Integrable representations of affine A(m, n) and C(m) superalgebras
aa r X i v : . [ m a t h . R T ] A p r INTEGRABLE REPRESENTATIONS OFAFFINE A p m , n q AND C p m q SUPERALGEBRAS
YUEZHU WU AND R. B. ZHANGA
BSTRACT . Rao and Zhao classified the irreducible integrable modules with finite di-mensional weight spaces for the untwisted affine superalgebras which are not ˆ A p m , n q ( m ‰ n ) or ˆ C p m q . Here we treat the latter affine superalgebras to complete the classi-fication. The problem boils down to classifying the irreducible zero-level integrablemodules with finite dimensional weight spaces for these affine superalgebras, whichis solved in this paper. We note in particular that such modules for ˆ A p m , n q ( m ‰ n )and ˆ C p m q must be of highest weight type, but are not necessarily loop modules. Thisis in sharp contrast to the cases of ordinary affine algebras and the other types of affinesuperalgebras. Key words: integrable modules; highest weight modules; evaluation modules, loopmodules.
1. I
NTRODUCTION
A finite dimensional simple Lie superalgebra g over the field C of complex numbersis called basic classical [4] if its even subalgebra g ¯0 is reductive and g carries an evennon-degenerate supersymmetric invariant bilinear form p¨ | ¨q . The full list of such sim-ple Lie superalgebras can be found in [4]. Fix a simple basic classical Lie superalgebra g and let L “ C r t , t ´ s be the algebra of Laurent polynomials in the indeterminate t .The untwisted affine superalgebra G associated to g [3] is G “ g b L ‘ C c ‘ C d with the commutation relations defined as follows. For any X P g and m P Z , we denote X p m q “ X b t m . Then for all a , b P g and m , n P Z , r c , G s “ , r d , a p m qs “ ma p m q , r a p m q , b p n qs “ r a , b sp m ` n q ` cm p a | b q d m ` n , . We shall also denote the affine superalgebra G by ˆ g following the convention of [5].Note that the even subalgebra of G is G ¯0 “ g ¯0 b L ‘ C c ‘ C d , which is the affinealgebra of g ¯0 .Let H “ h ‘ C c ‘ C d , where h is a Cartan subalgebra of g . If V is a Z -graded G -module, we denote by V l the weight space of V with weight l P H ˚ . The moduleis called integrable if (i) V “ ‘ l P H ˚ V l and (ii) when restricted to a G ¯0 -module, V isintegrable in the usual sense (see [1] and [5, § The irreducible integrable modules with finite dimensional weight spaces for affinealgebras associated with finite dimensional simple Lie algebras were classified byChari [1], who proved that such modules comprise of irreducible integrable highestweight modules, irreducible integrable lowest weight modules and loop modules. Theirreducible integrable modules for affine superalgebras were investigated systemati-cally by S. Rao and K. Zhao and others (see [12] and references therein).Let V be an irreducible integrable module for G with finite dimensional weightspaces. Since c is also the central extension of G ¯0 , it is known [5] that c must act on V by an integer, which is called the level of V . It has long been known [3, 6] that the affinesuperalgebras ˆ A p m , n q ( m ě , n ě B p m , n q ( m ě , n ě D p m , n q ( m ě , n ě q ,ˆ D p , , a q , ˆ F p q and ˆ G p q do not admit any integrable modules of nonzero level. In thecases when G is ˆ A p , n q ( n ě B p , n q ( n ě
1) and ˆ C p m q ( m ě V is a highest (resp. lowest) weightmodule with respect to the Borel subalgebra b ‘ g b t C r t s ‘ C c ‘ C d of G , where b isa Borel subalgeba of g . Therefore, the classification of irreducible integrable modulesreduces to the classification of those of zero-level.It is proved in [12] that any irreducible zero-level integrable module with finite di-mensional weight spaces for the affine superalgebra G is an loop module provided that G is not ˆ A p m , n q ( m ‰ n ) or ˆ C p m q . The method used to prove this in [12] is an adaptionto the affine superalgebra context of the method developed by Chari [1] for ordinary(i.e., non-super) affine algebras. Semi-simplicity of g ¯0 was used in a crucial way inproving that irreducible integrable G -modules were of highest weight type. This con-dition is not met in the cases of A p m , n q ( m ‰ n ) and C p m q . This is a main reason whythe method of [12] failed to produce a complete classification.The aim of this paper is to complete the classification of irreducible integrable mod-ules for untwisted affine superalgebras started by Rao and Zhao by treating the affinesuperalgebras ˆ A p m , n q ( m ‰ n ) and ˆ C p m q . This is achieved in Theorem 5.1. The irre-ducible zero-level integrable modules with finite dimensional weight spaces for ˆ A p m , n q ( m ‰ n ) and ˆ C p m q are all highest weight modules with respect to the triangular decom-position (2.1) (see Theorem 3.3), but in sharp contrast to Theorem 2.7 for ordinaryaffine algebras and the other types of affine superalgebras, such modules are not nec-essarily loop modules. The necessary and sufficient conditions for a simple highestweight module to be a loop module is given in Lemma 4.4 and Remark 5.2.Let us briefly describe the content of this paper. In Section 2 we recall the construc-tion of loop modules for affine superalgebras. In Section 3 we prove that any irre-ducible zero-level integrable module with finite dimensional weight spaces for ˆ A p m , n q ( m ‰ n ) and ˆ C p m q must be a highest weight module. The result is given in Theorem3.3. In Section 4 we construct irreducible integrable modules (see Definition 4.5) forthese affine superalgebras, which include the irreducible loop modules as a specialcase. In the last section we prove the main result of this paper, that is, Theorem 5.1,which states that any irreducible zero-level integrable module with finite dimensional NTEGRABLE REPRESENTATIONS OF AFFINE SUPERALGEBRAS 3 weight spaces for ˆ A p m , n q ( m ‰ n ) and ˆ C p m q is one of the modules given in Definition4.5.We point out that the method used in this paper to classify irreducible integrablemodules is very much inspired by the work [11] of Rao on finite dimensional modulesfor multi-loop superalgebras. It is very different from that of [12]. As far as we areaware, the method of [12] has not been improved to deal with ˆ A p m , n q ( m ‰ n ) andˆ C p m q . 2. L OOP MODULES
We recall results from [12, 10] on irreducible integral modules for affine superalge-bras, which will be needed later.2.1.
Highest weight modules.
Given a simple basic classical Lie superalgebra g , welet g “ n ´ ‘ h ‘ n ` be the triangular decomposition with b “ n ` ‘ h being a distin-guished Borel subalgebra and h a Cartan subalgebra.Denote L p g q “ g b L and L p h q “ h b L with L “ C r t , t ´ s . Let G ˘ “ n ˘ b L , T “ L p h q ‘ C c , T “ T ‘ C d . The affine superalgebra G “ L p g q ‘ C c ‘ C d associated with g contains the subalgebra G “ L p g q ‘ C c . We have the following triangular decompositions for G and G :(2.1) G “ G ´ ‘ T ‘ G ` , G “ G ´ ‘ T ‘ G ` . We shall deal only with elements l P H ˚ such that l p c q “
0, i.e., l P p h ‘ C d q ˚ . Amodule V of G (resp. G ) is called a highest weight module if there exists a weightvector v P V with respect to h ‘ C c ‘ C d (resp. h ‘ C c ) such that (1) U p G q v “ V (resp.U p G q v “ V ), (2) G ` v “
0, and (3) U p T q v (resp. U p T q v ) is an irreducible T -module(resp. T -module). The vector v is called a highest weight vector of V .Let ˜ j : U p T q Ñ L be a Z -graded algebra homomorphism such that ˜ j p c q “ j | h P h ˚ . Then for any given b P C , we can turn L into a T -module via ˜ j defined forall f P L by d f “ ˆ t ddt ` b ˙ f , c f “ , h p m q f “ ˜ j p h p m qq f , h p m q P L p h q . (2.2)We write j “ p ˜ j , b q and denote by L j the image of ˜ j regarded as a Z -graded T -submodule. It was shown in [1, §
3] that if L j is a simple T -module, it must be L : “ C or a Laurent subring L r : “ C r t r , t ´ r s for some integer r ą L j is a simple T -module. We extend L j to a module over B : “ G ` ‘ T with G ` acting trivially, and construct the induced G -module M p j q “ U p G q b U p B q L j . (2.3)This has a unique irreducible quotient, which we denote by ˆ V p j q . Then every irre-ducible highest weight G -module is isomorphic to some ˆ V p j q . YUEZHU WU AND R. B. ZHANG
Remark 2.1.
Let C a denote the -dimensional G-module with G acting trivially andd acting by multiplication by a P C , then ˆ V p ˜ j , b ` a q – ˆ V p ˜ j , b q b C C a . Define the evaluation map S : L Ñ C , t ÞÑ y “ S ˝ ˜ j : U p T q Ñ C . Let U p T q act on the one dimensional vector space C y “ C by y . We extend C y to a module over B : “ G ` ‘ T by letting G ` act trivially. Construct the induced G -module M p y q “ U p G q b U p B q C y , which also has a unique simple quotient V p y q .Form the vector space V p y q b L and denote w p s q “ w b t s for any w P V p y q and s P Z . We now turn V p y q b L into a G -module by defining the action cw p s q “ , dw p s q “ p s ` b q w p s q , x p m q w p s q “ p x p m q w qp s ` m q , x p m q P L p g q . (2.4)The following results due to Rao and Zhao [12, 10] will be important later. In[12, 10] only the b “ Theorem 2.2. [12, 10]
Let j and y be as above. Assume that L j – L r is an irreducibleT -module. Let v be a highest weight vector of V p y q and denote v p i q “ v b t i for anyi P Z . Then (1) V p y q b L – ‘ r ´ i “ U p G q v p i q as G-modules, where U p G q v p i q are irreducible G-submodules. Furthermore, U p G q v p q – ˆ V p j q . (2) ˆ V p j q has finite dimensional weight spaces with respect to h ‘ C d if and only ifV p y q has finite dimensional weight spaces with respect to h . (3) V p y q has finite dimensional weight spaces if and only if y factors through h b L { I for some co-finite ideal I of L. In this case p g b I q V p y q “ . We note that U p G q v p i q – V p ˜ j , b ` i q . In the case r “
0, the formula in part (1) of thetheorem should be understood as V p y q b L – ‘ i P Z U p G q v p i q . Remark 2.3.
Similar arguments as those in [12, 10] can show that Theorem 2.2 stillholds when g is a semi-simple Lie algebra. Remark 2.4.
The co-finite ideal I can be chosen to be generated by a polynomial P p t q (see [10] ). By multiplying it by t m p m P Z q , we can also assume that P p t q has non-zeroroots. Loop modules.
Now we recall the construction of loop modules. Denote by V p l q the irreducible highest weight g -module with highest weight l P h ˚ . Let K be apositive integer, and fix a K -tuple a “ p a , . . . , a K q of complex numbers, which are alldistinct and non-zero. Define a Lie superalgebra homomorphism z : L p g q Ñ g K “ g ‘ ¨ ¨ ¨ ‘ g loooomoooon K , z p x b t m q “ p a m x , ¨ ¨ ¨ , a mK x q , (2.5)for all x P g and m P Z . Then z is surjective under the given conditions for a . NTEGRABLE REPRESENTATIONS OF AFFINE SUPERALGEBRAS 5
Remark 2.5.
Let I be the ideal of L generated by P p t q “ K ś i “ p t ´ a i q . It was shown in [10] that ker z “ g b I and g b L { I – g K . Given irreducible g -modules V p l q , . . . , V p l K q with integral dominant highest weights l , . . . , l K respectively, we let V p l , a q “ V p l q b V p l q b ¨ ¨ ¨ b V p l K q , (2.6)where l “ p l , . . . , l K q . Then V p l , a q is an irreducible highest weight L p g q -modulevia z , where irreducibility follows from the surjectivity of z . Such modules are calledevaluation modules for L p g q .Let v i be a highest weight vector of V p l i q for each i . Then v : “ v b v b ¨ ¨ ¨ b v K isa highest weight vector of V p l , a q satisfying p h b t m q v “ K ÿ j “ a mj l j p h q v , @ h P h . (2.7)Define an algebra homomorphism y : U p T q Ñ C by y p c q “ y p h b t m q “ K ÿ j “ a mj l j p h q , @ h P h , m P Z . Then it follows from (2.7) that V p y q – V p l , a q since V p y q is determined by y .Introduce the Z -graded algebra homomorphism ˜ j : U p T q Ñ L defined by˜ j p c q “ , ˜ j p h b t m q “ K ÿ j “ a mj l j p h q t m , @ h P h , m P Z . (2.8)Then im p ˜ j q is a simple T -module via (2.2) for any fixed b P C , and there exists aninteger r ě im p ˜ j q “ L r by [2]. [Note that r “ l i “ i .] It follows from Theorem 2.2 that V p l , a q b L “ ‘ r ´ i “ ˆ V i p l , a q , where ˆ V i p l , a q : “ U p G qp v b t i q . Note in particular that ˆ V p l , a q “ ˆ V p j q with j “ p ˜ j , b q . Definition 2.6.
Call the G-modules ˆ V i p l , a q simple loop modules. If r “
1, then ˆ V p l , a q b L is simple. When r ą
1, assuming l i ‰ i , we have K “ rS for some positive integer S , and there exists a permutation s such that s p l q “p l s p q , l s p q , . . . , l s p K q q and s p a q “ p a s p q , a s p q , . . . , a s p K q q are respectively given by s p l q “ p µ , µ , . . . , µ loooooomoooooon r , µ , µ , . . . , µ loooooomoooooon r , . . . , µ S , µ S , . . . , µ S loooooomoooooon r q , s p a q “ p b , w b , . . . , w r ´ b , b , w b . . . , w r ´ b , . . . , b S , w b S , . . . , w r ´ b S q , where w “ exp ´ p ?´ r ¯ , and b , . . . , b S P C ˚ such that b i { b j is not a power of w if i ‰ j . The ideal I E L in Theorem 2.2(3) associated with V p y q “ V p l , a q is generated YUEZHU WU AND R. B. ZHANG by the polynomial ś Ss “ ś r ´ i “ p t ´ w i b s q “ ś Ss “ p t r ´ b rs q . It immediately follows fromthe obvious fact ř r ´ i “ w i “ y p h b t m q “ j p h b t m q “ r | m . The loop modules ˆ V i p l , a q when r ą § Theorem 2.7. [1, 2, 12]
Suppose g is a simple Lie algebra or is a basic classical Liesuperalgebra not of type A p m , n q (m ‰ n) or C p m q . Then any irreducible zero-levelintegrable module for the associated affine (super)algebra G with finite dimensionalweight spaces is a simple loop module. The proof of the theorem was given in [1, 2] when g is a simple Lie algebra, and in[12] when g is a basic classical Lie superalgebra. Remark 2.8.
Similar arguments as those in [1, 2, 12] can show that Theorem 2.7 stillholds for semi-simple Lie algebras.
In the remainder of the paper, we classify the irreducible zero-level integrable mod-ules with finite dimensional weight spaces for the affine superalgebras ˆ A p m , n q ( m ‰ n )and ˆ C p m q . This requires algebraic methods quite different from those used in [1, 2]and [12]. 3. H IGHEST WEIGHT MODULES FOR ˆ A p m , n q AND ˆ C p m q We will show in this section that irreducible zero-level integrable modules with fi-nite dimensional weight spaces for the affine superalgebras ˆ A p m , n q ( m ‰ n ) and ˆ C p m q must be highest weight modules with respect to the triangular decomposition (2.1), seeTheorem 3.3 for the precise statement.Detailed structures of the underlying finite dimensional simple Lie superalgebras A p m , n q and C p m q will be required, which we describe below.3.1. Lie superalgebras sl p m , n q and C p m q . Recall that A p m , n q is sl p m ` , n ` q if m ‰ n , and is sl p n ` | n ` q{ C I n ` if m “ n . Also C p m q “ osp p | m ´ q . To simplifynotation, we consider sl p m , n q instead of A p m , n q in this section.3.1.1. The Lie superalgebra sl p m , n q . Let V “ V ¯0 ‘ V ¯1 be a Z -graded vector spacewith dim V ¯0 “ m and dim V ¯1 “ n . Then the space of C -linear endomorphisms End p V q on V is also Z -graded, End p V q “ p End p V qq ¯0 ‘ p End p V qq ¯1 , with p End p V qq j “ t f P End p V q| f p V k q Ă V k ` j for all k P Z u . The general linear Lie superalgebra gl p m , n q is End p V q endowed with the followingLie super bracket r f , g s “ f ¨ g ´ p´ q i j f ¨ g , f P p
End p V qq i , g P p
End p V qq j . NTEGRABLE REPRESENTATIONS OF AFFINE SUPERALGEBRAS 7
By fixing bases for V ¯0 and V ¯1 we can write X P End p V q as X “ ˆ A BC D ˙ , where A is an m ˆ m matrix, B is an m ˆ n matrix, C is an n ˆ m matrix and D is an n ˆ n matrix. Then ˆ A D ˙ is even and ˆ BC ˙ is odd. Denote by E ab the p m ` n q ˆ p m ` n q -matrix unit,which has zero entries everywhere except at the p a , b q position where the entry is 1.Then gl p m , n q has the homogeneous basis t E ab | ď a , b ď m ` n u with E ab being even if1 ď a , b ď m , or m ` ď a , b ď m ` n , and odd otherwise. Let e a ( a “ , , . . . , m ` n ) beelements in the dual space of ˜ h : “ ř m ` na “ C E aa such that e a p E bb q “ d ab . There exists anon-degenerate bilinear form p¨ , ¨q : ˜ h ˚ ˆ ˜ h ˚ ÞÑ C such that p e a , e b q “ p´ q r a s d ab where r a s “ a ď m and 1 if a ą m .The special linear Lie superalgebra sl p m , n q is the Lie sub-superalgebra of gl p m , n q consisting of elements X P gl p m , n q such that str X “
0, where the supertrace is definedfor any X “ ˆ A BC D ˙ by str X “ tr A ´ tr D . It is well known that g “ sl p m , n q is simpleif m ‰ n . However, if m “ n , the identity matrix belongs to g , which clearly spans anideal.Let g “ "ˆ A D ˙* , g ` “ "ˆ B ˙* and g ´ “ "ˆ C ˙* . Then g ¯0 “ g and g ¯1 “ g ´ ‘ g ` . Note that g is reductive and g – sl p m q ‘ C z ‘ sl p n q , where C z is the center of g . Also, g admits a Z -grading g “ g ´ ‘ g ‘ g ` with g ˘ satisfying r g ` , g ` s “ “ r g ´ , g ´ s .Let h be the standard Cartan subalgebra consisting of the diagonal matrices in g .Denote d j “ e m ` j for 1 ď j ď n . The sets of the positive even roots, positive odd rootsand simple roots are respectively given by D ` ¯0 “ t e i ´ e j , d k ´ d l | ď i ă j ď m , ď k ă l ď n u , D ` ¯1 “ t e i ´ d j | ď i ď m , ď j ď n u , P “ t e ´ e , ¨ ¨ ¨ , e m ´ ´ e m , e m ´ d , d ´ d , ¨ ¨ ¨ , d n ´ ´ d n u , where P forms a basis of h ˚ . Set D ` “ D ` ¯0 Ť D ` ¯1 and D “ D ` Ť p´ D ` q . The rootsystem can be encoded in the Dynkin diagram ✐ ... ✐ ② ✐ ... ✐ ,where the grey node corresponds to the odd simple root.For any root a P D , we denote by g a the corresponding root space. Let n ˘ ¯ i “ À ˘ a P D ` ¯ i g a ( i “ ,
1) and n ˘ “ n ˘ ¯0 ‘ n ˘ ¯1 , then g “ n ´ ‘ h ‘ n ` . Note that n ˘ ¯1 “ g ˘ .3.1.2. The Lie superalgebra C p m q . The structure of g “ C p m q can be understood byregarding it as a Lie subalgebra of sl p , m ´ q that preserves a non-degenerate su-persymmetric bilinear form. To describe the root system of g , we let h be the Cartansubalgebra of g contained in the distinguished Borel subalgebra. Then h ˚ has a basis YUEZHU WU AND R. B. ZHANG e , d , . . . , d m ´ and is equipped with the bilinear form p¨ , ¨q : h ˚ ˆ h ˚ ÞÑ C such that p e , e q “ p d k , d l q “ ´ d kl and p e , d k q “ . The sets of positive even roots, positive oddroots and simple roots are respectively given by D ` ¯0 “ t d i ´ d j , d , d j | ď i ă j ď m ´ u , D ` ¯1 “ t e ˘ d j | ď j ď m ´ u , P “ t e ´ d , d ´ d , d m ´ ´ d m ´ , d m ´ u . We denote D ` “ D ` ¯0 Ť D ` ¯1 and D “ D ` Ť p´ D ` q . The Dynkin diagram of the rootsystem is given by ② ✐ ... ✐ ✐ ă ✐ .We have g “ n ´ ‘ h ‘ n ` , where n ˘ “ n ˘ ¯0 ‘ n ˘ ¯1 with n ˘ ¯ i “ À ˘ a P D ` ¯ i g a ( i “ , g ¯0 “ n ´ ¯0 ‘ h ‘ n ` ¯0 is C z ‘ sp m ´ , where C z is the center of g ¯0 . Fromthe root system one immediately sees that g admits a Z -grading g “ g ´ ‘ g ‘ g ` with g “ g ¯0 and g ˘ “ n ˘ ¯1 .It is known that C p m q – sl p , q if m “
2. Thus we may assume that m ě Highest weight modules.
Let G be the affine superalgebra associated with C p m q or sl p m , n q with m ‰ n . We retain notation of Section 2, and set G ˘ ¯0 “ n ˘ ¯0 b L . Define d P H ˚ by setting d | h ‘ C c “ d p d q “ . We write l ď µ for l , µ P H ˚ if p µ ´ l q| h “ ř k i a i with k i non-negative integers and a i P P .Let V be an irreducible zero-level integrable module for G with finite dimensionalweight spaces. In this subsection we will show that V has to be a highest weightmodule with respect to the triangular decomposition (2.1) of G . This will be done indetail for p sl p m , n q only as the proof for ˆ C p m q is similar.By the definition of integrable G -modules, V is integrable over the even subalgebra G ¯0 of G . It follows from Chari’s work [1] that there is a non-zero weight vector v P V such that G ` ¯0 v “
0. Denote by wt p v q the weight of v . Let X be the subspace of V spanned by the vectors E m , m ` p k q E m , m ` p´ k q v for all k ě
0, which is a subspace of V wt p v q` p e m ´ d q , thus dim X ă 8 . Therefore, there exists a finite positive integer N suchthat X “ span t E m , m ` p k q E m , m ` p´ k q v | ă k ă N u . Thus for any r P Z we have(3.1) E m , m ` p r q E m , m ` p´ r q v “ ÿ ă k ă N a p r q k E m , m ` p k q E m , m ` p´ k q v , a p r q k P C . Note that the elements E m , m ` p k q for all k P Z anti-commute among themselves andsatisfy E m , m ` p k q “
0. Thus equation (3.2) in the lemma below immediately followsfrom (3.1).
NTEGRABLE REPRESENTATIONS OF AFFINE SUPERALGEBRAS 9
Lemma 3.1.
Let V be an irreducible zero-level integrable G-module, and let v P V be anon-zero weight vector such that G ` ¯0 v “ . Then the following relations hold for largek: (3.2) E m , m ` p n q E m , m ` p´ n q ¨ ¨ ¨ E m , m ` p n k q E m , m ` p´ n k q v “ , @ n , . . . , n k P Z ;(3.3) E , m ` n p m q ¨ ¨ ¨ E , m ` n p m k q v “ , @ m , . . . , m k P Z . Proof.
Since (3.2) was proven already, we only need to consider (3.3). Assume that(3.2) holds for some k . If k “
1, by applying p E m p s q E m ` , m ` n p s qq to (3.2), we have E , m ` n p s ` n q E , m ` n p s ´ n q v “ . By setting p “ s ` n , q “ s ´ n , we obtain E , m ` n p p q E , m ` n p q q v “ p , q with p ” q p mod 2 q . Since two of any three integers p , q , a must have the same parity, we have E , m ` n p p q E , m ` n p q q E , m ` n p a q v “ p , q , a P Z , by noting that the elements E , m ` n p r q anti-commute for all r .For k ą
1, applying p E m p s q E m ` , m ` n p s qq k to (3.2), we have E , m ` p s ` n q E , m ` p s ´ n q ¨ ¨ ¨ E , m ` n p s ` n k q E , m ` n p s ´ n k q v “ . Set p i “ s ` n i and q i “ s ´ n i for i “ . . . . , k . Then for all p i , q i P Z with p i ” q i p mod 2 q , we have E , m ` n p p q E , m ` n p q q ¨ ¨ ¨ E , m ` n p p k q E , m ` n p q k q v “ . Therefore, for all p i , q i , a i P Z , E , m ` n p p q E , m ` n p q q E , m ` n p a q ¨ ¨ ¨ E , m ` n p p k q E , m ` n p q k q E , m ` n p a k q v “ . This proves (3.3). (cid:3)
By using the lemma, we can prove the following result.
Proposition 3.2.
Let V be an irreducible zero-level integrable G-module with finitedimensional weight spaces. Then there always exists a nonzero weight vector w P Vsuch that G ` ¯0 w “ , (3.4) E m ´ i , m ` j p r q w “ , p i , j q ‰ p , q , r P Z , (3.5) E m , m ` p n q ¨ ¨ ¨ E m , m ` p n k q w “ for large k, for all n i P Z . (3.6) Proof.
Let v be the vector in Lemma 3.1 with l bing the minimal value of k such thatthe equation (3.3) holds. Then there exist r , . . . , r l ´ P Z such that v , m ` n : “ E , m ` n p r q ¨ ¨ ¨ E , m ` n p r l ´ q v ‰ , E , m ` n p r q v , m ` n “ , @ r P Z . Since for any Y P G ` ¯0 , we have r Y , E , m ` n p r qs “ r , and hence G ` ¯0 v , m ` n “ . We observe that (3.2) still holds if we replace v by v , m ` n , namely, for large k and forall n , . . . , n k P Z ,(3.7) E m , m ` p n q E m , m ` p´ n q ¨ ¨ ¨ E m , m ` p n k q E m , m ` p´ n k q v , m ` n “ . Applying p E m p s q E m ` , m ` n ´ p s qq k , s P Z , to (3.7) and using the same arguments forthe proof of (3.3), we obtain(3.8) E , m ` n ´ p m q ¨ ¨ ¨ E , m ` n ´ p m k q v , m ` n “ m , . . . , m k P Z . Let l be the minimal integer such that (3.8) holds. Then there exist r , . . . , r l ´ P Z such that v , m ` n ´ : “ E , m ` n ´ p r q ¨ ¨ ¨ E , m ` n ´ p r l ´ q v , m ` n ‰ , E , m ` n ´ p r q v , m ` n ´ “ , @ r P Z . Therefore, G ` ¯0 v , m ` n ´ “ . Repeating the above arguments for a finite number oftimes, we will find a weight vector w such that(3.9) G ` ¯0 w “ , E i , m ` j p r q w “ , p i , j q ‰ p m , q , r P Z . Let µ be the weight of w . Observe that V , being irreducible, must be cyclicallygenerated by w over G . By using the PBW theorem for the universal enveloping algebraof G and equation (3.9), we easily show that any weight of V which is bigger than µ must be of the form µ ` a p e m ´ d q ` b d , a P Z ě , b P Z . (3.10)Now we prove (3.6). Suppose it is false, that is, for any positive integer p , therealways exist k ą p and n , . . . , n k P Z such that ˜ w : “ E m , m ` p n q ¨ ¨ ¨ E m , m ` p n k q w ‰ n : “ µ ` k p e m ´ d q ` k ř i “ n i d is the weight of ˜ w . But for large p , and hence large k , we have p n , e m ´ ´ e m q ă
0. Thus n ` p e m ´ ´ e m q is a weight of V by consideringthe action of the sl p q subalgebra generated by the root spaces g a and g ´ a where a “ e m ´ ´ e m . However, the weight n ` p e m ´ ´ e m q is not of the form (3.10). Thisproves (3.6) by contradiction. (cid:3) The following theorem is an easy consequence of Proposition 3.2.
Theorem 3.3.
Let V be an irreducible zero-level integrable G-module with finite di-mensional weight spaces. Then there exists a weight vector v P V such that G ` v “ .Furthermore, V is isomorphic to the irreducible quotient V p j q of the induced moduledefined by (2.3) for some j . NTEGRABLE REPRESENTATIONS OF AFFINE SUPERALGEBRAS 11
Proof.
Assume that g “ sl p m , n q . Consider the weight vector w of Proposition 3.2, andlet s be the minimal integer such that (3.6) holds. Then there exist r , . . . , r s ´ P Z suchthat v : “ E m , m ` p r q ¨ ¨ ¨ E m , m ` p r s ´ q w ‰ , E m , m ` p r q v “ , @ r P Z . It follows from (3.4) and (3.5) in Proposition 3.2 that G ` v “ . The existence of a highest weight vector can be proved similarly in the case of C p m q .We omit the details, but point out that the following property of C p m q plays a crucialrole: if a and b are positive odd roots, then r g a , g b s “ t u “ r g ´ a , g ´ b s . This propertyis shared by sl p m , n q .The last statement of the theorem will be verified in the proof of Theorem 5.1. Seein particular remarks below equation (5.1). (cid:3)
4. I
NTEGRABLE MODULES FOR ˆ A p m , n q AND ˆ C p m q We use g to denote A p m , n q with m ‰ n or C p m q . Recall that g ¯0 is reductive butnot semi-simple. Let g ss be the semi-simple part of g ¯0 and C z be the one dimensionalcenter of g ¯0 . Then g ¯0 “ g ss ‘ C z . Let g ss “ g ´ ss ‘ h ss ‘ g ` ss be the standard triangulardecomposition. Set L p g ss q “ g ss b L , T ss “ h ss b L ‘ C c , T ss “ T ss ‘ C d . Fix a K -tuple of integral dominant g ss -weights l “ p l , . . . , l K q , and take any a “p a , . . . , a K q P C K with distinct nonzero entries. Let ˜ j : U p T ss q Ñ L and y : U p T ss q Ñ C be algebra homomorphisms respectively defined by˜ j : h b t s Ñ p ÿ a sj l j p h qq t s , c Ñ , h P h ss , y : h b t s Ñ ÿ a sj l j p h q , c Ñ , h P h ss , (4.1)where ˜ j is Z -graded. Then im p ˜ j q – L r for some nonnegative integer r , and L r is anirreducible U p T ss q -module. By letting d act on im p ˜ j q by dt ir “ p ir ` b q t ir ( i P Z ) forany fixed b P C , we make im p ˜ j q into an irreducible T ss -module, which we denote by L j , where j “ p ˜ j , b q .Extend L j to a module over B ss : “ T ss ‘ g ` ss b L with g ` ss b L acting trivially, andconstruct the following induced module for L p g ss q ‘ C c ‘ C d : M p j q “ U p L p g ss q ‘ C c ‘ C d q b U p B ss q L j . This has a unique irreducible quotient, which we denote by V p j q . Then V p j q is anirreducible loop module by Theorem 2.7.Let U p T ss q act on the one dimensional vector space C y “ C by y . We extend C y to a module over B ss : “ T ss ‘ g ` ss b L with g ` ss b L acting trivially, and construct theinduced module M p y q “ U p L p g ss q ‘ C c q b U p B ss q C y for L p g ss q ‘ C c . This module has a unique simple quotient, which we denote by V p y q .We note in particular that V p y q is integrable with finite dimensional weight spaces.Let I be the ideal of L generated by P p t q “ K ś j “ p t ´ a j q . Then it follows from Remark2.5 that p g ss b I q V p y q “ . Given any positive integers b j (1 ď j ď K ), we let P p t q “ K ś j “ p t ´ a j q b j and set q “ ř Ki “ b i . Denote by I the ideal generated by P p t q , which clearly is contained in I . Wewrite P p t q “ ř q i “ c i t i , where the c i are complex numbers determined by a j and b j . Let t be any element of the set T I : “ t t P p z b L q ˚ | t p z b I q “ u . (4.2)Set t s “ t p z b t s q for all s P Z , and we shall also denote t by the sequence p t i q i P Z .We have ř q i “ c i t i ` m “ m P Z . Conversely, we may regard this as a lineardifference equation of order q for p t i q i P Z . It uniquely determines the sequence, andhence an element t P T I , by fixing, e.g., t , t , . . . , t q ´ , to any complex numbers.One can extend y to an algebra homomorphism U p T q Ñ C by letting y p z b t s q “ t s for all s P Z . Define the action of z b L on V p y q by p z b t s q u “ t s u , s P Z , u P V p y q . Since r z b L , L p g ss q ‘ C c s “
0, this makes V p y q into a simple module for g ¯0 b L ‘ C c .It follows from (4.2) that p z b I q V p y q “
0. Extend V p y q to a simple module for theparabolic subalgebra ˆ p : “ g ¯0 b L ‘ C c ‘ n ` ¯1 b L by letting n ` ¯1 b L act on V p y q trivially,and denote the resulting module by V p y , t q . Now construct the induced module for G “ L p g q ‘ C c : M p y , t q “ U p G q b U p ˆ p q V p y , t q . Standard arguments show that M p y , t q has a unique quotient V p y , t q , which is irre-ducible over G . Proposition 4.1.
We have p g b I q V p y , t q “ . This in particular implies that V p y , t q has finite dimensional weight spaces.Proof. For any positive odd root a , we denote by x a and y a the root vectors of g corresponding to a and ´ a respectively. Then for all positive odd roots a , b , we have r x a b I , y b b I s Ă g ¯0 b I . Since p g ¯0 b I q V p y , t q “
0, it follows that p y b b I q V p y , t q “ b . Note that V p y , t q is spanned by t y a p n q ¨ ¨ ¨ y a k p n k q u | a i are odd roots , n i P Z , u P V p y , t q , k ě u . The Lie superalgebra g is type I, thus r n ´ ¯1 , n ´ ¯1 s “
0, and it immediately follows that p n ´ ¯1 b I q V p y , t q “ . NTEGRABLE REPRESENTATIONS OF AFFINE SUPERALGEBRAS 13
Now p g ¯0 b I q¨p y a p n q u q “ t u and p n ` ¯1 b I q¨p y a p n q u q “ a . By using induction on k , it is not difficult to show that p g ¯0 b I q¨p y a p n q ¨ ¨ ¨ y a k p n k q u q “ p n ` ¯1 b I q ¨ p y a p n q ¨ ¨ ¨ y a k p n k q u q “ a , . . . , a k . Hence p g ¯0 b I q V p y , t q “ p n ` ¯1 b I q V p y , t q “
0. This proves that p g b I q V p y , t q “ I is co-finite, U p n ´ ¯1 b L { I q “ ^p n ´ ¯1 b L { I q is finite dimensional. This immedi-ately leads to the second statement. (cid:3) Remark 4.2.
Since V p y , t q is an integrable module for L p g ss q ‘ C c, the induced mod-ule M p y , t q is integrable with respect to G , and so is also V p y , t q . Proposition 4.3.
The G -modules V p y , t q and V p y , t q are isomorphic if and onlyif t “ t , and there exists a permutation s of t , , . . . , K u such that l “ s p l q anda “ s p a q .Proof. The given conditions are necessary and sufficient in order for one to have anisomorphism V p y , t q – V p y , t q of simple p -modules. Since V p y , t q and V p y , t q respectively determine V p y , t q and V p y , t q uniquely, the proposition follows. (cid:3) Lemma 4.4.
The V p y , t q is an evaluation L p g q -module if and only if y p z b I q “ .Proof. If y p z b I q “
0, then g b I acts on V p y , t q by zero. It follows from equation(2.5) and Remark 2.5 that V p y , t q is an evaluation module. Given any simple evalu-ation module V p l , a q (we may and will assume that the entries of l are all nonzero),we have y p z p m qq “ ř Ki “ a mi l i p z q for all m P Z by (2.5). Simplicity of V p l , a q requiresthat the entries of a “ p a , . . . , a K q are all distinct. It is clear that y p z b I q “ I “ ś Ki “ p t ´ a i q . (cid:3) We turn V p y , t q b L into a G -module using (2.4). Then by Theorem 2.2 we have thefollowing G -module isomorphism V p y , t q b L – ‘ r ´ i “ U p G q w p i q , where w is a highest weight vector of V p y , t q and w p i q “ w b t i with i P Z . Note thatU p G q w p i q are irreducible G -submodules. Definition 4.5.
We denote by ˆ V p j , t q the irreducible G-module U p G q w p q . Here ˆ V p j , t q is a highest weight module, where w p q is a highest weight vector. Wehave cw p q “ , dw p q “ bw p q , p z b t s q w p q “ t s w p s q , p h b t s q w p q “ ÿ a sj l j p h q w p s q , h P h ss (4.3)for all s P Z , where the t s satisfy (4.2). This induces a graded algebra homomorphismU p T q{ U p T q c ÝÑ L since c acts by zero. The image of this map is L r for some integer r ě § r ą
1, we necessarily have t s “ r | s . Lemma 4.6.
Under the given conditions on l , a and t , the irreducible G-module ˆ V p j , t q is integrable with finite dimensional weight spaces.Proof. From equation (2.4) one can see that ˆ V p j , t q is integrable over G if and only if V p y , t q is integrable over G . Similarly, ˆ V p j , t q has finite dimensional weight spaces(with respect to h ‘ C c ‘ C d ) if and only if V p y , t q has finite dimensional weightspaces (with respect to h ‘ C c q . It therefore immediately follows from Remark 4.2that ˆ V p j , t q is integrable. By Proposition 4.1, V p y , t q has finite dimensional weightspaces, thus ˆ V p j , t q also has finite dimensional weight spaces. (cid:3) Proposition 4.7.
The modules ˆ V p j , t q and ˆ V p j , t q are isomorphic if and only if all ofthe following conditions are satisfied:(a) b ” b p mod r q when L j “ L r ,(b) for r ą , there exists k P C zt u and a permutation s of t , , . . . , K u such that l “ s p l q , a “ ks p a q , p t i q i P Z “ p k i t i q i P Z . Remark 4.8. If t P T I for the ideal I “ p P p t qq of L generated by P p t q “ ś Kj “ p t ´ a j q b j ,then t P T J , where J “ p Q p t qq with Q p t q “ ś Kj “ p t ´ k a j q b j .Proof of Proposition 4.7. The r “ r ą G on ˆ V p j , t q isthe composition of the representation on ˆ V p j , t q and the algebra automorphism of L given by t ÞÑ k t . Thus the representations are isomorphic.To prove the converse, we note that the necessity of condition (a) is clear. The ac-tions of U p T q on the spaces of highest weight vectors of ˆ V p j , t q and ˆ V p j , t q respec-tively induce surjective graded algebra homomorphisms j , j : U p T q ÝÑ L r for some r . As we mentioned before, this in particular implies that j p h p m qq “ j p h p m qq “ h p m q P L p h q if r ffl m . Let ¯ j : U p T q{ Ker p j q ÝÑ L r and ¯ j : U p T q{ Ker p j q ÝÑ L r be the corresponding canonical isomorphisms. Let ¯ x “ ¯ j ´ p t r q , then ¯ j ´ p t rm q “ ¯ x m for all m P Z . If x is a representative of ¯ x in U p T q , we have j p x q “ r t r for some r P C zt u . Furthermore, given any h p rm q P L p h q , we have some h p h p rm qq P C suchthat h p rm q ” h p h p rm qq x m mod Ker p j q for all m . Thus j p h p rm qq “ h p h p rm qq t mr , j p h p rm qq “ r m h p h p rm qq t mr . These relations lead to the conditions (b) with r “ k r . (cid:3)
5. C
LASSIFICATION THEOREM
In this section, we classify the irreducible zero-level integrable modules with finitedimensional weight spaces for the untwisted affine superalgebras ˆ A p m , n q ( m ‰ n ) andˆ C p m q . The following theorem is the main result. NTEGRABLE REPRESENTATIONS OF AFFINE SUPERALGEBRAS 15
Theorem 5.1.
Let G be either ˆ A p m , n q (m ‰ n) or ˆ C p m q . Any irreducible zero-levelintegrable G-module with finite dimensional weight spaces is isomorphic to ˆ V p j , t q (see Definition 4.5) for some j and t .Proof. Let V be an irreducible zero-level integrable G -module with finite dimensionalweight spaces. By Theorem 3.3, there exists a highest weight vector v with weight l P H ˚ , that is, G ` v “ hv “ l p h q v for all h P H . Claim 1.
Let M “ U p T q v. Then M is an irreducible T -module. Let w , w P M be two weight vectors. Then by the irreducibility of V , there ex-ists g P G such that gw “ w . Write g “ ř i g ´ i h i g ` i , where g ´ i P U p G ´ q , h i P U p T q , g ` i P U p G ` q . Note that G ` w i “ . Hence w “ ř i g ´ i h i w , which forces all g ´ i to bescalars by weight considerations. Hence w “ hw for some h P U p T q , that is, M is anirreducible T -module. Observe that M is also an irreducible T -module.Since c acts as zero on M it follows that there exists a maximal graded ideal N of S “ U p T q{ U p T q c such that M – S { N as T -modules. It is known from [1] that M – S { N – L r for some integer r ě
0. Let ˜ j be the natural map defined by thefollowing composition(5.1) ˜ j : U p T q action ÝÝÝÝÑ M – S { N – L r Ă L . Clearly, ˜ j is Z -graded, ˜ j p c q “ j | h P h ˚ . Moreover V is isomorphic to ˆ V p j q ,the irreducible quotient of the induced module defined by (2.3) with j “ p ˜ j , l p d qq . If r “
0, the module is trivial. Thus we shall assume r ą V p y q for G “ L p g q ‘ C c from V – ˆ V p j q by setting y “ S ˝ ˜ j . Since V has finite dimensional weight spaces, it followsfrom Theorem 2.2 that V p y q has finite dimensional weight spaces and there exists aco-finite ideal I of L such that(5.2) p g b I q V p y q “ . This ideal can be determined as follows (for more detail see [10, Lemma 3.7]). Let w be a highest weight vector of V p y q and let µ be its weight. For each simple root a P P , we let y a be a root vector for the root ´ a . Consider t y a p s q w | s P Z u , which iscontained in the same weight space V p y q µ ´ a . Since dim V p y q µ ´ a ă 8 , there exists anon-zero polynomial P a p t q such that p y a b P a p t qq w “ . Set P p t q “ ś a P P P a p t q . Then I is the ideal generated by P p t q .To avoid confusion, we change the notation of V p y q to V ev . Recall that g ss is thesemi-simple part of g ¯0 . Regard V ev as a module for L p g ss q ‘ C c , and set V p y q “ U p L p g ss q ‘ C c q w . Claim 2. V p y q is an irreducible module for L p g ss q ‘ C c. Recall the standard triangular decomposition g ss “ g ´ ss ‘ h ss ‘ g ` ss for g ss . Since p g ` ss b L q w “ r g ` ss b L , T s Ă g ` ss b L , we have V p y q “ U p g ´ ss b L q w . Let u P V p y q be a weight vector. So we can write u “ ř i g i h i w for some g i P U p g ´ ss b L q and h i P U p h ss b L q . Since V ev is an irreducible G -module, there exists x P U p G ` q such that x p ř i g i h i w q “ w . Weight considerations require x P U p g ` ss b L q . This proves that V p y q is an irreducible module for L p g ss q ‘ C c .Since V p y q has finite dimensional h -weight spaces, so does also V p y q . As z is inthe center of g ¯0 ` C c , and acts on w by the multiplication by the scalar y p z q , it actson the entire V p y q by the multiplication by y p z q . It follows that V p y q has finitedimensional h ss -weight spaces. Therefore, there exists a co-finite ideal J of L such that p g ss b J q V p y q “
0, where J is generated by Q p t q : “ ś a P D ` ¯0 X P P a p t q .By remark 2.4 we can assume that Q p t q has non-zero roots. The module V p y q istrivial when Q p t q is a constant. Thus we will further assume that Q p t q is not a constant.Then up to a scalar multiple, Q p t q has the unique factorisation Q p t q “ S ś i “ p t ´ a i q s i ,where a i are distinct non-zero complex numbers, s i and S are positive integers. Let Q p t q “ S ś i “ p t ´ a i q and let J be the ideal generated by Q p t q . Similar arguments asthose in [12, Proposition 5.2] show that p g ss b J q V p y q “ . Therefore V p y q is a g ss b L { J -module. Because of the particular form of the gen-erator Q p t q of J , we have g ss b L { J – g ss ‘ ¨ ¨ ¨ ‘ g ss loooooomoooooon S by Remark 2.5 . Since V p y q is an irreducible integrable module, it is finite dimensional. Clearly any finite dimen-sional irreducible module for g ss ‘ ¨ ¨ ¨ ‘ g ss loooooomoooooon S is isomorphic to V p l q b ¨ ¨ ¨ b V p l S q forsome l i P h ˚ ss , which are integral dominant with respect to g ss . Here V p l i q denotesan irreducible g ss -module with highest weight l i . Therefore, V p y q is isomorphic to V p l q b ¨ ¨ ¨ b V p l S q as an irreducible module for L p g ss q ‘ C c via the map (2.5).Again by Remark 2.4 we can assume that P p t q has non-zero roots. We can alsoassume that P p t q is not a constant. Since P p t q has Q p t q as a factor, it factorises into P p t q “ S ź i “ p t ´ a i q b i K ź j “ S ` p t ´ a j q b j for some K ě S . Here all the a i are distinct nonzero complex numbers, and b i ě s i if1 ď i ď S .Let I be the ideal of L generated by P p t q . Set P p t q “ K ś i “ p t ´ a i q , and let I be theideal of L generated by P p t q . Clearly, I Ă J . For S ` ď j ď K , let V p l j q “ V p q “ C be the one-dimensional g ss -module. We have V p y q – V p l q b ¨ ¨ ¨ b V p l K q NTEGRABLE REPRESENTATIONS OF AFFINE SUPERALGEBRAS 17 with the action given by equation (2.5). This is also an isomorphism of g ss b L { I -modules.Let t P p z b L q ˚ be defined by t p x q “ y p x q for all x P z b L . Then t P T I (see (4.2)).Note that t p z b t s q “ y p z b t s q “ r ffl s since j p z b t s q “ r ffl s . Now z b t s acts onthe highest weight vector w by p z b t s q w “ t p z b t s q w . Since r z b L , L p g ss q ‘ C c s “ u P V p y q , p z b t s q u “ t p z b t s q u , s P Z . This makes V p y q into an irreducible g ¯0 b L ‘ C c -module.As p n ` ¯1 b L q w “ r n ` ¯1 b L , L p g ss q ‘ C c s Ă n ` ¯1 b L , we have p n ` ¯1 b L q V p y q “ . Regard V p y q as a module for p b L { I ‘ C c where p : “ g ¯0 ‘ n ` ¯1 , and denote it by V p y , t q . Construct the induced module M p y , t q “ U p g b L { I ‘ C c q b U p p b L { I ‘ C c q V p y , t q . Then the unique irreducible quotient module V p y , t q of M p y , t q is isomorphic to V ev .It then follows from part (1) of Theorem 2.2 that V – ˆ V p j , t q for some j and t as inDefinition 4.5. (cid:3) Remark 5.2.
From Lemma 4.4 we see that not all ˆ V p j , t q are loop modules, but thelemma provides the necessary and sufficient conditions for ˆ V p j , t q to be one.
6. C
ONCLUDING REMARKS
Theorem 5.1 and Theorem 2.7 together classify the irreducible zero-level integrablemodules with finite dimensional weight spaces for ˆ A p m , n q ( m ‰ n ) and ˆ C p m q . In viewof results of [12] and [3, 6] discussed in Section 1, this completes the classificationof the irreducible integrable modules with finite dimensional weight spaces for all theuntwisted affine superalgebras. An interesting fact in the case of ˆ A p m , n q ( m ‰ n ) andˆ C p m q is that such modules are are not necessarily loop modules. Acknowledgement.
We thank S. Rao for helpful comments on an earlier version ofthis paper. This work was supported by the Chinese National Natural Science Foun-dation grant No. 11271056, Australian Research Council Discovery-Project grant No.DP0986349, Qing Lan Project of Jiangsu Province, and Jiangsu Overseas Researchand Training Program for Prominent Young and Middle Aged University Teachers andPresidents. This paper was completed while both authors were visiting the Universityof Science and Technology of China, Hefei.R
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HINA (Wu, Zhang) S
CHOOL OF M ATHEMATICS AND S TATISTICS , U
NIVERSITY OF S YDNEY , S
YDNEY ,NSW 2006, A
USTRALIA S CHOOL OF M ATHEMATICAL S CIENCES , U
NIVERSITY OF S CIENCE AND T ECHNOLOGY OF C HINA ,H EFEI , C
HINA
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