Some categories of modules for toroidal Lie algebras
aa r X i v : . [ m a t h . R T ] S e p Some categories of modules for toroidal Lie algebras
Hongyan Guo, Shaobin Tan and Qing Wang School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Abstract
In this paper, we use basic formal variable techniques to study certain categoriesof modules for the toroidal Lie algebra τ . More specifically, we define and study twocategories E τ and C τ of τ -modules using generating functions, where E τ is proved tocontain the evaluation modules while C τ contains certain restricted τ -modules, theevaluation modules, and their tensor product modules. Furthermore, we classifythe irreducible integrable modules in categories E τ and C τ . Let ˆ g = g ⊗ C [ t, t − ] ⊕ C k be the untwisted affine Lie algebra of a finite dimensional simpleLie algebra g , and let e g = g ⊗ C [ t, t − ] ⊕ C k ⊕ C d be the affine Lie algebra with the derivationadded. The irreducible integrable modules with finite dimensional weight spaces for e g wereclassified by Chari [C]. In particular, Chari proved that every irreducible integrable ˆ g -module of level zero with finite dimensional weight spaces must be a finite dimensionalevaluation module. Furthermore, in [CP], they studied tensor products of modules ofdifferent types which were obtained in [C]. In particular, they proved that the tensorproduct of an irreducible integrable highest ˆ g -module and a finite dimensional irreducibleevaluation ˆ g -module is irreducible. Such irreducible tensor product modules are integrablebut with infinite dimensional weight spaces in general. This provides a new family ofirreducible integrable ˆ g -modules. Motivated by this, a canonical characterization for thetensor product modules of this form was given in [L1]. More specifically, two categories E and C of ˆ g -modules were defined and studied by using generating functions and formalvariables in [L1], where the category C unifies highest weight modules, evaluation modules,and their tensor product modules. Furthermore, all irreducible integrable ˆ g -modules incategories E and C were classified in [L1].Toroidal Lie algebras, which are central extensions of multi-loop Lie algebras, arenatural generalizations of affine Kac-Moody Lie algebras. For toroidal Lie algebras, irre-ducible integrable modules with finite-dimensional weight spaces have been classified andconstructed in [R]. It was proved therein that there are two classes of evaluation modulesdepending on the action of the center. One class consists of the evaluation modules arisingfrom finite dimensional irreducible modules for the finite dimensional simple Lie algebrawith the full center acting trivially, while the other consists of the evaluation modulesarising from irreducible integrable highest weight modules for the affine Lie algebra witha nontrivial action of the center. Then a natural question is whether one can apply the Partially supported by NSF of China (No.10931006) and a grant from the PhD Programs Foundationof Ministry of Education of China (No.20100121110014). Partially supported by NSF of China (No.11371024), Natural Science Foundation of Fujian Province(No.2013J01018) and Fundamental Research Funds for the Central University (No. 2013121001). E τ and C τ of modules for the toroidal Lie algebra τ . It is shown that the category E τ contains evaluation modules. As for the modules incategory C τ , we need a notion of restricted τ -module, defined in [LTW]. In [LTW], atheory of toroidal vertex algebras and their modules was developed, and we associatedtoroidal vertex algebras and their modules to toroidal Lie algebras, and proved that thereexists a canonical correspondence between restricted modules for toroidal Lie algebrasand modules for toroidal vertex algebras. In view of this, restricted modules for toroidalLie algebras are also important in the study of representation theory of toroidal vertexalgebras. The category C τ defined in this paper contains the evaluation modules, certainrestricted τ -modules where we denote the corresponding module category by e R , and theirtensor product modules.Note that the category e R is a mixed product of the notion of restricted modulesand the notion of evaluation modules for an affine Lie algebra, which is significantlydifferent from the category R of restricted modules for an affine Lie algebra. Due tothis, the treatment for the classification of irreducible integrable τ -modules in e R is morecomplicated than that for the case of the classification of irreducible integrable ˆ g -modulesin R . In the later case, we know from [DLM] that every nonzero restricted integrableˆ g -module is a direct sum of irreducible highest weight integrable modules. While in ourcase with e R , we have no such good result for restricted integrable modules for toroidalLie algebras τ . Thus a new method is needed to deal with this problem. As our mainresults, we prove that irreducible integrable τ -modules in the category E τ are exactlythe evaluation modules arising from finite-dimensional irreducible g -modules and thatirreducible integrable τ -modules in category e R are exactly the evaluation modules arisingfrom irreducible integrable highest weight modules for the affine Lie algebra. In this way,the two classes of modules which were classified and constructed in [R] are unified throughirreducible integrable modules in category C τ .This paper is organized as follows. In Section 2, we review some basic notions andfacts about the toroidal Lie algebras. In Section 3, we define the category E τ and classifyall irreducible integrable modules in this category. In Section 4, we define the category C τ and classify all irreducible integrable modules in C τ . In this section, we recall the definitions of restricted modules and integrable modules forthe toroidal Lie algebra. We also present some basic facts about them.Throughout this paper, the symbols x, x , x , x , . . . will denote mutually commutingindependent formal variables. All vector spaces are considered to be over C , the field ofcomplex numbers. For a vector space U , U (( x )) is the vector space of lower truncatedintegral power series in x with coefficients in U , U [[ x ]] is the vector space of nonnegativeintegral power series in x with coefficients in U , and U [[ x, x − ]] is the vector space ofdoubly infinite integral power series of x with coefficients in U . Multi-variable analogues2f these vector spaces are defined in the same way.Recall from [FLM] the formal delta function δ ( x ) = X n ∈ Z x n ∈ C [[ x, x − ]]and the basic property f ( x ) δ (cid:16) ax (cid:17) = f ( a ) δ (cid:16) ax (cid:17) for any f ( x ) ∈ C [ x, x − ] and for any nonzero complex number a .Let g be a finite dimensional simple Lie algebra equipped with the normalized Killingform h· , ·i . Let r be a positive integer and let C [ t ± , . . . , t ± r ] be the algebra of Laurentpolynomials in r commuting variables. For m = ( m , . . . , m r ) ∈ Z r , set t m = t m · · · t m r r .Define a Lie algebra τ = ( g ⊗ C [ t , t − ] ⊕ C K ) ⊗ C [ t ± , . . . , t ± r ] ⊕ r X i =1 C K i , (2.1)where [ a ⊗ t n t n , b ⊗ t m t m ] = [ a, b ] ⊗ t n + m t n + m + n h a, b i δ n + m , K ⊗ t n + m + h a, b i δ n + m , δ n + m, r X i =1 n i K i (2.2)for a, b ∈ g , n , m ∈ Z , n, m ∈ Z r , and where K ⊗ t m , K i , i = 1 , . . . , r are central. TheLie algebra τ is the toroidal Lie algebra which we consider in this paper. From now on, τ always stands for this particular toroidal Lie algebra .For a ∈ g , we denote a ( n , n ) = a ⊗ t n t n . For K , we denote K ( n ) = K ⊗ t n . Thenthe Lie bracket relation (2.2) amounts to[ a ( n , n ) , b ( m , m )] = [ a, b ]( n + m , n + m ) + n h a, b i δ n + m , K ( n + m )+ h a, b i δ n + m , δ n + m, r X i =1 n i K i . (2.3)For a ∈ g , form a generating function a ( x , x ) = X n ∈ Z X n ∈ Z r a ( n , n ) x − n − x − n − ∈ τ [[ x ± , x ± , . . . , x ± r ]]and set K ( x ) = X n ∈ Z r K ( n ) x − n − ∈ τ [[ x ± , . . . , x ± r ]] , x − n − = x − n − · · · x − n r − r . In terms of the generating functions, the defining rela-tions (2.2) are equivalent to[ a ( x , x ) , b ( y , y )] = [ a, b ]( y , y ) r Q i =0 y − i δ ( x i y i ) + h a, b i K ( y )( ∂∂y y − δ ( x y )) r Q i =1 y − i δ ( x i y i )+ h a, b i x − y − δ ( x y ) r P j =1 ( K j ( ∂∂y j y − j δ ( x j y j )) r Q i =1 ,i = j x − i y − i δ ( x i y i )) . (2.4) Definition 2.1. A τ -module W is said to be restricted if for any w ∈ W, a ∈ g , a ( n , n ) w = 0 for n sufficiently large, or equivalently a ( x , x ) w ∈ W [[ x ± , . . . , x ± r ]](( x )) . Let h be a Cartan subalgebra of g and let h be the linear span of h , K ( n ) and K i for n ∈ Z r , ≤ i ≤ r . We denote by h ∗ the dual space of h . Let ∆ be the root system of g and for each root α ∈ ∆, we fix a nonzero root vector x α . Definition 2.2. A τ -module W is called integrable if(i) W = L λ ∈ h ∗ W λ , where W λ = { w ∈ W | hw = λ ( h ) w, ∀ h ∈ h } .(ii) for any α ∈ ∆ , m ∈ Z , m ∈ Z r and w ∈ W , there exists a nonnegative integer k = k ( α, m , m, w ) such that ( x α ( m , m )) k w = 0 . The following lemma is similar to Lemma 2.3 in [L1].
Lemma 2.3.
There exists a basis { a , . . . , a l } of g such that [ a i ( n , n ) , a i ( m , m )] = 0 for ≤ i ≤ l, m , n ∈ Z , m, n ∈ Z r , and all a i ( m , m ) act locally nilpotently on everyintegrable τ -module. E τ of τ -modules In this section, we define and study a category E τ of modules for the toroidal Lie algebra τ ,which is analogous to the category E studied in [L1] of modules for the affine Lie algebraˆ g . The category E τ is proved to contain the evaluation modules studied by S. EswaraRao. Moreover, it is proved that every irreducible integrable τ -module in category E τ isisomorphic to a finite dimensional evaluation module. Definition 3.1.
Category E τ is defined to consist of τ -modules W satisfying the conditionthat there exist nonzero polynomials p i ( x ) ∈ C [ x ] , i = 0 , , . . . , r , such that p i ( x i ) a ( x , x ) w = 0 for i = 0 , , . . . , r, a ∈ g , w ∈ W. Lemma 3.2.
The central elements K ( n ) and K i of τ for i = 1 , . . . , r, n ∈ Z r act triviallyon every τ -module in category E τ . roof. Let W be a τ -module in category E τ with nonzero polynomials p i ( x ) such that p i ( x i ) a ( x , x ) w = 0 for any a ∈ g , w ∈ W, i = 0 , , . . . , r. First, we consider the case that p i ( x ) ∈ C for some i ∈ { , , . . . , r } . In this case, we have a ( x , x ) w = 0, i.e., a ( n , n ) w = 0for all a ∈ g , w ∈ W , n ∈ Z , n ∈ Z r . Then by using the Lie relations (2.3), we have0 = n h a, b i δ n + m , K ( n + m ) + h a, b i δ n + m , δ n + m, r X i =1 n i K i ! w (3.1)for all b ∈ g , n , m ∈ Z , n, m ∈ Z r , w ∈ W . Taking n = 0 in (3.1), we get0 = h a, b i δ m , δ n + m, r X i =1 n i K i ! w. (3.2)For each fixed i ∈ { , . . . , r } , by taking n j = δ ij for j ∈ { , . . . , r } , m = 0 and n + m = 0, we get K i = 0 on W . Thus the identity (3.1) becomes0 = n h a, b i δ n + m , K ( n + m ) w. (3.3)Taking n = 0 and n + m = 0 in (3.3), we get K ( n + m ) w = 0 for all n, m ∈ Z r , w ∈ W. Now we consider the case that p i ( x ) are of positive degrees for all i = 0 , , . . . , r. Thatis, p ′ i ( x ) = 0 for all i = 0 , , . . . , r. Pick a, b ∈ g such that h a, b i = 1. For s = 1 , . . . , r , byusing the relations (2.4), we have0 = p s ( x s ) p s ( y s )[ a ( x , x ) , b ( y , y )]= p s ( x s ) p s ( y s ) K ( y )( ∂∂y y − δ ( x y )) r Y i =1 y − i δ ( x i y i )+ p s ( x s ) p s ( y s ) x − y − δ ( x y ) r X j =1 ( K j ( ∂∂y j y − j δ ( x j y j )) r Y i =1 ,i = j x − i y − i δ ( x i y i )) . (3.4)Notice that Res y ( ∂∂y y − δ ( x y )) = 0 and Res y i p i ( x i ) p i ( y i )( ∂∂y i y − i δ ( x i y i )) = − p i ( x i ) p ′ i ( x i )for i = 0 , , . . . , r. Taking Res y Res y Res y · · · Res y r of the identity (3.4), we obtain K s p s ( x s ) p ′ s ( x s ) x − x − · · · x − s − x − s +1 · · · x − r = 0 . Thus we have K s = 0 on W for s = 1 , . . . , r . Now multiplying the identity (2.4) by p ( x ) p ( y ), we get 0 = p ( x ) p ( y )[ a ( x , x ) , b ( y , y )]= p ( x ) p ( y ) K ( y )( ∂∂y y − δ ( x y )) r Y i =1 y − i δ ( x i y i ) . (3.5)Taking Res y Res x · · · Res x r of the identity (3.5) and using y − i δ ( x i y i ) = x − i δ ( y i x i ), we get0 = p ( x ) p ′ ( x ) K ( y ) . Thus K ( n ) = 0 on W for all n ∈ Z r . 5e now give some examples of τ -modules in category E τ . Let U be a g -module andlet z = ( z , . . . , z r ) ∈ ( C ∗ ) r +1 , where C ∗ denotes the set of nonzero complex numbers. Wedefine an action of τ on U by a ( n , n ) u = z n · · · z n r r ( au ) (3.6)for a ∈ g , n ∈ Z , n = ( n , . . . , n r ) ∈ Z r , and by letting the whole center act trivially on U .Clearly, this makes U a τ -module, which is denoted by U ( z ) . More generally, let U , . . . , U s be g -modules and let z j = ( z j , . . . , z rj ) ∈ ( C ∗ ) r +1 for j = 1 , . . . , s. Then U = U ⊗ · · · ⊗ U s is a τ - module where the whole center act trivially and a ( n , n )( u ⊗ · · · ⊗ u s ) = s X j =1 z n j · · · z n r rj ( u ⊗ · · · ⊗ au j ⊗ · · · ⊗ u s ) (3.7)for a ∈ g , n ∈ Z , n ∈ Z r , u j ∈ U j . We denote this τ -module by ⊗ sj =1 U j ( z j ) and call it an evaluation module. Next, we show that ⊗ sj =1 U j ( z j ) belongs to the category E τ . For a ∈ g , u j ∈ U j ( z j ), wewrite (3.7) in terms of generating functions to get a ( x , x )( u ⊗ · · · ⊗ u s ) = X n ∈ Z X n ∈ Z r a ( n , n ) x − n − x − n − ( u ⊗ · · · ⊗ u s )= s X j =1 x − · · · x − r δ ( z j x ) · · · δ ( z rj x r )( u ⊗ · · · ⊗ au j ⊗ · · · ⊗ u s ) . Since ( x i − z ij ) δ ( z ij x i ) = 0 for i = 0 , , . . . , r , j = 1 , . . . , s, we have( x i − z i ) · · · ( x i − z is ) a ( x , x )( u ⊗ · · · ⊗ u s ) = 0for i = 0 , , . . . , r. To summarize we have:
Lemma 3.3.
Let U , . . . , U s be g -modules and let z ij be nonzero complex numbers, i =0 , , . . . , r , j = 1 , . . . , s. Set p i ( x ) = ( x − z i ) · · · ( x − z is ) for i = 0 , , . . . , r . Then the tensorproduct τ -module ⊗ sj =1 U j ( z j ) is in the category E τ , where z j = ( z j , . . . , z rj ) ∈ ( C ∗ ) r +1 . In the following we shall classify irreducible integrable τ -modules in category E τ . For a ∈ g , we have a ( x , x ) = X n ∈ Z X n ∈ Z r ( a ⊗ t n t n ) x − n − x − n − = a ⊗ r Y i =0 x − i δ ( t i x i ) ! . For each f k ( x ) ∈ C [ x ] , k ∈ { , , . . . , r } , m ∈ Z , m = ( m , . . . , m r ) ∈ Z r , a ∈ g , wehave ( r Y j =0 x m j j ) f k ( x k ) a ( x , x ) = a ⊗ ( r Y j =0 x m j j ) f k ( x k )( r Y i =0 x − i δ ( t i x i ))= a ⊗ ( r Y j =0 t m j j ) f k ( t k )( r Y i =0 x − i δ ( t i x i )) . x r · · · Res x Res x ( r Q j =0 x m j j ) f k ( x k ) a ( x , x ) = a ⊗ ( r Q j =0 t m j j ) f k ( t k ), we have thatfor any τ -module W , if f k ( x k ) a ( x , x ) W = 0, then ( a ⊗ f k ( t k ) C [ t ± , t ± , . . . , t ± r ]) W = 0 . For nonzero polynomials p i ( x ), let W be a τ -module such that p i ( x i ) a ( x , x ) w = 0for a ∈ g , w ∈ W, i = 0 , . . . , r.
We denote by P = h p ( t ) , . . . , p r ( t r ) i the ideal of C [ t ± , t ± , . . . , t ± r ] generated by p i ( t i ) , i = 0 , . . . , r. Then W is a module for the Liealgebra g ⊗ C [ t ± , t ± , . . . , t ± r ] /P . We have the following result. Proposition 3.4.
Any finite dimensional irreducible τ -module W from the category E τ is isomorphic to a τ -module U ( z ) ⊗ · · · ⊗ U s ( z s ) for some finite dimensional irreducible g -modules U , . . . , U s , and for s distinct ( r + 1) -tuples z i = ( z i , z i , . . . , z ri ) ∈ ( C ∗ ) r +1 ( i = 1 , . . . , s ).Proof. Let p i ( x ) be nonzero polynomials such that p i ( x i ) a ( x , x ) w = 0 for i = 0 , , . . . , r , w ∈ W . Assume that z i, , . . . , z i,N i ∈ C ∗ are distinct nonzero roots of p i ( x i ), where N i is the number of distinct nonzero roots of the polynomial p i ( x i ) for 0 ≤ i ≤ r . Let s = N · · · N r . Set τ r = g ⊗ C [ t ± , t ± , . . . , t ± r − ], τ r − = g ⊗ C [ t ± , t ± , . . . , t ± r − ], . . . ,τ = g ⊗ C [ t ± , t ± ], τ = g ⊗ C [ t ± ]. Let b τ i = τ i ⊗ C [ t i ] for 1 ≤ i ≤ r . By Lemma 3.2,the while center of τ act trivially on W , thus W can be viewed as a b τ r -module with theproperty p r ( x r ) a ( x , x ) w = 0 for a ∈ g , w ∈ W , and [ τ r , τ r ] w = τ r w for w ∈ W . Fromthe Proposition 3.9 of [L1], we have that W is isomorphic to a τ -module U ( r )1 ( z r, ) ⊗· · · ⊗ U ( r ) N r ( z r,N r ) for some finite dimensional irreducible τ r -modules U ( r )1 , . . . , U ( r ) N r , anddistinct nonzero complex numbers z r, , . . . , z r,N r . Since p r − ( x r − ) a ( x , x ) w = 0, i.e., p r − ( x r − ) a ( x , x ) u ⊗ · · · ⊗ u N r = 0 for u i ∈ U ( r ) i , i = 1 , . . . , N r . It follows that N r X j =1 p r − ( x r − ) z n r r,j ( u ⊗ · · · ⊗ a ( x , x , . . . , x r − ) u j ⊗ · · · ⊗ u N r ) = 0for any n r ∈ Z , where a ( x , x , . . . , x r − ) = X n ,...,n r − ∈ Z ( a ⊗ t n · · · t n r − r − ) x − n − · · · x − n r − − r − . Note that z r,j are distinct for 1 ≤ j ≤ N r . By taking n r = 0 , . . . , N r − p r − ( x r − ) a ( x , x , . . . , x r − ) U ( r ) j = 0for 1 ≤ j ≤ N r . Thus for every 1 ≤ j ≤ N r , U ( r ) j is a finite dimensional τ r -module with p r − ( x r − ) a ( x , x , . . . , x r − ) U ( r ) j = 0. Since [ τ r − , τ r − ] = τ r − and τ r = d τ r − , by usingthe Proposition 3.9 of [L1] again, we have that U ( r ) j is isomorphic to U ( r − j, ( z r − , ) ⊗ · · · ⊗ U ( r − j,N r − ( z r − ,N r − ) for some finite dimensional τ r − -module U ( r − j, , . . . , U ( r − j,N r − . Therefore W is isomorphic to a module of the from U ( r − , ( z r, , z r − , ) ⊗ · · · ⊗ U ( r − ,N r − ( z r, , z r − ,N r − ) ⊗ · · · ⊗ U ( r − N r ,N r − ( z r,N r , z r − ,N r − )7or some finite dimensional irreducible τ r − -modules U ( r − , , . . . , U ( r − ,N r − , . . . , U ( r − N r ,N r − . Thenthe proposition follows from recursion. Proposition 3.5.
Irreducible integrable τ -modules in the category E τ up to isomorphismare exactly the evaluation modules U ( z ) ⊗ · · · ⊗ U s ( z s ) , where U , . . . , U s are finite di-mensional irreducible g -modules and z i = ( z i , z i , . . . , z ri ) ∈ ( C ∗ ) r +1 for i = 1 , . . . , s aredistinct ( r + 1) -tuples.Proof. Let W be an irreducible integrable τ -module in the category E τ . Then there existnonzero polynomials p i ( x ) such that a ⊗ p i ( t i ) C [ t ± , . . . , t ± r ] W = 0 for a ∈ g , i = 0 , , . . . , r. Let I be the annihilating ideal of the τ -module W . Since a ⊗ h p ( t ) , · · · , p r ( t r ) i ⊂ I for all a ∈ g , it follows that τ /I is finite dimensional. By Lemma 2.3, there is a basis { a , . . . , a l } of g such that a i ( m , m ) acts locally nilpotently on W for any i ∈ { , . . . , l } .Let 0 = w ∈ W . We have W = U ( τ ) w = U ( τ /I ) w by the irreducibility of W . From thePBW theorem, W is finite dimensional. It then follows from Proposition 3.4. C τ of τ -modules In this section, we define and study a category C τ of modules for the toroidal Lie algebra τ , we also introduce a category e R of τ -modules and a category E ′ τ of τ -modules, where e R is a subcategory of the category of restricted τ -modules and E ′ τ is a subcategory of thecategory E τ . We prove that every irreducible τ -module in category C τ is isomorphic tothe tensor product of modules from categories e R and E ′ τ . Definition 4.1.
We define category C τ to consist of τ -modules W for which there existnonzero polynomials p i ( x ) for ≤ i ≤ r such that that the nonzero roots of p i ( x ) for ≤ i ≤ r are multiplicity-free, p ( x ) a ( x , x ) ∈ Hom ( W, W [[ x ± , . . . , x ± r ]](( x ))) for a ∈ g , and p i ( x i ) a ( x , x ) w = 0 , p i ( x i ) K ( x ) w = 0 for ≤ i ≤ r , w ∈ W. Definition 4.2.
We define category e R to consist of τ -modules W for which there ex-ist nonzero polynomials p i ( x ) for ≤ i ≤ r such that the nonzero roots of p i ( x ) aremultiplicity-free, a ( x , x ) ∈ Hom ( W, W [[ x ± , . . . , x ± r ]](( x ))) for a ∈ g , and p i ( x i ) a ( x , x ) w = 0 , p i ( x i ) K ( x ) w = 0 for ≤ i ≤ r , w ∈ W. efinition 4.3. We define a category E ′ τ to consist of τ -modules W for which there existnonzero polynomials p i ( x ) ∈ C [ x ] for ≤ i ≤ r such that the nonzero roots of p i ( x ) for ≤ i ≤ r are multiplicity-free and p i ( x i ) a ( x , x ) w = 0 for ≤ i ≤ r, a ∈ g , w ∈ W. Remark 4.4.
From the proof of Proposition 3.9 in [L1] and Proposition 3.5, we seethat irreducible integrable τ -modules in category E ′ τ are exactly those irreducible integrable τ -modules in category E τ , i.e., irreducible integrable τ -modules in category E ′ τ up to iso-morphism are the evaluation modules U ( z ) ⊗ · · · ⊗ U s ( z s ) , where U , . . . , U s are finitedimensional irreducible g -modules and z i = ( z i , z i , . . . , z ri ) ∈ ( C ∗ ) r +1 for ≤ i ≤ s aredistinct ( r + 1) -tuples. It is obvious that every τ -module in category e R and every τ -module in category E ′ τ are in C τ and the tensor products of τ -modules from e R and E ′ τ are in C τ . From the proof of Lemma 3.2, we obtain
Lemma 4.5.
The central elements K i for ≤ i ≤ r act trivially on every module fromcategories C τ and e R . Next, we define some vector spaces of formal series which will be used later.
Definition 4.6.
Let W be any vector space. Following [LTW] we set E ( W, r ) =
Hom ( W, W [[ x ± , . . . , x ± r ]](( x ))) . Define E ( W, r ) to be the subspace of (End W ) [[ x ± , . . . , x ± r ]] , consisting of formal series α ( x , x ) satisfying the condition that there exist nonzero polynomials p ( x ) , . . . , p r ( x ) suchthat the nonzero roots of p i ( x ) for ≤ i ≤ r are multiplicity-free, p ( x ) α ( x , x ) ∈ E ( W, r ) , and p i ( x i ) α ( x , x ) w = 0 for ≤ i ≤ r, w ∈ W . Define e E ( W, r ) to be the subspace of (End W ) [[ x ± , . . . , x ± r ]] ,consisting of formal series α ( x , x ) satisfying the condition that there exist nonzero poly-nomials p ( x ) , . . . , p r ( x ) whose nonzero roots are multiplicity-free such that α ( x , x ) ∈ E ( W, r ) , p i ( x i ) α ( x , x ) w = 0 for ≤ i ≤ r, w ∈ W . Furthermore, define E ( W, r ) to be the subspace of E ( W, r ) consist-ing of formal series α ( x , x ) satisfying the condition that there exist nonzero polynomials p ( x ) , . . . , p r ( x ) such that the nonzero roots of p i ( x ) for ≤ i ≤ r are multiplicity-freeand p i ( x i ) α ( x , x ) w = 0 for ≤ i ≤ r , w ∈ W . Definition 4.7.
For a vector space W , we define a linear map ψ e R : E ( W, r ) → e E ( W, r )9 y ψ e R ( α ( x , x )) w = l x ;0 ( f ( x ) − )( f ( x ) α ( x , x ) w ) , (4.1) for α ( x , x ) ∈ E ( W, r ) , w ∈ W , where f ( x ) is any nonzero polynomial such that f ( x ) α ( x , x ) ∈ E ( W, r ) . Notice that (
EndW )[[ x ± , . . . , x ± r ]] ⊂ E ( W, r ) ⊂ E ( W, r ) . (4.2)Thus for any β ( x ) ∈ ( EndW )[[ x ± , . . . , x ± r ]], we have ψ e R ( β ( x )) = β ( x ) . Remark 4.8.
Just as in [L1], one can show that ψ e R is well defined. From definition,for α ( x , x ) ∈ E ( W, r ) , we have f ( x ) ψ e R ( α ( x , x )) = f ( x ) α ( x , x ) for any nonzeropolynomial f ( x ) such that f ( x ) α ( x , x ) ∈ E ( W, r ) . Furthermore, if f i ( x i ) for ≤ i ≤ r are nonzero polynomials such that f i ( x i ) α ( x , x ) = 0 , then f i ( x i ) ψ e R ( α ( x , x )) = 0 for ≤ i ≤ r. Proposition 4.9.
For any vector space W , we have E ( W, r ) = e E ( W, r ) ⊕ E ( W, r ) (4.3)
Furthermore, ψ e R | e E ( W,r ) = 1 and ψ e R | E ( W,r ) = 0 . Proof.
Let α ( x , x ) ∈ e E ( W, r ). By Definition 4.6, we know α ( x , x ) ∈ E ( W, r ). Taking f ( x ) = 1 in Definition 4.7, we have ψ e R ( α ( x , x )) w = α ( x , x ) w for any w ∈ W , that is, ψ e R ( α ( x , x )) = α ( x , x ). Therefore ψ e R | e E ( W,r ) = 1.Let α ( x , x ) ∈ E ( W, r ). By definition there exist nonzero polynomials p i ( x ) for0 ≤ i ≤ r such that the nonzero roots of p i ( x ) for 1 ≤ i ≤ r are multiplicity-freeand p i ( x i ) α ( x , x ) w = 0 for w ∈ W . Then by (4.1) in Definition 4.7, we have ψ e R ( α ( x , x )) w = l x ;0 ( p ( x ) − )( p ( x ) α ( x , x ) w ) = 0 . Thus ψ e R ( α ( x , x )) = 0, i.e., ψ e R | E ( W,r ) = 0 . It is clear that e E ( W, r ) + E ( W, r ) is a directsum.Let α ( x , x ) ∈ E ( W, r ) and let f i ( x ) for 0 ≤ i ≤ r be nonzero polynomials suchthat the nonzero roots of f i ( x ) for 1 ≤ i ≤ r are multiplicity-free, f ( x ) α ( x , x ) ∈E ( W, r ), f i ( x i ) α ( x , x ) = 0 for 1 ≤ i ≤ r . From Remark 4.8, we have f i ( x i )( α ( x , x ) − ψ e R ( α ( x , x ))) = 0 for 0 ≤ i ≤ r . Thus α ( x , x ) − ψ e R ( α ( x , x )) ∈ E ( W, r ), which implies α ( x , x ) ∈ e E ( W, r ) ⊕ E ( W, r ). This proves E ( W, r ) ⊂ e E ( W, r ) + E ( W, r ). Therefore, wehave (4.3), and the proof is completed. 10n what follows, we denote by ψ E ′ the projection map of E ( W, r ) onto E ( W, r ) withrespect to the decomposition (4.3). For α ( x , x ) ∈ E ( W, r ), we set e α ( x , x ) = ψ e R ( α ( x , x )) , ˇ α ( x , x ) = ψ E ′ ( α ( x , x )) = α ( x , x ) − e α ( x , x ) . To prove our main result we need the following lemmas:
Lemma 4.10.
For α ( x , x ) ∈ E ( W, r ) , n ∈ Z , n ∈ Z r , w ∈ W , we have ψ e R ( α ( x , x ))( n , n ) w = l X i =0 β i α ( n + i, n ) w for some l ∈ N , β , . . . , β l ∈ C , depending on α ( x , x ) , w and n , n , where ψ e R ( α ( x , x )) = X n ∈ Z ,n ∈ Z r ψ e R ( α ( x , x ))( n , n ) x − n − x − n − Proof.
Since α ( x , x ) ∈ E ( W, r ), there exists a polynomial p ( x ) such that p (0) = 0 and p ( x ) α ( x , x ) ∈ E ( W, r ). Let k be a nonnegative integer such that x k p ( x ) α ( x , x ) w ∈ W [[ x ± , . . . , x ± r ]][[ x ]] . Set l x ;0 (cid:18) p ( x ) (cid:19) = X i ≥ α i x i ∈ C [[ x ]] . Note that Res x x k + m p ( x ) α ( x , x ) w = 0 for m ≥
0. We have X n ∈ Z r ψ e R ( α ( x , x ))( n , n ) x − n − w = Res x x n ψ e R ( α ( x , x )) w = Res x X ≤ i ≤ k − n − α i x n + i ( p ( x ) α ( x , x ) w )= X n ∈ Z r l X i =0 β i α ( n + i, n ) x − n − w. Then it follows immediately that ψ e R ( α ( x , x ))( n , n ) = l P i =0 β i α ( n + i, n ).The following delta function properties can be found in [L2], [LL]: For m > n ≥
0, wehave ( x − x ) m (cid:18) ∂∂x (cid:19) n x − δ (cid:18) x x (cid:19) = 0 , (4.4)and for 0 ≤ m ≤ n , we have( x − x ) m n ! (cid:18) ∂∂x (cid:19) n x − δ (cid:18) x x (cid:19) = 1( n − m )! (cid:18) ∂∂x (cid:19) n − m x − δ (cid:18) x x (cid:19) . (4.5)11 emma 4.11. Let W be any vector space, let α ( x , x ) , β ( x , x ) ∈ E ( W, r ) , and let γ ( x , x ) , . . . , γ n ( x , x ) be formal series in (End W ) [[ x ± , . . . , x ± r ]] such that on W , [ α ( x , x ) , β ( y , y )] = n X j =0 j ! γ j ( y , y ) (cid:18) ∂∂y (cid:19) j x − δ (cid:18) y x (cid:19) r Y i =1 x − i δ (cid:18) y i x i (cid:19) . (4.6) Then γ ( x , x ) , . . . , γ n ( x , x ) ∈ E ( W, r ) , and [ e α ( x , x ) , e β ( y , y )] = n X j =0 j ! e γ j ( y , y ) (cid:18) ∂∂y (cid:19) j x − δ (cid:18) y x (cid:19) r Y i =1 x − i δ (cid:18) y i x i (cid:19) . (4.7) Proof.
First we note that0 = Res x γ j ( y , y ) (cid:18) ∂∂y (cid:19) n x − δ (cid:18) y x (cid:19) r Y i =1 x − i δ (cid:18) y i x i (cid:19) = ( − n Res x γ j ( y , y ) (cid:18) ∂∂x (cid:19) n x − δ (cid:18) y x (cid:19) r Y i =1 x − i δ (cid:18) y i x i (cid:19) for n ≥
1. Then from (4.4), (4.5), (4.6), we obtain γ j ( y , y ) r Y i =1 x − i δ (cid:18) y i x i (cid:19) = Res x ( x − y ) j [ α ( x , x ) , β ( y , y )]Thus γ j ( y , y ) = Res x r · · · Res x Res x ( x − y ) j [ α ( x , x ) , β ( y , y )]for 0 ≤ j ≤ n . Since β ( x , x ) ∈ E ( W, r ), we see obviously that γ j ( x , x ) ∈ E ( W, r ) for j = 0 , . . . , n . Let 0 = f ( x ) ∈ C [ x ] be such that f ( x ) α ( x , x ) , f ( x ) β ( x , x ) , f ( x ) γ j ( x , x ) ∈ E ( W, r )for j = 0 , . . . , n. So we have f ( x ) α ( x , x ) = f ( x ) e α ( x , x ) , f ( x ) β ( x , x ) = f ( x ) e β ( x , x ) f ( x ) γ j ( x , x ) = f ( x ) e γ j ( x , x )for j = 0 , . . . , n. Multiplying both sides of (4.6) by f ( x ) f ( y ) we obtain f ( x ) f ( y )[ e α ( x , x ) , e β ( y , y )] = n X j =0 i ! f ( x ) f ( y ) e γ j ( y , y )( ∂∂y ) j x − δ ( y x ) r Y i =1 x − i δ ( y i x i ) . Then we multiply both sides by l x ;0 ( f ( x ) − ) l y ;0 ( f ( y ) − ) to get (4.7).12 heorem 4.12. Let π : τ → End W be a representation of toroidal Lie algebra τ incategory C τ . Define linear maps π e R and π E ′ from τ to End( W ) in terms of generatingfunctions by π e R a ( x , x ) + α K ( x ) + r X i =1 α i K i ! = ψ e R ( π ( a ( x , x )) + α π ( K ( x )) ,π E ′ a ( x , x ) + β K ( x ) + r X i =1 β i K i ! = ψ E ′ ( π ( a ( x , x ))) for a ∈ g , α i , β i ∈ C , where we extend π to τ [[ x ± , x ± , . . . , x ± r ]] canonically. Then π = π e R + π E ′ (4.8) and the linear map ϕ ( u, v ) = π e R ( u ) + π E ′ ( v ) defines a representation of τ ⊕ τ on W .If ( W, π ) is irreducible, we have that W is an irreducible τ ⊕ τ -module. Furthermore, ( W, π e R ) is a module in category e R and ( W, π E ′ ) is a module in category E ′ τ . At last, if ( W, π ) is integrable, then ( W, π e R ) is integrable in e R and ( W, π E ′ ) is integrable in E ′ τ .Proof. Firstly, the relation (4.8) follows from Proposition 4.9. By the commutator relation(2.4) and Lemma 4.11, it is straightforward to check that (
W, π e R ) is a τ -module and it isa restricted τ -module in category e R .Let f i ( x ) for i = 0 , , . . . , r be nonzero polynomials satisfying that the nonzero rootsof f i ( x ) for 1 ≤ i ≤ r are multiplicity-free, such that f ( x ) π ( a ( x , x )) ∈ E ( W, r ), f i ( x i ) π ( a ( x , x )) = 0, f i ( x i ) π ( K ( x )) = 0 for all 1 ≤ i ≤ r , a ∈ g . Then f ( x ) ψ e R ( π ( a ( x , x ))) = f ( x ) π ( a ( x , x )) , f i ( x i ) ψ e R ( π ( a ( x , x ))) = 0for 1 ≤ i ≤ r , and f i ( x i ) ψ E ( π ( a ( x , x ))) = 0for 0 ≤ i ≤ r . Thus we have f ( x ) π e R (( a ( x , x ))) = f ( x ) ψ e R ( π ( a ( x , x ))) = f ( x ) π ( a ( x , x )) , (4.9) f i ( x i ) π e R ( a ( x , x )) = 0 (4.10)for 1 ≤ i ≤ r , and f i ( x i ) π E ( a ( x , x )) = 0 . (4.11)for 0 ≤ i ≤ r . 13or a, b ∈ g , w ∈ W , by using the commutator relation (2.4) and delta-functionsubstitution property we have f ( x )[ π e R ( a ( x , x )) , π E ( b ( y , y ))] w = f ( x )[ π e R ( a ( x , x )) , π ( b ( y , y ))] w − f ( x )[ π e R ( a ( x , x )) , π e R ( b ( y , y ))] w = f ( x )[ π ( a ( x , x )) , π ( b ( y , y ))] w − f ( x )[ π e R ( a ( x , x )) , π e R ( b ( y , y ))] w = f ( x ) π ([ a, b ]( y , y )) r Y i =0 x − i δ ( y i x i ) w + f ( x ) π ( K ( y ))( ∂∂y y − δ ( x y )) r Y i =1 x − i δ ( y i x i ) w − f ( x ) π e R ([ a, b ]( y , y )) r Y i =0 x − i δ ( y i x i ) w − f ( x ) π e R ( K ( y ))( ∂∂y y − δ ( x y )) r Y i =1 x − i δ ( y i x i ) w = f ( y ) π ([ a, b ]( y , y )) r Y i =0 x − i δ ( y i x i ) w + f ( x ) π ( K ( y ))( ∂∂y y − δ ( x y )) r Y i =1 x − i δ ( y i x i ) w − f ( y ) π e R ([ a, b ]( y , y )) r Y i =0 x − i δ ( y i x i ) w − f ( x ) π e R ( K ( y ))( ∂∂y y − δ ( x y )) r Y i =1 x − i δ ( y i x i ) w =0 , where we have used identity f ( y ) π ([ a, b ]( y , y )) = f ( y ) π e R ([ a, b ]( y , y )) from (4.9).Since π e R ( a ( x , x )) ∈ E ( W, r ), we can multiply both sides by l x ;0 1 f ( x ) to get[ π e R ( a ( x , x )) , π E ( b ( y , y ))] w = 0 . (4.12)Since π E = π − π e R , we have[ π E ( a ( x , x )) , π E ( b ( y , y ))] w = [ π ( a ( x , x )) , π ( b ( y , y ))] w − [ π e R ( a ( x , x )) , π e R ( b ( y , y ))] w. This together with the identity (4.11) shows that (
W, π E ) is a τ -module in category E τ ′ . From (4.12), we see that ϕ ( u, v ) = π e R ( u ) + π E ( v ) defines a representation of τ ⊕ τ on W .Since π = π e R + π E ′ , it is easy to see that if W is an irreducible τ -module, then W is anirreducible τ ⊕ τ -module.Finally, we prove that ( W, π e R ) and ( W, π E ′ ) are integrable when ( W, π ) is integrable,i.e., we need to prove that for a ∈ g α with α ∈ ∆ and n ∈ Z , n ∈ Z r , e a ( n , n ) and ˇ a ( n , n )act locally nilpotently on W , where π e R ( a ( n , n )) = e a ( n , n ) and π E ′ ( a ( n , n )) = ˇ a ( n , n ).Since [ a, a ] = 0 and h a, a i = 0, we have [ a ( n , n ) , a ( m , m )] = 0 for all n , m ∈ Z , n, m ∈ Z r . For w ∈ W , we have a ( n , n ) e a ( x , x ) w = a ( n , n ) l x ;0 (1 /f ( x ))( f ( x ) a ( x , x ) w )= l x ;0 (1 /f ( x ))( f ( x ) a ( x , x ) a ( n , n ) w )= e a ( x , x ) a ( n , n ) w. Thus a ( n , n ) e a ( m , m ) = e a ( m , m ) a ( n , n ) for all n , m ∈ Z , n, m ∈ Z r . Fix a vector w ∈ W . By Lemma 4.10, we know that e a ( n , n ) w = P li =0 β i a ( n + i, n ) w for some14 ∈ N , β , . . . , β l ∈ C . Thus we get e a ( n , n ) p w = ( β a ( n , n ) + · · · + β l a ( n + l, n )) p w (4.13)for any p ≥
0. Since (
W, π ) is an integrable τ -module, there exists a positive integer k such that a ( m , n ) k w = 0for m = n , n + 1 , . . . , n + l . This together with 4.13 gives e a ( n , n ) k ( l +1) w = 0. Sinceˇ a ( n , n ) = a ( n , n ) − e a ( n , n ) and [ a ( n , n ) , e a ( n , n )] = 0, we haveˇ a ( n , n ) k ( l +2) w = ( a ( n , n ) − e a ( n , n )) k ( l +2) w = X i ≥ (cid:18) k ( l + 2) i (cid:19) ( − i a ( n , n ) ( k ( l +2) − i ) e a ( n , n ) i w = 0 . So we have proved the integrability. This completes the proof.We now consider irreducible integrable modules in category e R . Let W be an irreducibleintegrable τ -module in category e R . Then there exist nonzero polynomials p i ( x ) for 1 ≤ i ≤ r satisfying that the nonzero roots of p i ( x ) are multiplicity-free, p i ( x i ) a ( x , x ) w = 0,and p i ( x i ) K ( x ) w = 0. Let J = h p ( t ) , . . . , p r ( t r ) i be the ideal generated by p i ( t i ) , i =1 , . . . , r. By the same argument as with category E τ , we see that a τ -module W in category e R is a module for the Lie algebra ( g ⊗ C [ t ± ] ⊕ C K ) ⊗ C [ t ± , . . . , t ± r ] /J. We denote thisquotient algebra by ¯ τ .For each 1 ≤ i ≤ r , let { z ij ∈ C ∗ , ≤ j ≤ M i } be the set of distinct nonzeroroots of p i ( x i ), where the positive integer M i is the number of distinct nonzero roots of p i ( x i ). Set M = M · · · M r . Let I i = ( i , . . . , i r ), where 1 ≤ i j ≤ M j , 1 ≤ j ≤ r. Set z I i = ( z ,i , . . . , z r,i r ) and z nI i = z n ,i · · · z n r r,i r for n = ( n , . . . , n r ) ∈ Z r . Let φ be the Liealgebra homomorphism φ : ( g ⊗ C [ t ± ] ⊕ C K ) ⊗ C [ t ± , . . . , t ± r ] → ˆ g M := ( g ⊗ C [ t ± ] ⊕ C K ) M defined by φ ( X ⊗ t n ) = ( z nI i X )where ˆ g M = ⊕ M − copies( g ⊗ C [ t ± ] ⊕ C K ), X ∈ ˆ g = g ⊗ C [ t ± ] ⊕ C K . From the proofof Lemma 3.11 in [R], we see that φ is surjective and Ker φ = J , so that ¯ τ ∼ = ˆ g M .Consequently, W is a module for Lie algebra ˆ g M . Since τ -module W is in category e R , we also have a ( x , x ) ∈ E ( W, r ) for a ∈ g , i.e., a ( n , n ) w = 0 for n sufficientlylarge. Thus W is an irreducible restricted integrable ˆ g M -module. From [DLM] (also seethe Theorem 2.4 of [L1]), an irreducible restricted integrable ˆ g -module is an irreduciblehighest integrable module. Then by the above discussion and by Lemmas 2.6 and 2.7 in[L1], W is isomorphic to a tensor product module V ⊗ · · · ⊗ V M , where V , . . . , V M areirreducible highest integrable ˆ g -modules, with the action given by a ( n , n )( v ⊗ · · · ⊗ v M ) = M X i =1 z nI i ( v ⊗ · · · ⊗ a ( n ) v i ⊗ · · · ⊗ v M )15nd K ( n )( v ⊗ · · · ⊗ v M ) = M X i =1 z nI i ( v ⊗ · · · ⊗ K v i ⊗ · · · ⊗ v M ) , where z nI i is defined as above, v i ∈ V i for 1 ≤ i ≤ M , and a ( n ) = a ⊗ t n . We denote thismodule by V ( z I ) ⊗ · · · ⊗ V M ( z I M ) . Theorem 4.13.
Let τ be the toroidal Lie algebra. Then every irreducible integrable τ -module in category C τ is isomorphic to a module of the form V ( z I ) ⊗ · · · ⊗ V M ( z I M ) ⊗ U ( z ) ⊗ · · · ⊗ U s ( z s ) , where each V i is an irreducible highest weight integrable ˆ g -module, and each U j is afinite-dimensional irreducible g -module, z I j for j = 1 , . . . , M are defined above, and z i = ( z i , z i , . . . , z ri ) ∈ ( C ∗ ) r +1 for ≤ i ≤ s are distinct ( r + 1) -tuples.Proof. Let (
W, π ) be an irreducible integrable representation of τ in category C τ . ByTheorem 4.12, W is an irreducible τ ⊕ τ -module with the action π ( u, v ) = π e R ( u ) + π E ′ ( v )for u, v ∈ τ . Moreover, ( W, π e R ) is integrable in category e R and ( W, π E ′ ) is integrablein category E ′ τ . We view W as a τ -module in e R , then W is a restricted integrable τ -module. From the above discussion, we see that W is a restricted integrable ˆ g M -module.Furthermore, from the definition of Lie homomorphism φ and the proof of Lemma 3.11in [R], it follows that W is a restricted integrable ˆ g -module. From [DLM] (also see theTheorem 2.4 of [L1]), W is a direct sum of irreducible integrable highest ˆ g -modules. Thusby using the Lemma 2.7 of [L1] with A = A = U ( τ ), and from the above discussion andRemark 4.4, we get our conclusion. Remark 4.14.
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