 0 , using the algebra homomorphism ∆ in (3.3), we can define a functor − ⊗ ˜ B ˙ z : O ( sl ) → O [ ˙ z ] . Moreover, this functor is an equivalence of categories.Proof.
By the definition of category O , the functor F := − ⊗ ˜ B ˙ z maps modules in O ( sl ) tomodules in O [ ˙ z ]. By Lemma 7, the functor F is essentially surjective.Next, we will show that for any M, N ∈ Obj( O ( sl ) ), the map F M,N : Hom O ( sl ( M, N ) → Hom O [ ˙ z ] ( F ( M ) , F ( N ))is a bijection.For f ∈ Hom O ( sl ( M, N ) such that F M,N ( f ) = f ⊗ f = 0. So F M,N isinjective.For any g ∈ Hom O [ ˙ z ] ( F ( M ) , F ( N )), suppose that g ( w ⊗ v ) = k X i =0 w i ⊗ v i , w ∈ M, w k ∈ N, where v i is the image of Y i in B ˙ z . Since ˜ B ˙ z is a simple H q -module, by the density theorem,there exists u ∈ H q such that uv = v , uv i = 0 , i = 1 , . . . , k. Form ug ( w ⊗ v ) = g ( u ( w ⊗ v )) = g ( w ⊗ uv ) , we have: u ( k X i =0 w k ⊗ v k ) = k X i =0 w k ⊗ uv k = w ⊗ v = g ( w ⊗ v ) . Define the map f : M → N such that f ( w ) = w , i.e., g ( w ⊗ v ) = f ( w ) ⊗ v . From Y i g ( w ⊗ v ) = g ( w ⊗ Y i v ), we have g ( w ⊗ v i ) = f ( w ) ⊗ v i , ∀ i ∈ Z + . By Eg ( w ⊗ v ) = g ( E ( w ⊗ v )) and Ev = 0, we have that Ef ( w ) ⊗ Kv = f ( w ) ⊗ Ev + f ( Ew ) ⊗ Kv = f ( Ew ) ⊗ Kv , GENQIANG LIU, YANG LI i.e., Ef ( w ) = f ( Ew ). Similarly, we can check that F f ( w ) = f ( F w ), Kf ( w ) = f ( Kw ).Then f ∈ Hom O ( sl ( M, N ). So F M,N ( f ) = g , F M,N is surjective.Therefore F M,N is bijective, and hence F := − ⊗ ˜ B ˙ z is an equivalence. (cid:3) The description of O [ ξ, ˙ c, ˙ z ] . Let C = F E + qK + q − K − ( q − q − ) be the Casimir element of U q ( sl ). Define the following element in U q ( s ):˜ C = Z (cid:0) (1 + q − ) C + 1 q − K − (cid:1) + X F − Y EK − + XY ( q − F E − qEF ) . In [CCL], the following lemma was proved.
Lemma 9.
The element ˜ C belongs to the center of U q ( s ) . For a module M in O , from that the weight spaces of M are finite dimensional, we can seethat the action of C [ ˜ C, Z ] on M is locally finite.For λ ∈ C , ˙ z ∈ C , let v λ be a highest vector of the U q ( s )-module L ( λ, ˙ z ) . We denote by˜ c λ the scalar corresponding to the action of the central element ˜ C on v λ . Similarly, for the U q ( sl )-module L sl ( λ ), we denote scalar corresponding to the action of C by c λ . Lemma 10.
We have the following:(1) ˜ c λ = ˙ z ( q − q − ) (cid:16) ( q + q ) λ + ( q − + q − ) λ − (cid:17) .(2) ˜ c λ = ˜ c λq − k iff λ = q k − . Let ξ ∈ C ∗ /q Z . We denote by O [ ξ ] the full subcategory of O consisting of all M such thatsupp( M ) ⊂ ξ . For ˙ z, ˙ c ∈ C , we denote by O [ ξ, ˙ c, ˙ z ] the full subcategory of O [ ξ ] consistingof all M such that M is annihilated by some power of the maximal ideal h ˜ C − ˙ c, Z − ˙ z i of C [ Z, ˜ C ]. Since the action of C [ ˜ C, Z ] on M is locally finite, we have O [ ξ ] ∼ = M ˙ c, ˙ z ∈ C O [ ξ, ˙ c, ˙ z ] . Similarly, we can define the subcategory O ( sl ) [ ξ, ˙ c ] of O ( sl ) , where ˙ c is defined by the actionof Casimir element C of U q ( sl ).Using the equivalence between O ( sl ) and O [ ˙ z ] given in Theorem 8, we have the followingequivalence. Lemma 11.
The restriction functor − ⊗ ˜ B ˙ z : O ( sl ) [ ξ, c λ ] → O [ q − ξ, ˜ c λq − , ˙ z ] is an equivalence of categories, where λ ∈ ξ, c λ = qλ + q − λ − ( q − q − ) . GG CATEGORY FOR THE QUANTUM SCHR ¨ODINGER ALGEBRA 9
Using the structure of O ( sl ) (see Section 5.3 in [Maz]), we give descriptions of each block O [ ξ, ˙ c, ˙ z ] as follows. Proposition 12.
Let ξ = q + Z , ˙ z ∈ C ∗ . Then the following claims hold. ( ) The module ∆( q − , ˙ z ) is the unique simple object in O [ q + Z , ˜ c q − , ˙ z ] . Moreover, the block O [ q + Z , ˜ c q − , ˙ z ] is semisimple. ( ) For n ∈ Z + , ∆( q − n − , ˙ z ) , L ( q n − , ˙ z ) are all the simple objects in O [ q + Z , ˜ c q n − , ˙ z ] . ( ) For n ∈ Z + , the subcategory O [ q + Z , ˜ c q n − , ˙ z ] is equivalent to the category of finite di-mensional representations over C of the following quiver with relations: • a * * • b j j ab = 0 . Proposition 13.
Let ξ = q Z , ˙ z ∈ C ∗ . Then we have the following: ( ) For any n ∈ Z , the Verma module ∆( q n , ˙ z ) is simple. ( ) The modules ∆( q n , ˙ z ) , ∆( q n +3 , ˙ z ) are simple objects in O [ q Z , ˜ c q n , ˙ z ] . ( ) The block O [ q Z , ˜ c q n , ˙ z ] is equivalent to C ⊕ C -mod for any n ∈ Z . Proposition 14.
For ξ ∈ C ∗ /q Z , ˙ z ∈ C ∗ , λ ∈ ξ . If ξ = ± q Z , ± q + Z , then the block O [ ξ, ˜ c λ , ˙ z ] is semisimple with the unique simple object ∆( λ, ˙ z ) . Finite dimensional U q ( s ) -modules In this section, we study finite dimensional U q ( s )-modules. A Weight U q ( s )-module M is oftype 1 if the Supp( M ) ⊂ q Z . Note that L ( λ, ˙ z ) is infinite dimensional when ˙ z = 0. So Z actstrivially on any finite dimensional simple U q ( s )-module. Consequently Z acts nilpotentlyon any finite dimensional U q ( s )-module. Let A denote the category of finite dimensional U q ( s )-modules of type 1 with zero action of Z . Thus any module in A is a module over thesmash product algebra A := C q [ X, Y ] ⋊ U q ( sl ), where C q [ X, Y ] = C h X, Y | XY = qY X i is the quantum plane. The quantum plane C q [ X, Y ] is a U q ( sl )-module on the followingaction.(4.1) K · X = qX E · X = 0 F · X = Y,K · Y = q − Y E · Y = X F · Y = 0 . We will use the completely reducibility of finite dimensional U q ( sl )-modules and the Clebsch-Gordon rule to discuss the category A . For convenience, let L ( i ) denote the finite dimen-sional U q ( sl )-module with the highest weight q i , i ∈ Z + . In fact, we can assume that L (1) = C X + C Y , whose U q ( sl )-module structure was defined by (4.1). It is well knownthat the tensor product L ( m ) ⊗ L ( n ) is a U q ( sl )-module under the action defined by the following co-multiplication:∆ ′ ( E ) = E ⊗ K ⊗ E, ∆ ′ ( F ) = F ⊗ K − + 1 ⊗ F, ∆ ′ ( K ) = K ⊗ K, ∆ ′ ( K − ) = K − ⊗ K − . Remark:
The above co-multiplication ∆ ′ is different from ∆ in (3.3). Now ∆ ′ can guaranteethat τ ( θ i ) defined in Lemma 16 is a U q ( sl )-module homomorphism, however ∆ in (3.3) cannot. Because we consider the left action of L (1) on L ( i ) in Lemma 16 .Next, we will introduce two lemmas which are used in the proof of Theorem 17. Lemma 15. ( ) L (1) ⊗ L ( i ) ∼ = L ( i + 1) ⊕ L ( i − , for i > ; ( ) Suppose that v i is a highest weight vector of L ( i ) . Then X ⊗ v i is a highest weight vectorof L (1) ⊗ L ( i ) whose highest weight is q i +1 , and [ i ] Y ⊗ v i − q − X ⊗ F v i is a highest weightvector of L (1) ⊗ L ( i ) whose highest weight is q i − ; ( ) In L (1) ⊗ L (1) ⊗ L ( i ) , the elements [ i + 1] Y ⊗ X ⊗ v i − q − X ⊗ X ⊗ F v i − q − i − X ⊗ Y ⊗ v i and [ i ] X ⊗ Y ⊗ v i − q − X ⊗ X ⊗ F v i are highest weight vectors with the highest weight q i .Proof. (1) follows from the Clebsch-Gordon rule [J]: L ( m ) ⊗ L ( n ) ∼ = L ( m + n ) ⊕ L ( m + n − ⊕ · · · ⊕ L ( m − n ) , m > n. (2) We can check that E ( X ⊗ v i ) = ( E · X ) ⊗ v i + ( K · X ) ⊗ Ev i = 0 ,E ([ i ] Y ⊗ v i − q − X ⊗ F v i )= [ i ]( E · Y ) ⊗ v i + [ i ]( K · Y ) ⊗ Ev i − q − ( E · X ) ⊗ F v i − q − ( K · X ) ⊗ EF v i = [ i ] X ⊗ v i − X ⊗ EF v i = 0 . Then (2) holds.(3) follows from (1) and (2). (cid:3)
Lemma 16.
For any module V ∈ A , θ i ∈ Hom U q ( sl ) ( L ( i ) , V ) , the following map τ ( θ i ) : L (1) ⊗ L ( i ) −→ V ( aX + bY ) ⊗ v ( aX + bY ) θ i ( v ) a U q ( sl ) -module homomorphism, where a, b ∈ C , v ∈ L ( i ) . GG CATEGORY FOR THE QUANTUM SCHR ¨ODINGER ALGEBRA 11
Proof.
For a, b ∈ C , v ∈ L ( i ), we can check that τ ( θ i ) (cid:16) K (cid:0) ( aX + bY ) ⊗ v (cid:1)(cid:17) = τ ( θ i ) (cid:16) ( aK · X + bK · Y ) ⊗ Kv (cid:17) = aqXKv + bq − Y Kv = aKXv + bKY v = Kτ ( θ i ) (cid:16) ( aX + bY ) ⊗ v (cid:17) ,τ ( θ i ) (cid:16) E (cid:0) ( aX + bY ) ⊗ v (cid:1)(cid:17) = τ ( θ i ) (cid:16) ( aK · X + bK · Y ) ⊗ Ev (cid:17) + τ ( θ i ) (cid:16) ( aE · X + bE · Y ) ⊗ v (cid:17) = τ ( θ i ) (cid:16) ( aqX + bq − Y ) ⊗ Ev (cid:17) + τ ( θ i ) (cid:16) bX ⊗ v (cid:17) = aqXEv + bq − Y Ev + bXv = aEXv + bEY v = Eτ ( θ i ) (cid:16) ( aX + bY ) ⊗ v (cid:17) , .τ ( θ i ) (cid:16) F (cid:0) ( aX + bY ) ⊗ v (cid:1)(cid:17) = τ ( θ i ) (cid:16) ( aF · X + bF · Y ) ⊗ K − v (cid:17) + τ ( θ i ) (cid:16) ( aX + bY ) ⊗ F v (cid:17) = τ ( θ i ) (cid:16) aY ⊗ K − v (cid:17) + τ ( θ i ) (cid:16) ( aX + bY ) ⊗ F v (cid:17) = aY K − v + aXF v + bY F v = aF Xv + bF Y v = F τ ( θ i ) (cid:16) ( aX + bY ) ⊗ v (cid:17) . . So τ ( θ i ) is a U q ( sl )-module homomorphism. (cid:3) Consider the following quiver. Q ∞ : a * * b j j a * * b j j a + + . . . b k k The following theorem is inspired by the ideas in [DLMZ, Mak].
Theorem 17.
The category A is equivalent to the category B of finite dimensional repre-sentations for the quiver Q ∞ satisfying the following condition: (4.2) b a = 0 , a i b i = b i +1 a i +1 , i ∈ Z + . Proof.
By Lemma 15, we can define U q ( sl )-module homomorphisms t i +1 : L ( i + 1) → L (1) ⊗ L ( i ) , t ′ i − : L ( i − → L (1) ⊗ L ( i )such that t i +1 ( v i +1 ) = X ⊗ v i , , t ′ i − ( v i − ) = [ i ] Y ⊗ v i − q − X ⊗ F v i , where each v i is a fixed highest weight vector of L ( i ).We will prove the theorem in three steps. Step 1.
We define a functor F from A to B . Let V be a U q ( s )-module which belongs to A .(1) For every i , we can associate it with a vector space V i := Hom U q ( sl ) ( L ( i ) , V ).(2) For arrows a i , b i , we can define linear maps V ( b i ) : V i +1 → V i , V ( a i ) : V i → V i +1 asfollows: V ( b i )( θ i +1 ) = τ ( θ i +1 ) t ′ i , V ( a i )( θ i ) = τ ( θ i ) t i +1 . Next we check that: V ( b ) V ( a ) = 0 , V ( a i − ) V ( b i − ) = V ( b i ) V ( a i ) , i ∈ N . For θ ∈ V , v ∈ L (0), from F v = 0, we have (cid:16) V ( b ) V ( a )( θ ) (cid:17) ( v ) = τ (cid:0) τ ( θ ) (cid:1) ( Y ⊗ X ⊗ v − q − X ⊗ X ⊗ F v − q − X ⊗ Y ⊗ v )= τ (cid:0) τ ( θ ) (cid:1) ( Y ⊗ X ⊗ v − q − X ⊗ Y ⊗ v )= Y Xθ ( v ) − q − XY θ ( v ) = 0 . For θ i ∈ V i , from XY = qY X, [ i + 1] q − − q − i − = [ i ], we have (cid:16) V ( a i − ) V ( b i − )( θ i ) (cid:17) ( v i )= τ (cid:0) τ ( θ i ) (cid:1)(cid:0) [ i ] X ⊗ Y ⊗ v i − q − X ⊗ X ⊗ F v i (cid:1) = [ i ] XY θ i ( v i ) − q − XXF θ i ( v i )= [ i + 1] Y Xθ i ( v i ) − q − XXF θ i ( v i ) − q − i − XY θ i ( v i )= τ (cid:0) τ ( θ i ) (cid:1)(cid:0) [ i + 1] Y ⊗ X ⊗ v i − q − X ⊗ X ⊗ F v i − q − i − X ⊗ Y ⊗ v i (cid:1) = (cid:16) V ( b i ) V ( a i )( θ i ) (cid:17) ( v i ) . By the fact that the U q ( sl )-module L ( i ) is generated by v i and V ( b i ) V ( a i )( θ i ), V ( a i − ) V ( b i − )( θ i )are U q ( sl )-module homomorphisms, we see that V ( b i ) V ( a i )( θ i ) = V ( a i − ) V ( b i − )( θ i ).Thus ( V i , V ( a i ) , V ( b i ) , i ∈ Z + ) is a representation of Q ∞ satisfying the relation (4.2).(3) We define a functor F from A to B . For V, W ∈ Obj( A ) , f ∈ Hom U q ( s ) ( V, W ), define F ( V ) = ( V i , V ( a i ) , V ( b i ) , i ∈ Z + ) ,F ( f ) = ( f ∗ i : V i → W i , i ∈ Z + ) , where f ∗ i satisfies f ∗ i ( θ i ) = f θ i , θ i ∈ V i . GG CATEGORY FOR THE QUANTUM SCHR ¨ODINGER ALGEBRA 13
We check the following diagram V i − f ∗ i − (cid:15) (cid:15) V if ∗ i (cid:15) (cid:15) V ( b i − ) o o W i − W i , W ( b i − ) o o V if ∗ i (cid:15) (cid:15) V ( a i ) / / V i +1 f ∗ i +1 (cid:15) (cid:15) W i W ( a i ) / / W i +1 is commutative.Since τ ( θ i ) is a U q ( sl )-module homomorphism, W ( b i − ) f ∗ i ( θ i ) = W ( b i − )( f θ i )= τ ( f θ i ) t ′ i − = f τ ( θ i ) t ′ i − = f ∗ i − V ( b i − )( θ i ) , and W ( a i ) f ∗ i ( θ i ) = W ( a i )( f θ i )= τ ( f θ i ) t i +1 = f τ ( θ i ) t i +1 = f ∗ i +1 V ( a i )( θ i ) . So f ∗ i V ( b i ) = W ( b i ) f ∗ i +1 , f ∗ i +1 V ( a i ) = W ( a i ) f ∗ i . Therefore, F ( f ) = ( f ∗ i : V i → W i , i ∈ Z + ) is a morphism from the representation F ( V ) to F ( W ). Step 2.
For a representation ( V i , V ( a i ) , V ( b i ) , i ∈ Z + ) of the quiver Q ∞ satisfying therelation (4.2), there is a U q ( s )-module V such that F ( V ) ∼ = ( V i , V ( a i ) , V ( b i ) , i ∈ Z + ).Let V = L i ∈ Z + V i ⊗ L ( i ). Next we define the action of U q ( s ) on V . Since v i is a highestweight vector of L ( i ), F s v i , s ∈ Z + is a basis of L ( i ). For u ∈ U q ( sl ) , θ i ∈ V i , s ∈ Z + , define(4.3) u ( θ i ⊗ F s v i ) = θ i ⊗ uF s v i ,X ( θ i ⊗ F s v i ) = q s [ i + 1 − s ][ i + 1] V ( a i )( θ i ) ⊗ F s v i +1 − q s − i [ s ][ i + 1] V ( b i − )( θ i ) ⊗ F s − v i − ,Y ( θ i ⊗ F s v i ) = q [ i + 1] V ( b i − )( θ i ) ⊗ F s v i − + 1[ i + 1] V ( a i )( θ i ) ⊗ F s +1 v i +1 . The reason for defining the action of U q ( s ) on V by (4.3) coming from the definitions of V ( a i ) , V ( b i ) in Step 1 . We can check that the action (4.3) indeed defines a U q ( s )-modulethrough a little cumbersome calculation. For the verification process, one can see the ap-pendix of the present paper.From the definition of F , we can see that F ( V ) ∼ = ( V i , V ( a i ) , V ( b i ) , i ∈ Z + ). Step 3.
The functor F is completely faithful, i.e., the map F V,W : Hom A ( V, W ) → Hom B ( F ( V ) , F ( W ))is a bijection. If f ∈ Hom U q ( s ) ( V, W ) satisfying that F V,W ( f ) = 0, then for any θ i ∈ Hom U q ( sl ) ( L ( i ) , V ), i ∈ Z + , we have: f θ i = 0. Since V is a sum of simple submodules L ( i ). So f = 0, and F V,W is injective.From the completely reducibility of finite dimensional U q ( sl )-modules, V = X i ∈ Z + X θ ∈ V i θ i ( L ( i )) . For g = ( g i : V i → W i , i ∈ Z + ) ∈ Hom B ( F ( V ) , F ( W )), we define f : V → W as follows: f (cid:0) θ i ( w i ) (cid:1) = g i ( θ i )( w i ) , θ i ∈ V i , w i ∈ L ( i ) . Since θ i , g i ( θ i ) is a U q ( sl )-module homomorphism, for u ∈ U q ( sl ), we have f (cid:0) u ( θ i ( w i )) (cid:1) = f (cid:0) θ i ( uw i ) (cid:1) = g i ( θ i )( uw i ) = u ( g i ( θ i ))( w i ) = uf (cid:0) θ i ( w i ) (cid:1) . At the same time, using the following commutative diagram: V i − g i − (cid:15) (cid:15) V ig i (cid:15) (cid:15) V ( b i − ) o o W i − W iW ( b i − ) o o , V ig i (cid:15) (cid:15) V ( a i ) / / V i +1 g i +1 (cid:15) (cid:15) W i W ( a i ) / / W i +1 we obtain that f (cid:0) Xθ i ( v i ) (cid:1) = f (cid:16) ( V ( a i )( θ i ))( v i +1 ) (cid:17) = g i +1 V ( a i )( θ i )( v i +1 )= W ( a i ) g i ( θ i )( v i +1 ) = Xf (cid:0) θ i ( v i ) (cid:1) and f (cid:0) Y θ i ( v i ) (cid:1) = f (cid:0) q [ i + 1] V ( b i − )( θ i )( v i − ) + 1[ i + 1] V ( a i )( θ i )( F v i +1 ) (cid:1) = q [ i + 1] g i − V ( b i − )( θ i )( v i − ) + 1[ i + 1] g i +1 V ( a i )( θ i )( F v i +1 )= q [ i + 1] W ( b i − ) g i ( θ i )( v i − ) + 1[ i + 1] W ( a i ) g i ( θ i )( F v i +1 )= Y g i ( θ i )( v i ) = Y f (cid:0) θ i ( v i ) (cid:1) . So f is a U q ( s )-module homomorphism such that F ( f ) = g .Therefore, F is an equivalence. (cid:3) Let C h x , x i be the free associative algebra over C generated by two variables x , x . Recallthat an abelian category C is wild if there exists an exact functor from the category ofrepresentations of the algebra C h x , x i to C which preserves indecomposability and takesnonisomorphic modules to nonisomorphic ones, see Definition 2 in [Mak].By [DLMZ], the category B for the quiver Q ∞ is wild. Hence we have the following corol-lary. Corollary 18.
The representation type of the category A is wild. GG CATEGORY FOR THE QUANTUM SCHR ¨ODINGER ALGEBRA 15
Let O [0] be the full subcategory O consisting of modules with locally nilpotent action of Z .Since A is a subcategory of O [0] , O [0] is also wild.5. Appendix
In this appendix, we check that the action (4.3) indeed defines a U q ( s )-module. Let v i be afixed highest weight vector of L ( i ), θ i ∈ V i , s ∈ Z + .From the action of U q ( sl ) on V , it suffices to check the relations XY = qY X and therelations between H q and U q ( sl ).Firstly it is easy to see that F Y ( θ i ⊗ F s v i ) = Y F ( θ i ⊗ F s v i ) from F Y = Y F .Next we can compute that: XY ( θ i ⊗ F s v i ) = q [ i + 1] X (cid:16) V ( b i − )( θ i ) ⊗ F s v i − (cid:17) + 1[ i + 1] X (cid:16) V ( a i )( θ i ) ⊗ F s +1 v i +1 (cid:17) = q s +1 [ − s + i ][ i ][ i + 1] V ( a i − ) V ( b i − )( θ i ) ⊗ F s v i − q s − i +2 [ s ][ i + 1][ i ] V ( b i − ) V ( b i − )( θ i ) ⊗ F s − v i − − q s − i [ s + 1][ i + 1][ i + 2] V ( b i ) V ( a i )( θ i ) ⊗ F s v i + q s +1 [ − s + i + 1][ i + 1][ i + 2] V ( a i +1 ) V ( a i )( θ i ) ⊗ F s +1 v i +2 ,Y X ( θ i ⊗ F s v i ) = q s [ − s + i + 1][ i + 1] Y (cid:16) V ( a i )( θ i ) ⊗ F s v i +1 (cid:17) − q s − i [ s ][ i + 1] Y (cid:16) V ( b i − )( θ i ) ⊗ F s − v i − (cid:17) = q s +1 [ − s + i + 1][ i + 1][ i + 2] V ( b i ) V ( a i )( θ i ) ⊗ F s v i + q s [ − s + i + 1][ i + 1][ i + 2] V ( a i +1 ) V ( a i )( θ i ) ⊗ F s +1 v i +2 − q s − i +1 [ s ][ i + 1] 1[ i ] V ( b i − ) V ( b i − )( θ i ) ⊗ F s − v i − − q s − i [ s ][ i + 1] 1[ i ] V ( a i − ) V ( b i − )( θ i ) ⊗ F s v i . Then using V ( a i − ) V ( b i − ) = V ( b i ) V ( a i ) and q s +1 [ − s + i ][ i ][ i + 1] − q s − i [ s + 1][ i + 1][ i + 2] = q (cid:16) q s +1 [ − s + i + 1][ i + 1][ i + 2] − q s − i [ s ][ i + 1] 1[ i ] (cid:17) , we have XY ( θ i ⊗ F s v i ) = qY X ( θ i ⊗ F s v i ).Furthermore, from EF s v i = [ s ][ i + 1 − s ] F s − v i , we have EX ( θ i ⊗ F s v i )= q s [ i + 1 − s ][ i + 1] V ( a i )( θ i ) ⊗ EF s v i +1 − q s − i [ s ][ i + 1] V ( b i − )( θ i ) ⊗ EF s − v i − =[ s ][ i + 2 − s ] q s [ i + 1 − s ][ i + 1] V ( a i )( θ i ) ⊗ F s − v i +1 − [ s − i + 1 − s ] q s − i [ s ][ i + 1] V ( b i − )( θ i ) ⊗ F s − v i − ,XE ( θ i ⊗ F s v i ) = X ( θ i ⊗ EF s v i )=[ s ][ i + 1 − s ] X ( θ i ⊗ F s − v i )=[ s ][ i + 1 − s ] q s − [ i + 2 − s ][ i + 1] V ( a i )( θ i ) ⊗ F s − v i +1 − [ s ][ i + 1 − s ] q s − i − [ s − i + 1] V ( b i − )( θ i ) ⊗ F s − v i − . Then EX ( θ i ⊗ F s v i ) = qXE ( θ i ⊗ F s v i ).Using EF s = F s E + [ s ] F s − q − ( s − K − q s − K − q − q − , we have EY ( θ i ⊗ F s v i )= q [ i + 1] V ( b i − )( θ i ) ⊗ EF s v i − + 1[ i + 1] V ( a i )( θ i ) ⊗ EF s +1 v i +1 = q [ s ][ i − s ][ i + 1] V ( b i − )( θ i ) ⊗ F s − v i − + [ s + 1][ i − s + 1][ i + 1] V ( a i )( θ i ) ⊗ F s v i +1 , ( X + q − Y E )( θ i ⊗ F s v i )= q s [ i + 1 − s ][ i + 1] V ( a i )( θ i ) ⊗ F s v i +1 − q s − i [ s ][ i + 1] V ( b i − )( θ i ) ⊗ F s − v i − + [ s ][ i + 1 − s ][ i + 1] V ( b i − )( θ i ) ⊗ F s − v i − + q − [ s ][ i + 1 − s ][ i + 1] V ( a i )( θ i ) ⊗ F s v i +1 . Then from [ i + 1 − s ] − q s − i = q [ i − s ] and q s + q − [ s ] = [ s + 1], we have EY ( θ i ⊗ F s v i ) = ( X + q − Y E )( θ i ⊗ F s v i ) . Next we have
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