aa r X i v : . [ m a t h . QA ] M a r FINITE DIMENSIONAL MODULES OVER QUANTUMTOROIDAL ALGEBRAS
LIMENG XIA
Institute of Applied System Analysis, Jiangsu UniversityZhenjiang 212013, Jiangsu Prov. China
Abstract.
In this paper, for all generic q ∈ C ∗ , if g is not of type A , we prove that the quantum toroidal algebra U q ( g tor ) has nonontrivial finite dimensional simple module. Key Words: quantum toroidal algebra, finite dimensionalmodule
AMS Subject Classification (2010): Introduction
Let g be a finite dimensional complex simple Lie algebra and b g = g ⊗ C [ t, t − ] ⊕ C c be the associated affine Lie algebra. Both quantumgroups U q ( g ) and U q ( b g ) have finite dimensional simple modules whichcan be viewed as the quantization of simple modules over classic Liealgebras ([2], [3], [4]).In 1987, Drinfeld gave an extremely important realization of quan-tum affine algebras, then it was applied to construct the affinizationof quantum affine algebras. Such new algebras are called quantumtoroidal algebras. Let g tor be the toroidal Lie algebra of g with nullity2. The quantum toroidal algebra U q ( g tor ) can be regarded as a quan-tum deformation of the enveloping algebra U ( g tor ) of g tor . In the pastdecades, the quantum toroidal algebras and their representations havebeen researched by many authors (see [6], [7], [8], [10], [11], [12], [13],[14]).Certainly, g tor has nontrivial finite dimensional modules. However,no one has constructed some finite dimensional modules for U q ( g tor )while q is generic. In this paper, we prove the following result. E-mail address: [email protected].
Theorem 1.1. If g is not of type A and q ∈ C ∗ is generic, thenquantum toroidal algebra U q ( g tor ) has no nontrivial finite dimensionalsimple module. Quantum toroidal algebras
Let C = ( c i,j ) ≤ i,j ≤ n be a symmetrizable Cartan matrix. So thereexists a diagonal matrix D = diag ( d , · · · , d n )with d i ∈ Z + such that gcd( d , · · · , d n ) = 1 and DC is symmetric.In this paper, we always assume that ( c i,j ) ≤ i,j ≤ n is of finite type X n ( = A ) and C is of affine type X (1) n . They are the Cartan matrices ofLie algebras g and b g , respectively.For convenience, we use the following standard notations: q i = q d i , [ m ] i = q mi − q − mi q i − q − i , m ∈ Z , [ m ] i ! = m Y k =1 [ k ] i , h mk i i = [ m ] i ![ k ] i ![ m − k ] i ! , m ≥ . quantum toroidal algebras.Definition 2.1. The quantum toroidal algebra U q ( g tor ) is an asso-ciative C ( q ) -algebra generated by elements x ± i ( k ) , a i ( l ) , K ± i , γ ± ( i =0 , , · · · , n, k ∈ Z , l ∈ Z \ { } satisfying γ ± are central and K i K j = K j K i , K ± i K ∓ i = 1 , (2.1)[ a i ( l ) , K ± i ] = 0 , (2.2)[ a i ( l ) , a j ( l ′ )] = δ l + l ′ , [ lc i,j ] i l · γ l − γ − l q − q − , (2.3) K i x ± j ( k ) K − i = q ± c i,j x ± j ( k ) , (2.4)[ a i ( l ) , x ± j ( k )] = ± [ lc i,j ] i l γ ∓ | l | x ± j ( l + k ) , (2.5)[ x ± i ( k + 1) , x ± j ( k ′ )] q ± ci,ji = − [ x ± j ( k ′ + 1) , x ± i ( k )] q ± ci,ji , (2.6)[ x + i ( k ) , x − j ( k ′ )] = δ i,j γ k − k ′ φ + i ( k + k ′ ) − γ − k − k ′ φ − i ( k + k ′ ) q i − q − i , (2.7) and the q -Serre relations Sym t ,...,t m m =1 − c i,j X k =0 ( − k (cid:20) − c i,j k (cid:21) i x ± i ( t ) · · · x ± i ( t k ) x j ( t ) x ± i ( t k +1 ) · · · x ± i ( t m ) = 0 , where Sym t ,...,t m denotes the symmetrization with respect to the indices ( t , · · · , t m ) and φ ± i ( z ) = ∞ X m =0 φ ± i ( ± m ) z ∓ m = K ± i exp ± ( q i − q − i ) ∞ X l =1 a i ( ± l ) z ∓ l ! . quantum affine algebras.Definition 2.2. The horizontal quantum affine algebra U q ( b g ) is thesubalgebra of U q ( g tor ) generated by elements { x ± i (0) , K ± i , ( i = 0 , , · · · , n ) } . Let α i be the simple root associated to x + i (0) and δ = P ni =0 s i α i theprimitive imaginary root. Then Γ = Q ni =0 K s i i is a central element. Definition 2.3.
The vertical quantum affine algebra U q ( b g ) is the sub-algebra of U q ( g tor ) generated by elements x ± i ( k ) , a i ( l ) , K ± i , γ ± ( i = 1 , · · · , n, k ∈ Z , l ∈ Z \ { } . Adding Γ to U q ( b g ), it is well known that the extended algebra isisomorphic to U q ( b g ). In particular, there exists an isomorphism Θ suchthat Θ( x ± i (0)) = x ± i (0) , i = 1 , · · · , n, Θ(Γ ) = γ . Highest weight modules over quantum affine algebras
In this section, we introduce the notion highest weight moduleof Kac-Moody type .For arbitrary given m , · · · , m n ∈ Z , let A ( m , · · · , m n ) denote thesubalgebra generated by { x ± i ( ± m i ) , ( γ m i K i ) ± , ( i = 0 , , · · · , n ) } . Lemma 3.1. A ( m , · · · , m n ) is isomorphic to U q ( b g ) .Proof. In fact, the map π : x ± i (0) x ± i ( ± m i ) , K ± i ( γ m i K i ) ± defines an isomorphism of algebras. In the following we shall identify A ( m , · · · , m n ) as U q ( b g ) by this isomorphism. (cid:3) XIA
Let N ± be the subalgebra generated by { x ± i (0) , ( i = 0 , , · · · , n ) } and let N be the Laurent polynomial algebra C ( q )[ K ± , · · · , K ± n ],then U q ( b g ) = N − · N · N + . Definition 3.2. (a) Suppose that V is a U q ( b g ) -module and v ∈ V . If x + i (0) v = 0 , K ± i v = a ± i v for all i , then v is called a highest weightvector of Kac-Moody type.(b) A module generated by a highest weight vector of Kac-Moody typeis called a highest weight module of Kac-Moody type.(c) If v is a highest weight vector of Kac-Moody type, then C v is aone dimensional module over N · N + . The induced module M ( v ) = U q ( b g ) ⊗ N · N + C v is called a Verma module of Kac-Moody type. Lemma 3.3.
Any highest weight module of Kac-Moody type is a quo-tient of some Verma module of Kac-Moody type.Proof.
It follows by the definition. (cid:3)
Lemma 3.4.
Assume that v is a highest weight vector of Kac-Moodytype and V is a simple U q ( b g ) -module generated by v . If dim V < ∞ ,then V = C v is trivial.Proof. First we claim that any simple finite dimensional U q ( b g )-module V is a simple U q ( b g )-module.Note that γ is central and it acts as a scalar over V . By relation[ a (1) , a ( − γ − γ − q − q − , if γ = ±
1, the subalgebra generated by a (1) | V , a ( − | V is isomorphic to the Weyl algebra A , which has nofinite dimensional module ([1]). So Γ | V = γ | V = ± i , v is a highest weight vector of the quantum group of type A generated by x ± i (0) , K ± i . So dim V < ∞ implies K i v = ε i q λ i v suchthat λ i ∈ N , ε i ∈ { , − } (see Theorem 2.6 of [9]).Moreover, we infer that q s λ + ··· + s n λ n ∈ { , − } . Since q is generic,we have s λ + · · · + s n λ n = 0 and λ = · · · = λ n = 0. Then theVerma module M ( v ) has a unique maximal submodule J generated by { x − i (0) v | i = 0 , , · · · } . So V = M ( v ) /J ∼ = C ( q ) v . (cid:3) Proof for main theorem
Throughout this section, we always assume that q is generic and V is a finite-dimensional simple U q ( g tor )-module. So γ | V = ±
1. For convenience, we assume γ | V = 1. The proof for γ | V = − C ( q, l ) = ([ lc i,j ] i ) ≤ i,j ≤ n for all l = 0. Then there exitsa polynomial det g such thatdet( C ( q, l )) = det g ( q l ) Q ni =0 ( q i − q − i ) . The polynomial det g is explicitly give by the following tabular:Type of g det g ( q ) A n ( n ≥
2) (1 − q − ) n +1 ( q n +1 − B n ( n ≥
2) ( q − q − n q − ( q − q − ) C n ( n ≥
3) ( q − q − ) n +1 ( q + q − ) q n − D n ( n ≥
4) ( q − q − ) n +2 ( q + q − ) ( q + 1)( q n − − q − n +2 ) E ( q − q − ) ( q − q − )( q − q − ) E ( q − q − ) ( q − q − )( q − q − )( q − q − ) E ( q − q − ) ( q − q − ) ( q − q − )( q − q − ) F ( q − q − )( q − q − ) ( q − q − )( q − q − )( q − q − ) G q − ( q − q − q − some useful lemmas.Lemma 4.1. C ( q, l ) is invertible for all l = 0 .Proof. Straightforward. (cid:3)
Lemma 4.2.
There exist elements ξ ( l ) for all l ∈ Z \ { } such that [ ξ ( l ) , x ± j ( k )] = ± x ± j ( l + k ) , ∀ k ∈ Z , j = 0 , , · · · , n. (4.1) Proof.
Because C ( q, l ) is invertible, let ( ξ , · · · , ξ n ) be the unique solu-tion of ( ξ , · · · , ξ n ) C ( q, l ) = ( l, · · · , l ) , and let ξ ( l ) = P ni =0 ξ i a i ( l ). Then[ ξ ( l ) , x ± j ( k )] = ± n X i =0 ξ i [ lC i,j ] i l x ± j ( l + k ) = ± x ± j ( l + k ) . (cid:3) XIA
For convenience, we write x ± i ⊗ f ( z ) := X f j x ± i ( j ) (4.2)for all 0 ≤ i ≤ n and f ( z ) = P f j z j ∈ C [ z, z − ]. Lemma 4.3.
There exists f ( z ) = P nj =0 f j z j ∈ C [ z ] such that f = − and x ± i ⊗ f ( z ) V = 0 for each i .Proof. Since dim
V < ∞ , there are polynomials g ± i ( z ) ∈ C [ z, z − ] suchthat x ± i ⊗ g ± i ( z ) V = 0 . If p ( z ) = g ± i ( z ) h ( z ), then x ± i ⊗ p ( z ) = ± [ P h l ξ ( l ) , x ± i ⊗ g ± i ( z )], then x ± i ⊗ p ( z ) V = 0. Let f ( z ) = cz m g − ( z ) · · · g − n ( z ) g +0 ( z ) · · · g + n ( z ), where c ∈ C ∗ and m ∈ Z such that f ( z ) ∈ C [ z ] and f (0) = − (cid:3) Let Ξ = P deg ( f ) l =1 f l ξ ( l ). Assume v ∈ V is an eigenvector of Ξ witheigenvalue λ . Lemma 4.4.
For all ≤ i , · · · , i k ≤ n and m , · · · , m k ∈ Z , we have Ξ · ( x ± i ( m ) · · · x ± i k ( m k ) v ) = ( λ ± k ) x ± i ( m ) · · · x ± i k ( m k ) v. (4.3) Proof.
It follows from x ± i ⊗ z m f ( z ) V = 0 and[Ξ | V , x ± i ( m ) | V ] = ± ( x ± i ⊗ z m f ( z ) + x ± i ( m )) | V = ± x ± i ( m ) | V . (cid:3) proof of main theorem. Because V is finite-dimensional, by(4.3), there exists v ′ of U q ( b g ) such that Ξ · v ′ = λ ′ v ′ and x + i ( m ) v ′ = 0 , ∀ m ∈ Z , i = 0 , · · · , n. (4.4)By (4.4), U q ( b g ) v ′ is a finite dimensional highest weight module of A ( m , m , · · · , m n ). By Lemma 3.4, dim A ( m , m , · · · , m n ) v ′ < ∞ and Theorem 2.6 of [9], we have K i v ′ = ǫ i v ′ , ǫ i ∈ { , − } for all i =0 , · · · , n .If there exists v ′′ := x − i ( m i ) v ′ = 0 for some index i and integer m i , then v ′′ is also a highest weight vector of U q ( b g ) and K i v ′′ = ± q − i v ′′ = ± v ′′ , this forces dim V = ∞ , a contradiction. So we alsohave x − i ( m ) v ′ = 0 for all m ∈ Z and i = 0 , , · · · , n . This by (2.7) alsoimplies a i ( l ) v ′ = 0 , l ∈ Z \ { } , i = 0 , · · · , n. So V = C v ′ is trivial. ACKNOWLEDGMENTS
The author gratefully acknowledges the partial financial supportfrom the NNSF (Nos. 11871249, 11771142) and the Jiangsu Natu-ral Science Foundation (No. BK20171294). Part of this work was doneduring the author’s visiting Paris Diderot. The author would like tothank Prof. Marc Rosso for warm hospitality and helpful discussions.
References [1] R. E. Block,
The irreducible representations of the Lie algebra sl (2) and ofthe Weyl algebra . Adv. Math. 39 (1981), 69-110.[2] V. ChariV, A. Pressley, Quantum affine algebras , Comm. Math. Phys. (1991), 261–283.[3] V. ChariV, A. Pressley, minimal affinizations of representations of quantumgroups: The nonsimply-laced case , Lett. Math. Phys. (1995), 99–114.[4] V. ChariV, A. Pressley, minimal affinizations of representations of quantumgroups: The simply-laced case , J. Algebra. (1996), 1–30.[5] Y. Gao and N. Jing, U q ( gl N ) action on gl N -modules and quantum toroidalalgebras , J. Algebra (2004), no. 1, 320–343.[6] V. Ginzburg, M. Kapranov and E. Vasserot, Langlands reciprocty for alge-bric surfaces,
Math. Res. Lett. (1995), 147–160.[7] I. B. Frenkel, N. Jing and W. Wang, Quantum vertex representations via fi-nite groups and the McKay correspondence , Comm. Math. Phys. (2000),365–393.[8] D. Hernandez,
Quantum toroidal algebras and their representations , SelectaMath. (N.S.) (2009), 701–725.[9] J. C. Jantzen, Lectures on Quantum groups , A.M.S. Providence, (1996).[10] K. Miki,
Toroidal and level 0 U ′ q ( b sl n +1 ) actions on U q ( b gl n +1 ) -modules J.Math. Phys. (1999), 3191–3210.[11] K. Miki, Representations of quantum toroidal algebra U q ( sl n +1 , tor )( n > (2000), 7079-7098.[12] K. Miki, Quantum toroidal algebra U q ( sl , tor ) and R matrices , J. Math.Phys. (2001), 2293-2308.[13] Y. Saito, Quantum toroidal algebras and their vertex representations , Publ.RIMS. Kyoto Univ., (1998), 155–177.[14] M. Varagnolo and E. Vasserot, Schur duality in the toroidal setting , Comm.Math. Phys.182