On quantum toroidal algebra of type A 1
aa r X i v : . [ m a t h . QA ] J un ON QUANTUM TOROIDAL ALGEBRA OF TYPE A FULIN CHEN , NAIHUAN JING , FEI KONG , AND SHAOBIN TAN Abstract.
In this paper we introduce a new quantum algebra which specializesto the 2-toroidal Lie algebra of type A . We prove that this quantum toroidal alge-bra has a natural triangular decomposition, a (topological) Hopf algebra structureand a vertex operator realization. Introduction
Let ˙ g be a finite dimensional simple Lie algebra over C . The universal centralextension t ( ˙ g ) of the 2-loop algebra ˙ g ⊗ C [ t ± , t ± ], called the toroidal Lie algebra,has a celebrated presentation given by Moody-Rao-Yokonuma [9] for constructingthe vertex representation for t ( ˙ g ). In understanding the Langlands reciprocity foralgebraic surfaces, Ginzburg-Kapranov-Vasserot [3] introduced a notion of quantumtoroidal algebra U ~ ( ˙ g tor ) associated to ˙ g . The algebra U ~ ( ˙ g tor ) specializes to theMoody-Rao-Yokonuma presentation of t ( ˙ g ) in general, except for ˙ g in type A when U ~ ( ˙ g tor ) specializes to a proper quotient of the latter [2]. The theory of quantumtoroidal algebras has been extensively studied, especially with a rich representationtheory developed by Hernandez [5, 6] and others, see [7] for a survey. One noticesthat two major structural properties of U ~ ( ˙ g tor ) have played a fundamental rolein Hernandez’s work: the triangular decomposition and the (deformed) Drinfeldcoproduct.Let A be the generalized Cartan matrix associated to the affine Lie algebra g of ˙ g .When A is symmetric, by using the vertex operators calculus, Jing introduced in [8]a quanutm affinization algebra U ~ (ˆ g ) associated to g . Meanwhile, it is remarkablethat finite dimensional representations of U ~ (ˆ g ) were studied by Nakajima in [10]using powerful geometric approach of quiver varieties. If A is of simply-laced type,then U ~ (ˆ g ) is nothing but the quantum toroidal algebra U ~ ( ˙ g tor ). However, for thecase that A is not of simply-laced type, the definition of U ~ (ˆ g ) is slightly differentfrom that of U ~ ( ˙ g tor ). Explicitly, one notices that A (1)1 is the unique symmetricbut non-simply-laced affine generalized Cartan matrix. In this case, the defining Mathematics Subject Classification.
Key words and phrases. quantum toroidal algebra, triangular decomposition, Hopf algebra. Partially supported by the Fundamental Research Funds for the Central Universities(No.20720190069) and NSF of China (No.11971397). Partially supported by NSF of China (No.11531004, No.11726016) and Simons Foundation(No.523868). Partially supported by NSF of China (No.11701183). Partially supported by NSF of China (Nos.11531004, 11971397). urrents x ± ( z ) , x ± ( z ) in U ~ (ˆ g ) satisfy the relation( z − q ∓ w )( z − w ) x ± ( z ) x ± ( w ) = ( q ∓ z − w )( z − w ) x ± ( w ) x ± ( z ) , (1.1)which appeared naturally in calculations of quantum vertex operators [8] and equi-variant K-homology of quiver varieties [10]. In particular, U ~ (ˆ g ) specializes to thetoroidal Lie algebra t ( ˙ g ) of type A as the classical limit of (1.1) holds in t ( ˙ g ). Onthe other hand, in U ~ ( ˙ g tor ) these two currents satisfy the relation( z − q ∓ w ) x ± ( z ) x ± ( w ) = ( q ∓ z − w ) x ± ( w ) x ± ( z ) . (1.2)This stronger relation was needed in verifying the compatibility with affine quantumSerre relations in U ~ ( ˙ g tor ) so that it processes a canonical triangular decomposition[5]. For the algebra U ~ (ˆ g ), we only know that it has a weak form of triangulardecomposition [10].From now on, we assume that ˙ g is of type A . The main goal of this paper is todefine a “middle” quantum algebra U ~ (ˆ g ) ։ U ։ U ~ ( ˙ g tor )of U ~ (ˆ g ) and U ~ ( ˙ g tor ), and prove that this new quantum toroidal algebra U processesthe “good” properties enjoyed by both of U ~ (ˆ g ) and U ~ ( ˙ g tor ). Explicitly, we firstintroduce in Section 2 a quantum algebra U which specializes to the toroidal Liealgebra t ( ˙ g ). By definition, U is the quotient algebra of U ~ (ˆ g ) obtained by modulothe relation[ x ± ( z ) , ( z − q ∓ w ) x ± ( z ) x ± ( w ) − ( q ∓ z − w ) x ± ( w ) x ± ( z )] = 0 . (1.3)One notices that U ~ ( ˙ g tor ) is a quotient algebra of U as the relation (1.2) implies therelations (1.1) and (1.3). In Section 3, we prove that U admits a triangular decom-position (see Theorem 3.1). In Section 4, we prove that U has a deformed Drinfeldcoproduct (see Theorem 4.1). As in [4], this allows us to define a (topological) Hopfalgebra structure on U (see Theorem 4.2). As usual, the crucial step in establish-ing Theorems 3.1 and 4.1 is to check the compatibility with affine quantum Serrerelations, in where the new relation (1.3) appeared naturally (see (3.10) and (4.6)).Finally, in Section 5 we point out that the quantum vertex operators constructed in[8] satisfy the relation (1.3), and so we obtain a vertex representation for U .Throughout this paper, we denote by C [[ ~ ]] the ring of complex formal series inone variable ~ . By a C [[ ~ ]]-algebra, we mean a topological algebra over C [[ ~ ]] withrespect to the ~ -adic topology. For n, k, s ∈ Z with 0 ≤ k ≤ s , we denote the usualquantum numbers as follows[ n ] q = q n − q − n q − q − , [ s ] q ! = [ s ] q [ s − q · · · [1] q , (cid:18) sk (cid:19) q = [ s ] q ![ k ] q ![ s − k ] q ! , where q = exp( ~ ) ∈ C [[ ~ ]] . . Quantum toroidal algebra of type A . In this section we introduce a new quantum algebra which specializes to toroidalLie algebra of type A .Let A = ( a ij ) i,j =0 , = (cid:18) − − (cid:19) be the generalized Cartan matrix of type A (1)1 . For i, j = 0 ,
1, let g ij ( z ) = q a ij − z − q a ij z (2.1)be the formal Taylor series at z = 0. The following is the main object of this paper: Definition 2.1.
The quantum toroidal algebra U is the C [[ ~ ]]-algebra topologicallygenerated by the elements h i,n , x ± i,n , c i = 0 , , n ∈ Z , (2.2)and subject to the relations in terms of generating functions in z : φ ± i ( z ) = q ± h i, exp ± ( q − q − ) X ± n> h i,n z − n ! , x ± i ( z ) = X n ∈ Z x ± i,n z − n . The relations are:(Q1) c is central , [ φ ± i ( z ) , φ ± j ( w )] = 0 , (Q2) φ + i ( z ) φ − j ( w ) = φ − j ( w ) φ + i ( z ) g ij ( q c w/z ) − g ij ( q − c w/z ) , (Q3) φ + i ( z ) x ± j ( w ) = x ± j ( w ) φ + i ( z ) g ij ( q ∓ c w/z ) ± , (Q4) φ − i ( z ) x ± j ( w ) = x ± j ( w ) φ − i ( z ) g ji ( q ∓ c z/w ) ∓ , (Q5) [ x + i ( z ) , x − j ( w )] = δ ij q − q − (cid:18) φ + i ( zq − c ) δ (cid:18) q c wz (cid:19) − φ − i ( zq c ) δ (cid:18) q − c wz (cid:19)(cid:19) , (Q6) ( z − q ± w ) x ± i ( z ) x ± i ( w ) = ( q ± z − w ) x ± i ( w ) x ± i ( z ) , (Q7) ( z − q ∓ w )( z − w ) x ± i ( z ) x ± j ( w ) = ( q ∓ z − w )( z − w ) x ± j ( w ) x ± i ( z ) , (Q8) [ x ± i ( z ) , (cid:0) ( z − q ∓ w ) x ± i ( z ) x ± j ( w ) − ( q ∓ z − w ) x ± j ( w ) x ± i ( z ) (cid:1) ] = 0 , (Q9) X σ ∈ S X r =0 ( − r (cid:18) r (cid:19) q x ± i ( z σ (1) ) · · · x ± i ( z σ ( r ) ) x ± j ( w ) x ± i ( z σ ( r +1) ) · · · x ± i ( z σ (3) ) = 0 , where i, j = 0 , i = j in (Q7), (Q8), (Q9) and δ ( z ) = P n ∈ Z z n is the usual δ -function. Remark 2.2.
As indicated in Introduction, in literature there are two other def-initions of quantum toroidal algebra of type A : the algebra U ~ (ˆ g ) introduced in[8, 10] and the algebra U ~ ( ˙ g tor ) introduced in [3, 5]. By definition, the algebra U ~ (ˆ g ) is the C [[ ~ ]]-algebra topologically generated by the elements as in (2.2) with elations (Q1)-(Q7) and (Q9), while the algebra U ~ ( ˙ g tor ) is the C [[ ~ ]]-algebra topo-logically generated by the elements as in (2.2) with relations (Q1)-(Q6), (Q9) andthe following relation( z − q ∓ w ) x ± i ( z ) x ± j ( w ) = ( q ∓ z − w ) x ± j ( w ) x ± i ( z ) , i = j ∈ { , } . (2.3)By definition we have that U is a quotient algebra of U ~ (ˆ g ), while U ~ ( ˙ g tor ) is aquotient algebra of U .Now we recall the definition of the toroidal Lie algebra of type A . Let K be the C -vector space spanned by the symbols t m t m k i , i = 1 , , m , m ∈ Z subject to the relations m t m t m k + m t m t m k = 0 . Let ˙ g = sl ( C ) be the simple Lie algebra of type A and h· , ·i the Killing form on˙ g . The toroidal Lie algebra (see [9]) t = t ( ˙ g ) = (cid:0) ˙ g ⊗ C [ t ± , t ± ] (cid:1) ⊕ K is the universal central extension of the double loop algebra ˙ g ⊗ C [ t ± , t ± ], where K is the center space and[ x ⊗ t m t m , y ⊗ t n t n ] = [ x, y ] ⊗ t m + n t m + n + h x, y i ( X i =1 m i t m + n t m + n k i ) , for x, y ∈ ˙ g and m , m , n , n ∈ Z .Let { e + , α, e − } be a standard sl -triple in ˙ g , that is,[ e + , e − ] = α, [ α, e ± ] = ± e ± . For i = 0 , m ∈ Z , set α ,m = α ⊗ t m , α ,m = t m k − α ⊗ t m , e ± ,m = e ± ⊗ t m , e ± ,m = e ∓ ⊗ t ± t m . Note that these elements generate the algebra t .Following [9], we have: Proposition 2.3.
The toroidal Lie algebra t is abstractly generated by the elements α i,m , e ± i,m , k for i = 0 , , m ∈ Z with relations (L1) [k , t ] = 0 , [ α i,m , α j,n ] = a ij δ m + n, m k , (L2) [ α i,m , e ± j,n ] = ± a ij e j,m + n , (L3) [ e + i,m , e − j,n ] = δ ij ( α j,m + n + mδ m + n, k ) , (L4) ( z − w )[ e ± i ( z ) , e ± i ( w )] = 0 , (L5) ( z − w ) [ e ± i ( z ) , e ± j ( w )] = 0 , i = j, (L6) ( z − w )[ e ± i ( z ) , [ e ± i ( z ) , e ± j ( w )]] = 0 , i = j, (L7) [ e ± i ( z ) , [ e ± i ( z ) , [ e ± i ( z ) , e ± j ( w )]]] = 0 , i = j, where i, j = 0 , , m, n ∈ Z and e ± i ( z ) = P n ∈ Z e ± i,n z − n . roof. Denote by L the Lie algebra abstractly generated by the elements α i,m , e ± i,m , k for i = 0 , , m ∈ Z with relations (L1)-(L7). One easily checks that the relations(L1)-(L7) hold in t and so we have a surjective Lie homomorphism ψ from L to t . On the other hand, denote by L ′ the Lie algebra abstractly generated by theelements α i,m , e ± i,m , k for i = 0 , , m ∈ Z with relations (L1)-(L4) and (L7). Thenthere is a quotient map, say ϕ , from L ′ to L . It was proved in [9] that the surjectivehomomorphism ψ ◦ ϕ : L ′ → L → t is an isomorphism, noting that the relation (L4)is equivalent to the relation [ e ± i ( z ) , e ± i ( w )] = 0. This in turn implies that the map ψ is an isomorphism, as required. (cid:3) By combing Definition 2.1 with Proposition 2.3, one immediate gets the followingresult.
Theorem 2.4.
The classical limit U / ~ U of U is isomorphic to the universal en-veloping algebra U ( t ) of the torodial Lie algebra t . Remark 2.5.
From the proof of Proposition 2.3, one knows that the algebra U ~ (ˆ g )also specializes t . On the other hand, it is straightforward to see that the current( z − w )[ e ± ( z ) , e ± ( w )]is nonzero in t and its components lie in the space ¯ K = P m ∈ Z ( C t m t k + C t m t − k ).Thus, the algebra U ( ˙ g tor ) specializes to the quotient algebra t / ¯ K of t (cf. [2]).3. Triangular decomposition of U In this section, we prove a triangular decomposition of U . By a triangular de-composition of a C [[ ~ ]]-algebra A , we mean a data of three closed C [[ ~ ]]-subalgebras( A − , H, A + ) of A such that the multiplication x − ⊗ h ⊗ x + x − hx + induces an C [[ ~ ]]-module isomorphism from A − b ⊗ H b ⊗ A + to A . Here and henceforth, for two C [[ ~ ]]-modules U, V , the notion U b ⊗ V stands for the ~ -adically completed tensorproduct of U and V .Let U + (resp. U − ; resp. H ) be the closed subalgebra of U generated by x + i,m (resp. x − i,m ; resp. h i,m , c ). The following is the main result of this section: Theorem 3.1. ( U − , H , U + ) is a triangular decomposition of U . Moreover, U + (resp. U − ; resp H ) is isomorphic to the C [[ ~ ]] -algebra topologically generated by x + i,m (resp. x − i,m ; resp. h i,m , c ), and subject to the relations (Q6-Q9) with “ + ” (resp. (Q6-Q9)with “ − ”; resp. (Q1-Q2)). The rest of this section is devoted to a proof of Theorem 3.1. We first introducesome algebras related to U that will be used later on. Definition 3.2.
Let e U be the C [[ ~ ]]-algebra topologically generated by the elementsin (2.2) with defining relations (Q1-Q5), b U the quotient algebra of e U modulo therelations (Q6-Q7), and ¯ U the quotient algebra of b U modulo the relation (Q8).Denote by e U + (resp. e U − ; resp. e H ) the closed subalgebra of e U generated by x + i,m (resp. x − i,m ; resp. h i,m , c ). The following result is standard. emma 3.3. ( e U − , e H , e U + ) is a triangular decomposition of e U . Moreover, e U + (resp. e U − ) isomorphic to the C [[ ~ ]] -algebra topologically free generated by x + i,m (resp. x − i,m )and e H is isomorphic to the C [[ ~ ]] -algebra topologically generated by h i,m , c withrelations (Q1-Q2). The following result was proved in (the proof of) [5, Lemma 8].
Lemma 3.4.
For i, j, k = 0 , , the following hold in e U : [( z − q ± a ij w ) x ± i ( z ) x ± j ( w ) − ( q ± a ij z − w ) x ± j ( w ) x ± i ( z ) , x ∓ k ( w )] = 0 . (3.1)Similarly, we have: Lemma 3.5.
For i, j, k = 0 , with i = j , the following hold in b U : [[ x ± i ( z ) , ( z − q ∓ w ) x ± i ( z ) x ± j ( w ) − ( q ∓ z − w ) x ± j ( w ) x ± i ( z )] , x ∓ k ( w )] = 0 . (3.2) Proof.
Let i, j be as in lemma. We first prove that for η = ± [ φ ηi ( q ∓ η c z ) , (cid:0) ( z − q ∓ w ) x ± i ( z ) x ± j ( w ) − ( q ∓ z − w ) x ± j ( w ) x ± i ( z ) (cid:1) ] = 0 . (3.3)Indeed, it follows from (Q3) and (Q7) that[ φ + i ( q ∓ c z ) , (cid:0) ( z − q ∓ w ) x ± i ( z ) x ± j ( w ) − ( q ∓ z − w ) x ± j ( w ) x ± i ( z ) (cid:1) ]= (cid:18) q ± z − z z − q ± z q ∓ z − wz − q ∓ w − (cid:19) · (cid:0) ( z − q ∓ w ) x ± i ( z ) x ± j ( w ) − ( q ∓ z − w ) x ± j ( w ) x ± i ( z ) (cid:1) φ + i ( q ∓ c z )= ( q ± − q ∓ ) z ( z − w )( z − q ± z )( z − q ∓ w ) · (cid:0) ( z − q ∓ w ) x ± i ( z ) x ± j ( w ) − ( q ∓ z − w ) x ± j ( w ) x ± i ( z ) (cid:1) φ + i ( q ∓ c z )=0 . Similarly, for the case η = − , (3.3) follows from (Q4) and (Q7). Now, in view of(3.3) and (Q5), we have[[ x ± i ( z ) , x ∓ k ( w )] , ( z − q ∓ w ) x ± i ( z ) x ± j ( w ) − ( q ∓ z − w ) x ± j ( w ) x ± i ( z )] = 0 . (3.4)This together with (3.1) (with i = j ) gives (3.2), which proves the lemma. (cid:3) Lemma 3.6.