Classification of level zero irreducible integrable modules for twisted full toroidal Lie algebras
aa r X i v : . [ m a t h . R T ] N ov CLASSIFICATION OF LEVEL ZERO IRREDUCIBLEINTEGRABLE MODULES FOR TWISTED FULLTOROIDAL LIE ALGEBRAS
SOUVIK PAL AND S. ESWARA RAO
Abstract.
In this paper, we first construct the twisted full toroidal Liealgebra by an extension of a centreless Lie torus LT which is a multiloopalgebra twisted by several automorphisms of finite order and equippedwith a particular grading. We then provide a complete classificationof all the irreducible integrable modules with finite dimensional weightspaces for this twisted full toroidal Lie algebra having a non-trivial LT -action and where the centre of the underlying Lie algebra acts trivially. MSC:
Primary: 17B67; Secondary: 17B65, 17B70.
KEY WORDS:
Lie torus, integrable, twisted toroidal, level zero. Introduction
It is well-known that the Virasoro algebra plays a prominent role in therepresentation theory of affine Lie algebras as it acts on almost every highestweight module for the affine Lie algebra via the famous Sugawara operators.This remarkable connection eventually resulted in the construction of the so-called affine-Virasoro algebra [18, 21] which is the semi-direct product of theVirasoro algebra and the derived affine Kac-Moody algebra. Subsequentlythis Lie algebra has emerged to be an extremely important object of studyin various branches of mathematics and physics. In particular, its relationto Conformal Field Theory has been explained in great detail in [14]. Thetwisted multivariable generalization of this classical object is the subject ofour current paper.Our pursuit of a suitable replacement for the twisted affine Lie algebraleads us to the notion of a centreless Lie torus satisfying f gc condition (which means that the Lie torus is finitely generated as a module over itscentroid). It was shown in [1] that such a Lie torus of non-zero nullity hasa multiloop realization . Moreover in the one variable case, this multiloopalgebra is simply the loop algebra twisted by a single Dynkin automorphismused in the construction of twisted affine Kac-Moody algebras [19]. It isworth mentioning here that centreless Lie tori satisfying fgc condition play a key role in the theory of extended affine Lie algebras (EALAs for short)as they turn out to be the centreless cores of almost every
EALA [29].We first start with a centreless Lie torus LT (see Definition 2.2) andhenceforth consider its universal central extension LT = LT ⊕ Z ( m ) (seeSubsection 2.4). The next step is to generalize the Virasoro algebra in themultivariable setup. To this end, let us consider the algebra of derivationson A ( m ) = C [ t ± m , · · · , t ± m n n ] which we shall denote by D ( m ). Unlike in thesingle variable case, this derivation algebra is centrally closed for n > Z ( m ) of LT which generalizes the Virasoro algebra [33]. We can now take the semi-direct product of LT with D ( m ) and finally obtain the twisted full toroidalLie algebra e τ (see Subsection 2.5). The centre of this resulting Lie algebrais spanned by only finitely many elements K , · · · , K n . If all these elementsact trivially on a e τ -module, we say that the level of this representation is zero , otherwise we say that the representation has non - zero level .The classification of irreducible integrable modules with finite dimensionalweight spaces for the untwisted as well as for the twisted affine Kac-Moodyalgebras have been carried out in [9, 10, 11, 12]. In the multivariable setup,if the n -tuple of automorphisms σ = ( σ , · · · , σ n ) corresponding to our Lietorus L ( g , σ ) (see (2.4)) are all taken to be identity, then e τ is simply the untwisted full toroidal Lie algebra (FTLA for short). In this context, a largeclass of modules have been explicitly constructed in [4, 5] through the useof vertex operator algebras. The irreducible integrable modules with finitedimensional weight spaces for FTLA have been classified in [36].The classification of some specif ic irreducible integrable representationsof non - zero level with finite dimensional weight spaces for the multiloopalgebra twisted by a single non-trivial automorphism first appeared in [15].Thereafter this was generalized for all irreducible integrable modules of non - zero level over the twisted full toroidal Lie algebra (TFTLA for short) in [2].Realizations of several modules of non - zero level have also been providedin [6] for the twisted case. The current paper addresses these problems forthe first time in case of level zero modules for TFTLA.The main purpose of our paper is to classify the irreducible integrablelevel zero modules with finite dimensional weight spaces for TFTLA havinga non-trivial action of the Lie torus. If the action of LT is trivial, then thesemodules eventually turn out to be irreducible over D ( m ) which have beenclassified in [7]. We now summarize the content of our paper.After constructing the TFTLA in Section 2, we observe that this Liealgebra e τ has a natural triangular decomposition (see (3.1)) given by e τ = e τ − ⊕ e τ ⊕ e τ + . NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 3
In Section 3, we prove that our module V is a highest weight module inthe sense of Definition 3.3 (see Proposition 3.4). In Section 4, we show thatthe action of Z ( m ) is in fact trivial on our level zero irreducible module (seeTheorem 4.1). We then recall some results from [13] and [22] in the followingsection and thereby utilize them in Section 6 where we construct a familyof level zero irreducible integrable modules with finite dimensional weightspaces for TFTLA and having a non-trivial LT -action (see Proposition 6.1).These are the first such realizations of level zero irreducible modules forTFTLA. In Section 7, we show that this collection completely exhausts allsuch irreducible integrable modules over TFTLA (see Corollary 7.9). Wethen proceed to Section 8 where we find the isomorphism classes of theaforementioned irreducible integrable modules (see Theorem 8.2). In the lastsection, we obtain the main result of our paper which conveys that the levelzero irreducible integrable modules with finite dimensional weight spacesand having a non-trivial action of the Lie torus are completely as well asuniquely determined by a finite dimensional irreducible gl n ( C )-module anda finite dimensional graded-irreducible g -module up to a shift of grading.We would like to emphasize that many of our results actually hold good fora more general class of irreducible highest weight modules that need not beintegrable (for instance, see Theorem 4.1). These can be used in the study ofhighest weight modules that may possibly help us to understand the famous Harish - Chandra modules . The irreducible Harish-Chandra modules havebeen classified over some well-known Lie algebras [16, 23, 25, 26]. Finallyobserve that our TFTLA is intimately connected to the twisted toroidalEALA (TTEALA for short) which is formed by just adjoining the subalgebra S ( m )= span { D ( u, r ) | ( u | r ) = 0 , u ∈ C n , r ∈ Γ } of D ( m ) to LT . Therepresentation theory of TTEALA is still in progress and most of the workhas been done only with regard to representations of non-zero level [6, 37].So we hope that this paper will also contribute to develop the representationtheory of EALAs, especially in case of level zero modules.2. Notations and Preliminaries
Throughout the paper, all the vector spaces, algebras and tensor productsare over the field of complex numbers C . We shall denote the set of integers,natural numbers, non-negative integers and non-zero complex numbers by Z , N , Z + and C × respectively.2.1. Full Toroidal Lie Algebras.
Consider a finite dimensional simple Liealgebra g equipped with a Cartan subalgebra h . Then g is endowed witha symmetric, non-degenerate bilinear form which remains invariant underevery automorphism of g . We shall denote this bilinear form by ( ·|· ). SOUVIK PAL AND S. ESWARA RAO
Let A = C [ t ± , · · · , t ± n ] be the algebra of Laurent polynomials in n variables.Consider the (untwisted) multiloop algebra given by L ( g ) = g ⊗ A which is a Lie algebra under the following pointwise bilinear operation:[ x ⊗ f, y ⊗ g ] = [ x, y ] ⊗ f g ∀ x, y ∈ g and f, g ∈ A. For any x ∈ g and k ∈ Z n , let x ( k ) = x ⊗ t k denote a typical element of themultiloop algebra L ( g ). For k ∈ Z n , write t k = t k · · · t k n n and define(2.1) Ω A = span { t k K i | ≤ i ≤ n, k ∈ Z n } , d A = span { n X i =1 k i t k K i } ⊆ Ω A . If we now consider the quotient space Z = Ω A /d A , then we know that L ( g ) = L ( g ) ⊕ Z is the universal central extension of L ( g ) (see [20, 27] for details). Henceforth L ( g ) also forms a Lie algebra under the following bracket operations:(1) [ x ( k ) , y ( l )] = [ x, y ]( k + l ) + ( x | y ) P ni =1 k i t k + l K i ,(2) Z is central in L ( g ).The final ingredient needed to define the (untwisted) full toroidal Lie algebrais the Lie algebra of derivations on A which we shall denote by D . Taking d i = t i ddt i for 1 i n which acts on A as derivations, we can also define D = span { t r d i : r ∈ Z n , i n } . It is now easy to verify that[ t r d i , t s d j ] = s i t r + s d j − r j t r + s d i . Again any d ∈ D can be extended to a derivation on L ( g ) by simply setting d ( x ⊗ f ) = x ⊗ df ∀ x ∈ g , f ∈ A which subsequently has a unique extension to L ( g ) via the action t r d i ( t s K j ) = s i t r + s K j + δ ij n X p =1 r p t r + s K p . Moreover it is well-known that D admits of two non-trivial 2-cocyles φ and φ with values in Z (see [3] for more details) where we have φ ( t r d i , t s d j ) = − s i r j n X p =1 r p t r + s K p ,φ ( t r d i , t s d j ) = r i s j n X p =1 r p t r + s K p . NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 5
Let φ be any arbitrary linear combination of φ and φ . Then we can definethe (untwisted) full toroidal Lie algebra (relative to g and φ ) by setting τ A = L ( g ) ⊕ Z ⊕ D (2.2)with the following bracket operations besides the commutator relations (1)and (2) recorded in Subsection 2.1.(1) [ t r d i , t s K j ] = s i t r + s K j + δ ij P np =1 r p t r + s K p ,(2) [ t r d i , t s d j ] = s i t r + s d j − r j t r + s d i + φ ( t r d i , t s d j ),(3) [ t r d i , x ⊗ t s ] = s i x ⊗ t r + s ∀ x ∈ g , r, s ∈ Z n , i, j n .2.2. Lie Tori and Twisted Full Toroidal Lie Algebras.
Let us fixany n ∈ N and suppose that we have n commuting automorphisms of afinite dimensional simple Lie algebra g given by σ , · · · , σ n with finite orders m , · · · , m n respectively. Put σ = ( σ , · · · , σ n ) , Γ = m Z ⊕ · · · ⊕ m n Z , G = Z n / Γ . Thus we have a natural map Z n −→ G ( ∼ = Z /m Z × · · · × Z /m n Z )( k , · · · , k n ) = k k = ( k , · · · , k n )(2.3)For 1 i n , let ξ i denote a m i − th primitive root of unity.Then we obtain an eigenspace decomposition of g given by g = M k ∈ G g k where g k = { x ∈ g | σ i x = ξ k i i x, i n } . It is well-known that g is a reductive Lie algebra (see [8, Proposition 4.1]),even with the possibility of being zero. Finally, let us define L ( g , σ ) = M k ∈ Z n g k ⊗ C t k (2.4)which is clearly a Lie subalgebra of the (untwisted) multiloop algebra L ( g ).The above Lie algebra L ( g , σ ) is known as the (twisted) multiloop algebraassociated to g and σ .For any finite dimensional simple Lie algebra g with a Cartan subalgebra h of g , we have the root space decomposition of g relative to h g = M α ∈ h ∗ g ,α where g ,α = { x ∈ g | [ h, x ] = α ( h ) x ∀ h ∈ h } . Set ∆( g , h ) = { α ∈ h ∗ | g ,α = (0) } . Then ∆ × = ∆( g , h ) \{ } is clearlyan irreducible reduced finite root system having at most two root lengths.Let ∆ × , sh denote the set of all non-zero short roots of ∆ . Define∆ × , en = ( ∆ × ∪ × , sh , if ∆ × is of type B l ∆ × , otherwise . SOUVIK PAL AND S. ESWARA RAO
Suppose that { α , · · · , α p } is a collection of simple roots of g with respectto h . Then Q = L pi =1 Z α i is the corresponding root lattice containingthe non-negative root lattice Q +1 = L pi =1 Z + α i . Also let ∆ × , +1 , en and ∆ × , − , en denote the respective set of positive and negative roots of the extended rootsystem. Also set ∆ , en = ∆ × , en ∪ { } .For λ, µ ∈ h ∗ , define a partial order relation ” ” on h ∗ by setting λ µ if and only if µ − λ = P pi =1 n i α i for some n , · · · , n p in Z + .Moreover, if ( · , · ) is a non-degenerate, symmetric and associative bilinearform on g , then we shall denote the set of all dominant integral weights of g with respect to h by P + g = { λ ∈ h ∗ | ( λ, α i ) ∈ Z + ∀ i = 1 , · · · , p } .Let us now define the notion of a Lie torus, which plays a central role in theconstruction of twisted full toroidal Lie algebras. In this paper, we shall usethe characterization of a Lie torus given in [1] and take it as our definitionrather than the axiomatic definition recorded in [40, 41]. For this purpose,we first need the following definition. Definition 2.1.
We say that a finite dimensional g -module V satisfiescondition ( M ) if(1) V is an irreducible module having dimension greater than one;(2) The weights of V relative to h are contained in ∆ , en . Definition 2.2.
A multiloop algebra L ( g , σ ) is called a Lie torus if(1) g ¯0 is a finite dimensional simple Lie algebra;(2) For ¯ k = 0 and g ¯ k = (0), g ¯ k ∼ = U ¯ k ⊕ W ¯ k , where U ¯ k is trivial as a g ¯0 -module and either W ¯ k is zero or it satisfies condition ( M );(3) | < σ , · · · , σ n > | = Q ni =1 | σ i | where | σ i | denotes the order of theautomorphism σ i (1 i n ) and | < σ , · · · , σ n > | is the order ofthe group generated by the σ i ’s.For the rest of this paper, we shall denote the Lie torus L ( g , σ ) by LT .Now by [6, Lemma A.1], we can choose a Cartan subalgabra h ¯0 of g suchthat h ¯0 ⊆ h . Then h ¯0 turns out to be an ad-diagonalizable subalgebra of g (see [28, Lemma 3.1.3]) and ∆ × = ∆ × ( g , h ¯0 ) is an irreducible (possibly non-reduced) finite root system (see [28, Proposition 3.3.5]). Let Q be the rootlattice corresponding to ∆ × . Finally put ∆ = ∆( g ¯0 , h ¯0 ) and ∆ = ∆ × ∪ { } .2.3. Properties of LT. [1](LT1) The Lie torus LT is Z n -graded as well as Q -graded. This is usuallyreferred to as a Lie Z n -torus of type ∆.(LT2) ∆ = ( ∆ , en , if ∆ × is of type B l ∆ , otherwise . NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 7
Universal Central Extension of Lie Torus.
Let LT be a Lie Z n -torus of type ∆ as defined earlier. Now just as in the case of τ A in (2.1), wecan similarly construct the spaces Ω A ( m ) and dA ( m ) for the smaller algebra A ( m ) = C [ t ± m , · · · , t ± m n n ]. Set Z ( m ) = Ω A ( m ) /dA ( m ) and define LT := LT ⊕ Z ( m ) . Then LT forms a Lie algebra under the following bracket operations:(1) [ x ( k ) , y ( l )] = [ x, y ]( k + l ) + ( x | y ) n X i =1 k i t k + l K i ;(2) Z ( m ) is central in LT . Remark 2.3. (1) Note that since ( . | . ) is invariant under automorphisms of g , it isevident that k + l ∈ Γ whenever ( x | y ) = 0 and thus the abovebracket operation on LT is well-defined.(2) LT is the universal central extension of LT (see [39, Corollary 3.27]).(3) LT is also a Lie Z n -torus of type ∆ (with respect to the axiomaticdefinition of a Lie torus recorded in [40]). This implies that LT isgenerated as a Lie algebra by the spaces ( LT ) α , α ∈ ∆.2.5. Twisted Full Toroidal Lie Algebras.
Consider the Lie algebra ofderivations on A ( m ) which we shall denote by D ( m ). Then similar to theuntwisted case, we can likewise take any linear span φ of the 2-cocyles φ and φ now defined on D ( m ) with values in Z ( m ). This subsequently givesus the corresponding twisted full toroidal Lie algebra by simply setting e τ = LT ⊕ Z ( m ) ⊕ D ( m )and prescribing the following commutator relations along with the bracketoperations (1) and (2) mentioned in Subsection 2.4.(1) [ t r d i , t s K j ] = s i t r + s K j + δ ij P np =1 r p t r + s K p ,(2) [ t r d i , t s d j ] = s i t r + s d j − r j t r + s d i + φ ( t r d i , t s d j ),(3) [ t r d i , x ⊗ t k ] = k i x ⊗ t r + k ∀ x ∈ g k , k ∈ Z n , r, s ∈ Γ , i, j n .The centre of e τ is clearly spanned by the elements K , · · · , K n . Remark 2.4.
Note that by definition, the universal central extension of g ¯0 ⊗ A ( m ) is simply g ¯0 ⊗ A ( m ) ⊕ Z ( m ) (see (2.1)). It is now immediate thatthe (untwisted) full toroidal Lie algebra τ A ( m ) = g ¯0 ⊗ A ( m ) ⊕ Z ( m ) ⊕ D ( m )is a subalgebra of e τ . Moreover if we take D =span { d , · · · , d n } , then e τ alsocontains the subalgebra, namely the graded Lie torus given by f LT = LT ⊕ Z ( m ) ⊕ D. SOUVIK PAL AND S. ESWARA RAO
Roots and Coroots.
Consider the abelian subalgebra of e τ given by e h = h ⊕ n X i =1 C K i ⊕ n X i =1 C d i . In order to describe the roots of e τ , let us first define δ i ∈ e h ∗ by setting δ i ( h ) = 0 , δ i ( K j ) = 0 and δ i ( d j ) = δ ij . Put δ β = n X i =1 β i δ i for β = ( β , · · · , β n ) ∈ C n . For k ∈ Z n , we shall refer tothe vector δ k + γ as the translate of δ k by the vector γ = ( γ , · · · , γ n ) ∈ C n .Set e ∆ = { α + δ k | α ∈ ∆ , en , k ∈ Z n } . Then we have the root spacedecomposition of e τ with respect to e h given by(2.5) e τ = M γ ∈ e ∆ ∪{ } e τ γ where e τ α + δ k = g ¯ k ( α ) ⊗ C t k , if α = 0 (cid:0) h ⊗ C t k (cid:1) ⊕ (cid:0) n X i =1 C t k K i (cid:1) ⊕ (cid:0) n X i =1 C t k d i (cid:1) , if α = 0 , k ∈ Γ g ¯ k (0) ⊗ C t k , if α = 0 , k / ∈ Γ . This shows that the the roots of e τ are given by e ∆. Furthermore, let e ∆ + = { α + δ k | α ∈ ∆ × , +0 , en , k ∈ Z n } , e ∆ − = { α + δ k | α ∈ ∆ × , − , en , k ∈ Z n } denote the set of all positive and negative roots of e τ respectively.A root γ = α + δ k is said to be real if α = 0, else we call it a null root. Let e ∆ re be the set of all real roots of e τ . For each γ ∈ e ∆ re , define the correspondingco-root γ ∨ := α ∨ + α | α ) n X i =1 k i K i where α ∨ is the co-root of α ∈ ∆ , en .3. Integrable Modules
In this section, we introduce the notions of integrable representations andlevel zero highest weight modules. We conclude this section by ultimatelyshowing that every irreducible integrable module over e τ of level zero is in facta highest weight module with respect to a suitable triangular decomposition. Definition 3.1. A e τ -module V is called integrable if(1) V = M λ ∈ e h ∗ V λ , where V λ = { v ∈ V | h.v = λ ( h ) v ∀ h ∈ e h } ,(2) For each x ∈ g k ( α ) ⊗ C t k ( α = 0) and every v ∈ V , there exists some m = m ( α, k, v ) ∈ N such that x m .v = 0. NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 9
For an integrable module V over e τ , we shall denote the set of all weightsof V by P ( V ) = { µ ∈ e h ∗ | V µ = (0) } . For any µ ∈ P ( V ), V µ is the called theweight space of V having weight µ and the elements of V µ are referred to asthe weight vectors of V with weight µ . In this paper, our goal is to classify allthose irreducible integrable modules over e τ having finite dimensional weightspaces with respect to e h where the central elements K , · · · , K n act trivially.These modules are commonly known as level zero modules. Lemma 3.2.
Let V be an integrable module for e τ with finite dimensionalweight spaces. Then (1) P ( V ) is invariant under the action of W . (2) dim( V λ ) = dim( V wλ ) ∀ λ ∈ P ( V ) and w ∈ W . (3) If λ ∈ P ( V ) and γ ∈ e ∆ re , then λ ( γ ∨ ) ∈ Z . (4) If λ ∈ P ( V ) and γ ∈ e ∆ re with λ ( γ ∨ ) > , then λ − γ ∈ P ( V ) .Proof. The proof is similar to [38, Lemma 2.3]. Note that the irreduciblityof V is not required. (cid:3) The root space decomposition of e τ in (2.5) automatically induces a naturaltriangular decomposition of e τ given by e τ = e τ − ⊕ e τ ⊕ e τ + (3.1)where e τ − = M α ∈ e ∆ − , k ∈ Z n (cid:18) g ¯ k ( α ) ⊗ C t k (cid:19) , e τ = (cid:18) X k ∈ Z n g ¯ k (0) ⊗ C t k (cid:19) M (cid:18) X ≤ i ≤ n, k ∈ Γ C t k K i (cid:19) M D ( m ) , e τ + = M α ∈ e ∆ + , k ∈ Z n (cid:18) g ¯ k ( α ) ⊗ C t k (cid:19) . We shall denote the universal enveloping algebra of e τ by U ( e τ ). Definition 3.3. V is said to be a level zero highest weight module for e τ ifthere exists a non-zero weight vector v in V such that(1) V = U ( e τ ) v .(2) e τ + .v = 0.(3) U ( e τ ) v is an irreducible module over e τ .(4) The elements K , · · · , K n act trivially on V .For the rest of this section, let us fix an irreducible integrable module V over e τ of level zero having finite dimensional weight spaces with respect to the extended Cartan subalgebra e h . Further set V + = { v ∈ V | e τ + .v = (0) } . Proposition 3.4.
The highest weight space V + is non-zero.Proof. By Remark 2.4, V is clearly an integrable module over f LT havingfinite dimensional weight spaces with respect to e h . The desired result isnow a direct consequence of [31, Proposition 3.7]. Also see [34, Lemma 2.6]and [9, Theorem 2.4(ii)]. Finally observe that the irreducibility conditionprescribed in [31, Proposition 3.7] is actually redundant. (cid:3) Lemma 3.5. (1) V + is an irreducible module over e τ . (2) The weights of V + are the same up to a translate of the null roots,i.e. there exists a unique λ ∈ h ∗ and some β ∈ C n (not necessarilyunique) such that P ( V + ) ⊆ { λ + δ r + β | r ∈ Z n } .Proof. (1) The required result easily follows from the irreducibility of V andby an application of the PBW theorem.(2) The proof proceeds verbatim as in the case of [31, Lemma 4.5(1)]. Justobserve that h commutes with D ( m ). (cid:3) Corollary 3.6. V is a level zero highest weight module for e τ .Proof. This is a trivial consequence of Proposition 3.4 and Lemma 3.5. (cid:3)
Remark 3.7. If λ = 0 in Lemma 3.5, then using the integrability of V ,it is easy to conclude that the only possible weights of V are δ γ ( γ ∈ C n ).Then from Remark 2.3, it trivially follows that LT acts trivially on V . Thisultimately reduces our problem to classifying all the irreducible modules forthe algebra D ( m ) having finite dimensional weight spaces with respect tothe space of degree derivations D =span { d , · · · , d n } . The classification ofthese irreducible modules can be found in [7]. In the present paper, we shallprimarily focus on those irreducible modules where λ = 0.4. Action Of The Central Operators
Throughout this section, unless otherwise explicitly stated, V will alwaysstand for a level zero irreducible (but not necessarily integrable) highestweight module for e τ (see Definition 3.3) having finite dimensional weightspaces with respect to e h . We shall also assume that the Cartan subalgebra h of g does not act trivially on the highest weight space V + . Under theseassumptions, our main aim in this section is to prove the following result. Theorem 4.1. Z ( m ) acts trivially on V . NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 11
We need some preparation to prove this theorem. First observe thatby the initial assumptions on our module V and Lemma 3.5, there existsΛ ∈ P ( V ) with Λ | h = 0 satisfying h.w = Λ( h ) w ∀ h ∈ h , w ∈ V + . Pick some h ∈ h such that Λ( h ) = 0. It is now clear that V + = M m ∈ Z n V + ( m )with V + ( m ) = { v ∈ V + | d i .v = (Λ( d i ) + m i ) v, i n } which shows that V + is a Z n -graded e τ -module with finite dimensional graded components. Lemma 4.2.
Let h ⊗ t k ∈ e τ where k ∈ Γ \ { } . Suppose there exists anon-zero element w ∈ V + such that ( h ⊗ t k ) .w = 0 . Then h ⊗ t k is locallynilpotent on V + .Proof. It is trivial to check that for any r ∈ Γ and i = 1 , · · · , n , we have( h ⊗ t k ) (( t r d i ) .w ) = 0 . Similarly we can show that if x r ⊗ t r ∈ (cid:0) g r (0) ⊗ C t r (cid:1) where r ∈ Z n , then( h ⊗ t k ) (( x r ⊗ t r ) .w ) = 0 . Consequently by induction on p , it can be further deduced that( h ⊗ t k ) p +1 (cid:0) ( t r d i · · · t r p d i p ) .w (cid:1) = 0 , ( h ⊗ t k ) p +1 (cid:18)(cid:0) ( x s ⊗ t s ) · · · ( x s p ⊗ t s p ) (cid:1) .w (cid:19) = 0for all p ∈ N , r · · · , r p ∈ Γ and s · · · , s p ∈ Z n with 1 i , · · · , i p n .Finally since h ⊗ A ( m ) commutes with Z ( m ), the lemma now directlyfollows from the irreducibility of V + over e τ . (cid:3) Next let us record some results which can be proved using the aboveLemma 4.2 and Remark 2.4 in a more or less similar manner as in [17] and[36] by considering h ⊗ t k instead of t m k and replacing c by Λ( h ). Thelast assertion follows using our Theorem 4.1. Lemma 4.3. (1) dim V + ( k ) = dim V + ( k + r ) = N k ∈ N ∀ k ∈ Z n , r ∈ Γ . Inparticular, we have dim V + ( r ) = dim V + ( s ) = N (say) ∀ r, s ∈ Γ . (2) Let w be a non-zero element in V + such that ( t r K i ) .w = 0 for some r ∈ Γ \ { } and i n . Then t r K i acts locally nilpotently on V + . (3) For each r ∈ Γ , ( t r K i ) N .V + = (0) ∀ i = 1 , · · · , n . (4) (cid:0) ( h ⊗ t r )( h ⊗ t s ) (cid:1) .v = Λ( h )( h ⊗ t r + s ) .v ∀ r, s ∈ Γ , v ∈ V + . Lemma 4.4.
Let n > and suppose that for each i n , there existsa fixed N ∈ N such that ( t r K i · · · t r N K i ) .V + = (0) ∀ r , · · · , r N ∈ Γ \ { } .Then there exists a non-zero vector v in the highest weight space V + suchthat t r K i .v = 0 ∀ r ∈ Γ , i n .Proof. The proof of this lemma is algorithmic, i.e. we shall provide a preciseprocedure that will enable us to acquire a common eigenvector. First notethat if N = 1, then we are done. So assume that N >
1. Now by ourhypothesis, we can find a non-zero v ∈ V + satisfying t r K .v = 0 ∀ r ∈ Γ.If t s K .v = 0 for some s ∈ Γ, then repeatedly act central operators ofthe form t p K to this vector until we obtain a non-zero vector of the form w = ( t r K · · · t r m K ) .v for some m ∈ N such that t p K .w = 0 ∀ p ∈ Γ. Ourhypothesis guarantees that such a vector indeed exists and in fact m N − t r K .w = 0 = t r K .w = 0 ∀ r ∈ Γ. Continuing this processfor the remaining operators t r K i , we finally get our common eigenvector. (cid:3) Proof of Theorem 4.1 . If n = 1, then the assertion clearly holds good.Let us first fix any 1 i n . Since we are assuming n >
2, we can nowchoose 1 j n such that i = j . Then using Lemma 4.3, we have0 = t r d j (cid:0) ( t s K i ) N .v (cid:1) = N s j (cid:18) t r + s K i (cid:0) ( t s K i ) N − .v (cid:1)(cid:19) ∀ r , s ∈ Γ , v ∈ V + from which it clearly follows that (cid:0) t r + s K i ( t s K i ) N − (cid:1) .V + = (0) ∀ r ∈ Γ and s ∈ Γ satisfying s j = 0 . This consequently implies that t r d j (cid:18) s j t r + s K i (cid:0) ( t s K i ) N − .v (cid:1)(cid:19) = 0 ∀ r , r , s ∈ Γ , v ∈ V + from which we directly get( N − s j (cid:18) t r + s K i t r + s K i (cid:0) ( t s K i ) N − .v (cid:1)(cid:19) = 0for all r , r ∈ Γ with s ∈ Γ satisfying s j = 0 and v ∈ V + .This immediately reveals that (cid:0) t r + s K i t r + s K i ( t s K i ) N − (cid:1) .V + = (0) ∀ r , r ∈ Γ and s ∈ Γ with s j = 0 . By repeating the above argument another ( N −
2) times, we thus obtain( t r + s K i · · · t r N + s K i ) .V + = (0)for all r , · · · , r N ∈ Γ and s ∈ Γ satisfying s j = 0.Let s , · · · , s N ∈ Γ \ { } be arbitrary. Choose any s ∈ Γ such that s j = 0and thereby set r k = s k − s ∀ k = 1 , · · · , N . This finally gives us( t s K i · · · t s N K i ) .V + = (0) ∀ i = 1 , · · · , n. NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 13
Subsequently by Lemma 4.4, there exists a non-zero vector v ∈ V + suchthat ( t r K i ) .v = 0 ∀ r ∈ Γ , i n . Now it is trivial to check that W = { v ∈ V | ( t r K i ) .v = 0 ∀ r ∈ Γ , i n } is a e τ -submodule of V . Therefore we are done by the irreducibility of V . Remark 4.5.
The above theorem thereby allows us to fully restrict ourattention to irreducible modules over b τ having finite dimensional weightspaces with respect to b h where b τ = LT ⊕ D ( m ) and b h = h ⊕ D .The next result follows from Corollary 3.6, Remark 3.7 and Theorem 4.1. Corollary 4.6.
Let V be a level zero irreducible integrable module for e τ having finite dimensional weight spaces with respect to e h . Then Z ( m ) actstrivially on V . Remark 4.7. (1) The proof of Corollary 4.6 for the untwisted full toroidal Lie algebraprovided in [36, Theorem 4.1] is not complete. Nevertheless, theirtheorem follows instantaneously from our Theorem 4.1.(2) The above corollary can be also derived independently based on thearguments sketched in [31, Theorem 4.9]. But we chose a differentapproach in this case as it helps us to conclude that the assertionactually holds good for a much bigger class of level zero highestweight modules that may not be integrable.5.
Finite Dimensional Graded-Irreducible Representations andEvaluation Modules
This section is mainly devoted to the recollection of a few notions andresults from [13] and [22] which will be useful for our classification problem.At the end of this section, we shall prove an important result that will beutilized in the next section.5.1.
Modules twisted by an automorphism. [13, Subsection 3.1]Recall that g is a finite dimensional simple Lie algebra also equipped witha G -grading. Putting b G = { f : G −→ C × | f is a group homomorphism } which is a finite group, it is immediate that any χ ∈ b G gives rise to anautomorphism of g defined by α χ ( x ) = χ ( k ) x ∀ x ∈ g k , k ∈ Z n . For any g -module V and φ ∈ Aut( g ), we also have the twisted g -module V φ ,which is nothing but the same vector space V , but now twisted by a g -actiongiven by x.v = φ ( x ) v for all x ∈ g and v ∈ V . Thus we find that the groupAut( g ) and subsequently b G acts (on the right) on the isomorphism classes of g -modules which thereby induces a natural action of b G on P + g . Consequentlythe action of b G on the isomorphism classes of finite dimensional irreducible g -modules can be represented as an action of the Dynkin automorphismsof g on the set of dominant integral weights of g by simply permuting thevertices of the Dynkin diagram and the corresponding fundamental weights.Let us denote this action by ’ ◦ ’. The finite dimensional irreducible highestweight module for g with highest weight λ will be denoted by V ( λ ).5.2. Finite dimensional graded-irreducible modules.
Suppose that λ ∈ P + g with graded Schur index S( λ ) ∈ N (see [13, Section 3] for theprecise definition of S( λ )). If b Gλ = { λ , · · · , λ l } , then it can be shown that L li =1 V ( λ i ) ⊕ S ( λ ) is a finite dimensional G -graded-irreducible module for g (see [13, Subsection 3.3]). Conversely, up to an isomorphism of ( ungraded )modules over g , every finite dimensional G -graded-irreducible module for g is of the form L li =1 V ( λ i ) ⊕ S ( λ ) where b Gλ = { λ , · · · , λ l } for some λ ∈ P + g (see [13, Theorem 8]). Moreover it is also known that b Gλ i = b Gλ j and S ( λ i ) = S ( λ j ) ∀ i, j l (see [13, Subsection 3.4]). We finally remarkthat if G is trivial, then we always have S( λ )=1.5.3. Finite dimensional evaluation modules.
For any a ∈ ( C × ) n and k ∈ N n , write a k = Q ni =1 a k i i and a ( k ) = ( a k , · · · , a k n n ). Consider m ∈ N n as mentioned in Subsection 2.2. We say that a ∈ C n is a root of unity for m if we have a ( m ) = (1 , · · · ,
1) = 1. Let U ( m ) represent the set of allroots of unity for m in C n . We can now put a LT -module structure on theirreducible g -module V ( λ ) by setting( x ⊗ t k ) .v = a k ( x.v ) ∀ x ∈ g k , k ∈ Z n , v ∈ V ( λ ) . We shall denote this LT -module by ev a V ( λ ). These modules are commonlyknown as evaluation modules in the literature which can be easily shownto be irreducible. Refer to [22, Section 4] for more details. Proposition 5.1. [22, Theorem 5.4]
The finite dimensional LT -modules ev a V ( λ ) and ev b V ( µ ) are isomorphic if and only if we have a ( m ) = b ( m ) and µ = λ ◦ γ where γ is the outer part of ω ∈ Aut( g ) that is defined by ω ( x ) = ( a k /b k ) x for all k ∈ Z n and x ∈ g k . Corollary 5.2.
Let λ, µ ∈ P + g such that µ ∈ b Gλ . Then for any given a ∈ ( C × ) n , there exists some b ∈ ( C × ) n such that ev b V ( λ ) ∼ = ev a V ( µ ) asirreducible LT -modules.Proof. We know that G ∼ = G × · · · × G n where G i = h g i i (say) ∀ i n .By hypothesis, there exists a Dynkin automorphism γ satisfying µ = λ ◦ γ . NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 15
Moreover γ is the outer part of some ω ∈ Aut( g ) given by ω ( x ) = χ ( k ) x ∀ k = ( k , · · · , k n ) ∈ Z n , x ∈ g k where χ ∈ b G . Now since the group b G is canonically isomorphic to thegroup c G × · · · × c G n , we can use this identification to get a unique element( χ , · · · , χ n ) ∈ c G × · · · × c G n corresponding to our χ ∈ b G . Moreover as | G i | = m i , we have χ i ( g i ) = ξ d i i for some d i ∈ Z . Finally setting b i = ξ d i i a i for each 1 i n , we thereby obtain w ( x ) = (cid:18) n Y i =1 χ i ( k i ) (cid:19) x = ( b k /a k ) x for all k ∈ Z n and x ∈ g k where b = ( b , · · · , b n ) ∈ ( C × ) n . The assertion isnow a direct consequence of Proposition 5.1. (cid:3) Remark 5.3.
The finite dimensional irreducible LT -modules have beencompletely classified in [22] and [30].6. An Explicit Realization
In this section, we shall extract a family of level zero integrable modulesfor e τ having finite dimensional weight spaces with respect to e h and eventuallyshow that they are all irreducible.Consider ψ ∈ P + sl n and c ∈ C . It is well-known that these parameters give riseto a unique finite dimensional irreducible gl n ( C )-module, say V = V ( c, ψ ).Again take any λ ∈ P + g \ { } (which we shall henceforth denote by ( P + g ) × )with b Gλ = { λ , · · · , λ l } and graded Schur index S ( λ ) = k (say). Then fromour discussion in Subsection 5.2, it is evident that V ′ = L li =1 V ( λ i ) ⊕ k is afinite dimensional G -graded-irreducible g -module. Observe that the numberof irreducible summands occuring in the above decomposition of V ′ is equalto kl = N (Say). Take V = L Nν =1 V ν where the irreducible components V ν of V are all coming from the decomposition of V ′ . For example, if k = l = 2,there are precisely five choices for our V . Now since b Gλ = { λ , · · · , λ l } and b Gλ i = b Gλ j for all 1 i, j l , we can invoke Corollary 5.2 to pick evaluationpoints a , · · · , a N ∈ U ( m ) such that the twisted g -module (which we shallagain denote by V ) defined by x. (cid:0) N X ν =1 v ν (cid:1) = N X ν =1 a νl ( x.v ν ) ∀ x ∈ g l , v ν ∈ V ν (6.1)is isomorphic to V ′ as ungraded g -modules. This immediately suggests thatthis twisted module is G -graded-irreducible over g . Let { E ij | i, j n } be the standard basis of gl n ( C ). Now pick any β = ( β , · · · , β n ) ∈ C n and define an action of e τ on V ⊗ V ⊗ A by setting( x ⊗ t l ) . (cid:0) N X ν =1 v ⊗ v ν ⊗ t k (cid:1) = N X ν =1 v ⊗ a νl ( x.v ν ) ⊗ t k + l ; t r d i . (cid:0) N X ν =1 v ⊗ v ν ⊗ t k (cid:1) = ( k i + β i ) (cid:0) N X ν =1 v ⊗ v ν ⊗ t k + r (cid:1) + N X ν =1 n X j =1 (cid:0) ( r j m j E ji ) .v (cid:1) ⊗ v ν ⊗ t k + r ; t r K i . (cid:0) N X ν =1 v ⊗ v ν ⊗ t k (cid:1) = 0for all x ∈ g l , k, l ∈ Z n , r = ( r m , · · · , r n m n ) ∈ Γ , v ∈ V , v ν ∈ V ν and 1 i n . It can be verified that V ⊗ V ⊗ A is a e τ -module. Taking V = L k ∈ G V ,k , let us further define V ′ := L k ∈ Z n V ⊗ V ,k ⊗ C t k which isagain a submodule of V ⊗ V ⊗ A . Proposition 6.1. V ′ is a level zero irreducible integrable e τ -module withfinite dimensional weight spaces having a non-trivial LT -action.Proof. By our construction, the twisted module V is a G -graded-irreduciblemodule over g which implies that L k ∈ Z n V ,k ⊗ C t k is an irreducible moduleover LT ⊕ D where D =span { d , · · · , d n } . The desired result can be nowdeduced using [35, Proposition 2.8] which conveys that V ⊗ A ( m ) is anirreducible module for D ( m ) ⋊ (cid:0) h ⊗ A ( m ) (cid:1) . Finally observe that the non-zero λ ∈ P + g ensures us that LT does not act trivially on V ′ . (cid:3) Remark 6.2.
The irreducible module V ′ over e τ is completely determinedby the quadruplet ( ψ, c, λ, β ) ∈ P + sl n × C × ( P + g ) × × C n together with thecorresponding multiplicities, say p , · · · , p l of V ( λ ) , · · · , V ( λ l ) respectivelythat add up to N ( p i is allowed to be 0) and some appropriately chosenevaluation points a , · · · , a N ∈ U ( m ) where N = kl , k = S ( λ ) and l = | b Gλ | .So we shall denote this module V ′ by V c,ψλ,β ( a , · · · , a N ; p , · · · , p l ). Nowfor any arbitrary choice of non-negative integers p , · · · , p l with P li =1 p i = N , we clearly have V c,ψλ,β ( a , · · · , a N ; p , · · · , p l ) ∼ = V c,ψλ,β (1 , · · · , k, · · · , k ) asirreducible e τ -modules where a , · · · , a N ∈ U ( m ) are chosen in a way (maynot be unique) such that the twisted g -module V (see (6.1)) used in ourconstruction of V c,ψλ,β ( a , · · · , a N ; p , · · · , p l ) becomes G -graded-irreducible.Thus we can simply drop the evaluation points and denote these irreduciblemodules over e τ by just V ( ψ, c, λ, β ). Finally note that S ( λ ) and b Gλ areautomatically fixed after our initial choice of λ . NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 17 Classification of Irreducible Modules
In this section, V will always denote an irreducible integrable level zeromodule for e τ with finite dimensional weight spaces and having a non-trivialaction of the Lie torus LT . Towards the end of this section, we shallshow that these irreducible integrable modules are essentially of the form V ( ψ, c, λ, β ) as described in the previous section.Recall that b τ = LT ⊕ D ( m ) which like e τ has a triangular decompositiongiven by b τ = b τ − ⊕ b τ ⊕ b τ + where b τ ± = e τ ± and b τ = e τ \ Z ( m ). Set ◦ g = { x ∈ g | [ h, x ] = 0 ∀ h ∈ h } . Now since ◦ g is invariant under the σ i ’s, this thereby induces a G -grading on ◦ g with ◦ g k = g k ∩ ◦ g ∀ k ∈ G . Denote the corresponding multiloop algebra by L ( ◦ g , σ ) = M k ∈ Z n ◦ g k ⊗ C t k . We now introduce a particular Lie subalgebra of D ( m ) which will play animportant role not only in this section, but also in the rest of this paper.Let D ( u, r ) = P ni =1 u i t r d i and I ( u, r ) = D ( u, r ) − D ( u, ∀ u ∈ C n , r ∈ Γ.It can be easily verfied that for u, v ∈ C n and r, s ∈ Γ,[ I ( u, r ) , I ( v, s )] = ( v, r ) I ( u, r ) − ( u, s ) I ( v, s ) + I ( w, s + r )where ( · , · ) is the standard inner product on C n and w = ( u, s ) v − ( v, r ) u .This immediately reveals that the subspace I := span { I ( u, r ) : u ∈ C n , r ∈ Γ } forms a subalgebra of D ( m ). Furthermore let us also set e L = I ⋉ L ( ◦ g , σ ) and W = span { ( h ⊗ t r ) .v − v | r ∈ Γ , v ∈ V + } . Now since Λ( h ) = 0, we may assume that Λ( h ) = 1 and therefore byLemma 4.3, it easily follows that W is a e L -module. This again implies that e V := V + /W is also a e L -module. Finally observe that b τ = L ( ◦ g , σ ) ⊕ D ( m ). Remark 7.1.
Note that e L is naturally G -graded with e L = I ⋉ h ⊗ A ( m ).Moreover the Z n -grading on V + again gives rise to a G -grading on both V + and W via the natural map given in (2.3). This in turn shows us that thequotient e V is also G -graded. Lemma 7.2. (1) W is a proper e L -submodule of V + . (2) e V is a finite dimensional e L -module. Proof.
Let z i = h ⊗ t m i i for each i = 1 , · · · , n . Since Λ( h ) = 1, it can beeasily deduced from Lemma 4.3 that W = span { z i .v − v | v ∈ V + , i n } . (1) With regard to the above discussion, the proof of the assertion followsverbatim as in [31, Proposition 5.4(3)].(2) From Lemma 4.3, it is evident that all the z i ’s are invertible on V + . Thedesired result can now be established by proceeding similarly as in Claim 2and Claim 3 in [34, Theorem 4.5] and then applying our Lemma 3.5. (cid:3) Consider β ∈ C n as obtained in Lemma 3.5. Then for any e L -module V ,we can now put a b τ -module structure on L ( β, V ) = V ⊗ A by setting x ⊗ t k . ( v ⊗ t s ) = (cid:0) ( x ⊗ t k ) .v (cid:1) ⊗ t k + s , (7.1) D ( u, r ) . ( v ⊗ t s ) = ( I ( u, r ) .v ) ⊗ t r + s + ( u, s + β )( v ⊗ t s + r )(7.2)for all v ∈ V , x ∈ ◦ g k with k, s ∈ Z n , u ∈ C n and r ∈ Γ. It can be verifiedthat L ( β, V ) is indeed a b τ -module (see [2, Section 8]).For v ∈ V + , let us denote its image in e V by v and define e φ : V + −→ L ( β, e V ) v v ⊗ t k , v ∈ V + ( k ).This map is clearly a non-zero b τ -module homomorphism and thus from theirreducibility of V + , it follows that V + ∼ = e φ ( V + ) is a b τ -submodule of L ( β, e V ).Recall that e V = L p ∈ G e V p and for every p ∈ G , set L ( β, e V )( p ) = { v ⊗ t k + r + p , v ∈ e V k , r ∈ Γ , k ∈ Z n } which thereby forms a b τ -module under the usual action induced from (7.1).The following results can be deduced similarly as in [2] by simply workingwith our Lie algebra b τ instead of L . Proposition 7.3. (1) V + ∼ = L ( β, e V )(0) as b τ -modules. (2) e V is a G -graded-irreducible module over e L . (3) e V is a completely reducible module over e L and all its irreduciblecomponents are mutually isomorphic as (cid:0) I ⋉ L ( ◦ g , σ ) (cid:1) -modules. For any d ∈ N , k ∈ Z n , x ∈ ◦ g k and r , · · · , r d ∈ Γ, define x ( k, r , · · · , r d ) := x ⊗ t k − P i x ⊗ t k + r i + P i Ker ( ev )= F . But since this map is onto, weclearly have L ( ◦ g , σ ) /F ∼ = ◦ g as Lie algebras. Lemma 7.4. Let k ∈ Z n , x ∈ ◦ g k , s , · · · , s d ∈ Γ and d ∈ N . If theelement x ( k, s , · · · , s d ) ∈ F d acts by the scalar λ ( k, s , · · · , s d ) on a finitedimensional representation ( ρ ′ , V ′ ) of e L , then λ ( k, s , · · · , s d ) = 0 .Proof. Let r ∈ Γ be arbitrary. Now we have[ I d ( u, r, s , · · · , s d ) , x ⊗ t k ] = ( u, k ) x ( k + r, s , · · · , s d ) ∀ u ∈ C n . But as V ′ is finite dimensional, we can simply take the trace operator afterapplying the map ρ ′ on both sides to finally obtain λ ( k + r, s , · · · , s d ) = 0 ∀ k ∈ Z n \ { } , r ∈ Γ . This immediately gives us the desired conclusion. (cid:3) We already know from our Proposition 7.3 that there exist irreducible e L -modules f M , · · · , g M N such that e V = L Nν =1 g M ν for some N ∈ N with f M i ∼ = f M j as (cid:0) I ⋉ L ( ◦ g , σ ) (cid:1) -modules ∀ i, j N . It is also clear that wehave L ( ◦ g , σ ) = h ⊗ A ( m ). Theorem 7.5. For each ν N, g M ν is a finite dimensional irreduciblemodule for the direct sum gl n ( C ) ⊕ ◦ g . Thus e V is the unique completelyreducible finite dimensional G -graded-irreducible quotient of V + over e L . Proof. Using our Lemma 7.4, we can give a similar argument as presented in[2, Theorem 9.4] to show that the Lie algebra I ⊕ F actually acts trivially onevery g M ν . The theorem now directly follows from (7.3). The final assertionis clear from the definitions of F and the e L -submodule W of V + . (cid:3) Remark 7.6. The second author would like to take this opportunity tocorrect some results in [2]. The result stated in [2, Proposition 8.12(1)] is notcorrect as the eigenspaces need not be isomorphic as g -modules, rather theyare mutually isomorphic only as g -modules where g is a graded Lie algebra.This is simply because the map defined at the beginning of Page-99 in [2] isa g -module homomorphism and not necessarily a g -module homomorphismas claimed by the authors. As a result, Theorem 8.3 in [2] is also affectedand we ultimately obtain e V to be a completely reducible e L -module with itsirreducible components being isomorphic as (cid:0) I ⋉ L ( ◦ g , σ ) (cid:1) -modules. Finallyalthough [2, Lemma 9.5] is correct, the proof provided there is not valid inthis context due to the reasons stated above and thereby cannot be employedto prove [2, Theorem 9.4]. Nonethetheless, the trace argument given in ourLemma 7.4 can be utilized to resolve this problem and complete the proof ofTheorem 9.4 in [2]. We would like to also point out that [2, Theorem 8.3] isagain recalled in [37] and [38], but not effectively used in both these papers.For each 1 ν N , consider any surjective Lie algebra homomorphism φ ν : e L −→ gl n ( C ) ⊕ ◦ g satisfying Kerφ ν = F ⊕ I and which maps L ( ◦ g , σ ) onto ◦ g and I onto gl n ( C ).Set φ ν | I = φ ′ ν and φ ν | L ( ◦ g ,σ ) = φ ′′ ν . Using this identification map φ ν and thenapplying [24, Lemma 2.7], we can thereby find finite dimensional irreduciblerepresentations, say f V ν and g W ν of gl n ( C ) and ◦ g respectively so that we havean isomorphism of irreducible (cid:0) gl n ( C ) ⊕ ◦ g (cid:1) -modules given by g M ν ∼ = f V ν ⊗ g W ν ∀ ν N. Note that due to (7.3), it suffices to only consider the same identifiation map φ ′ ν = π for defining a I -module structure on all the f V ν ’s. Consequently allthe gl n ( C )-modules f V ν remain isomorphic to each other which permits usto take f V ν ∼ = f V as gl n ( C )-modules for all 1 ν N . Also consider f W := P Nν =1 g W ν which is a L ( ◦ g , σ )-module where L ( ◦ g , σ ) sits diagonallyinside L ( ◦ g , σ ) ⊕ N which we can then identify with ◦ g ⊕ N . Since all the W ν ’sare irreducible, we can assume (without any loss of generality) that thissum is direct . We can now invoke Proposition 7.3 to infer that f W is a G -graded-irreducible L ( ◦ g , σ )-module where we are allowed to take f V to be NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 21 zero -graded as I lies in the zeroth graded component of e L .We now define a b τ -module structure on f V ⊗ f W ⊗ A by setting( x ⊗ t l ) . (cid:0) N X ν =1 v ⊗ w ν ⊗ t k (cid:1) = N X ν =1 v ⊗ (cid:0) φ ′′ ν ( x ⊗ t l ) .w ν (cid:1) ⊗ t k + l ; t r d i . (cid:0) N X ν =1 v ⊗ w ν ⊗ t k (cid:1) = ( k i + β i ) (cid:0) N X ν =1 v ⊗ w ν ⊗ t k + r (cid:1) + N X ν =1 n X j =1 (cid:0) ( r j m j E ji ) .v (cid:1) ⊗ w ν ⊗ t k + r for all x ∈ ◦ g l , k, l ∈ Z n , r = P nj =1 r j m j e j ∈ Γ , v ∈ f V , w ν ∈ f W ν and1 i n . Then it can be trivially verified that f V ⊗ f W ⊗ A is a b τ -module.Let f W = L k ∈ G f W ,k and define T ′ := L k ∈ Z n f V ⊗ f W ,k ⊗ C t k which isa submodule of f V ⊗ f W ⊗ A . One can now check that T ′ is in fact anirreducible module over b τ and henceforth directly define a natural non-zero b τ -module homomorphism between L ( β, e V )(0) and T ′ . This subsequentlybecomes an isomorphism due to the irreduciblity of both L ( β, e V )(0) and T ′ .Finally using Proposition 7.3, it is trivial to see that V + ∼ = T ′ as b τ -modules.Consider the following triangular decomposition of LT given by LT = e τ − ⊕ L ( ◦ g , σ ) ⊕ e τ + . Let f V stand for the unique G -graded-irreducible quotient of the inducedmodule for f W with respect to this triangular decomposition. Also for each1 ν N , let us denote the unique irreducible quotient of the inducedrepresentation of g W ν for the same triangular decomposition by ( f V ν , e ρ ν ).Now we already know that f W = L Nν =1 g W ν as L ( ◦ g , σ )-modules. Then byanalogous arguments presented in [38, Lemma 3.2], we can easily infer that f V = L Nν =1 f V ν and thus L Nν =1 f V ν is a G -graded-irreducible LT -module.Again define a b τ -module structure on f V ⊗ f V ⊗ A by setting( x ⊗ t l ) . (cid:0) N X ν =1 v ⊗ v ν ⊗ t k (cid:1) = N X ν =1 v ⊗ (cid:0)e ρ ν ( x ⊗ t l ) v ν (cid:1) ⊗ t k + l ; t r d i . (cid:0) N X ν =1 v ⊗ v ν ⊗ t k (cid:1) = ( k i + β i ) (cid:0) N X ν =1 v ⊗ v ν ⊗ t k + r (cid:1) + N X ν =1 n X j =1 (cid:0) ( r j m j E ji ) .v (cid:1) ⊗ v ν ⊗ t k + r for all x ∈ g l , k, l ∈ Z n , r = P nj =1 r j m j e j ∈ Γ , v ∈ f V , v ν ∈ e V ν and1 i n . One can readily check that f V ⊗ f V ⊗ A is a b τ -module. Put f V = L k ∈ G e V ,k and define V ′ := L k ∈ Z n f V ⊗ e V ,k ⊗ C t k which is again asubmodule of f V ⊗ f V ⊗ A . Proposition 7.7. V ′ is an irreducible module over b τ .Proof. The proof is similar to Proposition 6.1. (cid:3) Theorem 7.8. V ∼ = V ′ as b τ -modules.Proof. It is immediate that the highest weight space of V ′ is given by T ′ .The result now directly follows from Proposition 7.7 as V + ∼ = T ′ . (cid:3) Corollary 7.9. There exists a quadruplet ( ψ, c, λ, β ) ∈ P + sl n × C × ( P + g ) × × C n such that V ∼ = V ( ψ, c, λ, β ) as e τ -modules.Proof. From our construction of V ′ and Theorem 7.8, it is evident thatthe integrability of V also forces each e V ,k to be integrable over the finitedimensional simple Lie algebra g . Moreover as f W is finite dimensional,we obtain that e V ,k has finite dimensional weight spaces with respect to h .Subsequently by [31, Lemma 3.5], it readily follows that e V ,k must be finitedimensional which shows that f V is also finite dimensional as G is a finitegroup. In particular, e V ν is a finite dimensional irreducible LT -module foreach 1 ν N . Then by [22, Corollary 4.4], we can directly conclude thatthere exists a finite dimensional irreducible representation ( V ν , f ν ) of g and a ν ∈ ( C × ) n such that e V ν ∼ = ev a ν V ν as LT -modules for all 1 ν N . Thiseventually implies that it suffices to only consider e ρ ν to be the composition of ev a ν and f ν which consequently gives φ ′′ ν = ev a ν . But again as Kerφ ′′ ν = F ,it is clear that a ν ∈ U ( m ) ∀ ν N . Moreover since we know that f V is a G -graded-irreducible module over LT , we can infer that the twisted g -module under the action x. (cid:0) N X ν =1 v ν (cid:1) = N X ν =1 a νl ( x.v ν ) ∀ x ∈ g l , v ν ∈ V ν is also G -graded-irreducible. The corollary then follows from Theorem 7.8and Remark 6.2. Lastly observe that since LT does not act trivially on V ,we can never have λ = 0. (cid:3) Isomorphism Classes of Irreducible Modules In this section, our primary goal is to determine the isomorphism classesof the irreducible integrable modules that we have classified in Section 7. NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 23 We first note that in our construction of V ( ψ, c, λ, β ) in Section 6, the vectorspace V ⊗ V is endowed with a ( I ⋉ LT )-module structure given by( x ⊗ t l ) . ( v ⊗ v ) = v ⊗ ( x.v ) ,I ( u, r ) . ( v ⊗ v ) = n X i,j =1 (cid:0) ( u i r j m j E ji ) .v (cid:1) ⊗ v for all x ∈ g l , l ∈ Z n , u ∈ C n , r = P nj =1 r j m j e j ∈ Γ and v ∈ V , v ∈ V . Proposition 8.1. Let L k ∈ Z n V ⊗ V ,k ⊗ C t k ∼ = L k ∈ Z n V ′ ⊗ V ′ ,k ⊗ C t k as irreducible e τ -modules where V , V ′ are finite dimensional irreducible I -modules and V , V ′ are finite dimensional G -graded-irreducible LT -modulesused in the construction of V ( ψ, c, λ, β ) in Section 6. Then (1) V ⊗ V ∼ = V ′ ⊗ V ′ as ( I ⋉ LT ) -modules. (2) V ∼ = V ′ as gl n ( C ) -modules and V ∼ = V ′ as G -graded-irreducible g -modules up to a possible shif t of grading .Proof. (1) Let W and W ′ be the highest weight spaces of the G -graded-irreducible LT -modules V and V ′ respectively. We claim that both thehighest weight spaces are also G -graded-irreducible modules over L ( ◦ g , σ ).It suffices to prove our claim only for W . As V is G -graded, we have w = P k ∈ G w k for any w ∈ W . Applying an element of the positive rootspace e τ + (see (3.1)) on w , it is easy to see that w k ∈ W for all k ∈ G which in turn shows that W is also G -graded. The claim now directlyfollows from the G -graded-irreducibility of V and the PBW theorem. Nowthe isomorphism between the irreducible e τ -modules again gives rise to anisomorphism of the highest weight spaces, say V + and V ′ + over e τ F : M k ∈ Z n V ⊗ W ,k ⊗ C t k = V + −→ V ′ + = M k ∈ Z n V ′ ⊗ W ′ ,k ⊗ C t k . Moreover as W and W ′ are finite dimensional G -graded-irreducible modulesover L ( ◦ g , σ ) and [ h , ◦ g ] = (0), it follows that there exists a unique λ ∈ h ∗ such that P ( V + ) = { λ + δ k + β | k ∈ Z n } and P ( V ′ + ) = { λ + δ k + β ′ | k ∈ Z n } for some β, β ′ ∈ C n with β − β ′ ∈ Z n . From our construction, it is alsoevident that L ( ◦ g , σ ) does not act trivially on both W and W ′ and thus wemust have λ = 0. For the sake of notational convenience, we shall assumewithout loss of generality that β = 0 = β ′ . Again as F is an isomorphism of e τ -modules, the k -weight vectors in V + should map to k -weight vectors in V ′ + and therefore for some l ∈ N , we must have F ( v ⊗ w ⊗ t k ) = l X i =1 v ′ ,i ⊗ w ′ ,i ⊗ t k , v ′ ,i ∈ V ′ , w ′ ,i ∈ W ′ ,k , k ∈ Z n . Define φ : V ⊗ W −→ V ′ ⊗ W ′ by setting v ⊗ w l X i =1 v ′ ,i ⊗ w ′ ,i , v ∈ V , v ′ ,i ∈ V ′ and w ,k ∈ W ,k , w ′ ,i ∈ W ′ ,k . Now using the fact that λ = 0 and the relation F (cid:0) ( h ⊗ t r ) . ( v ⊗ w ⊗ t k ) (cid:1) = ( h ⊗ t r ) .F (cid:0) v ⊗ w ⊗ t k (cid:1) ∀ h ∈ h and r ∈ Γ,it is trivial to verify that φ is well-defined. Furthermore it is also clear that φ is in fact an isomorphism of ( I ⋉ L ( ◦ g , σ ))-modules. But since the G -graded-irreducible modules V ⊗ W and V ′ ⊗ W ′ are clearly the highest weightspaces of the G -graded-irreducible modules V ⊗ V and V ′ ⊗ V ′ respectively,this finally yields V ⊗ V ∼ = V ′ ⊗ V ′ as ( I ⋉ LT )-modules. Note that thisisomorphism is also a homomorphism of G -graded modules over ( I ⋉ LT ).(2) We already have V ⊗ W ∼ = V ′ ⊗ W ′ as ( I ⋉ L ( ◦ g , σ ))-modules. Let us referto this isomorphism as ψ . Now choose bases { w , · · · , w r } and { w ′ , · · · , w ′ s } of W and W ′ respectively. Then we obtain an isomorphism of I -modules r M i =1 V ⊗ C w i ∼ = s M j =1 V ′ ⊗ C w ′ j whence we get r = s and V ∼ = V ′ as gl n ( C )-modules. Again let us pick abasis { v , · · · , v k } of V and set ψ i = ψ | C v i ⊗ W for each 1 i k . Thenthere exists some 1 i k such that ψ i = 0. Composing ψ i with the obviousprojection map from V ′ ⊗ W ′ to W ′ , we eventually get a non-zero graded-module homomorphism, say ǫ , between the G -graded-irreducible L ( ◦ g , σ )-modules C v i ⊗ W and W ′ . This readily implies that ǫ must be injective.But as dim W = dim W ′ , it follows that ǫ is a L ( ◦ g , σ )-module isomorphism.Thus we have V ∼ = V ′ as LT -modules which in turn gives us the desiredresult from Proposition 5.1. (cid:3) Theorem 8.2. If V ( ψ, c, λ, β ) ∼ = V ( ψ ′ , c ′ , λ ′ , β ′ ) as e τ -modules, then (1) ψ = ψ ′ and c = c ′ ; (2) λ ′ ∈ b Gλ which in particular shows that b Gλ = b Gλ ′ and S ( λ ) = S ( λ ′ ) ; (3) β − β ′ ∈ Z n .Proof. This follows directly from Subsection 5.2 and Proposition 8.1. (cid:3) The Final Theorem Let us define an equivalence relation ’ ∼ ’ on C n by declaring β ∼ β ′ ⇐⇒ β − β ′ ∈ Z n . Also define another equivalence relation ’ ∼ ’ on ( P + g ) × bysetting λ ∼ λ ′ ⇐⇒ b Gλ = b Gλ ′ . We shall denote the equivalence classunder these equivalence relations by [ . ]. We are now ready to state the NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 25 main theorem of our paper which is a direct consequence of Proposition 6.1,Remark 6.2, Corollary 7.9 and Theorem 8.2. Theorem 9.1. If I fin denotes the set of all level zero irreducible integrablemodules over e τ (upto isomorphism) having finite dimensional weight spaceswith respect to e h and endowed with a non-trivial LT -action, then we have awell-defined bijective map κ : P + sl n × C × ( P + g ) × / ∼ × C n / ∼ −→ I fin ( ψ, c, [ λ ] , [ β ]) V ( ψ, c, [ λ ] , [ β ]) . Remark 9.2. Our Corollary 7.9 also recovers the classification result givenin [36, Theorem 4.3] for the untwisted case. In fact, we have further deducedthat any such level zero irreducible integrable module is uniquely determinedby the parameters ( ψ, c, λ, [ β ]) ∈ P + sl n × C × ( P + g ) × × C n / ∼ . Note that since S ( λ ) = 1 and | b Gλ | = 1, these parameters (which are dependent on λ ) donot make any appearance in the description of these irreducible integrablemodules in the untwisted case. Acknowledgements : The first author would like to thank Prof. MikhailKochetov for some helpful discussions. References [1] B. Allison, S. Berman, J. Faulkner, A. Pianzola, Multiloop realization of extended affineLie algebras and Lie tori, Trans. Amer. Math. Soc. 361 (2009) 4807–4842.[2] P. Batra, S. Eswara Rao, On integrable modules for the twisted full toroidal Lie algebra, J. Lie Theory 28 (2018), no. 1, 79–105.[3] S. Berman, Y. Billig, Irreducible representations for toroidal Lie algebras, J. Algebra221 (1999), no. 1, 188–231.[4] Y. Billig, A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not.2006, Art. ID 68395, 46 pp.[5] Y. Billig, Representations of toroidal extended affine Lie algebras, J. Algebra 308(2007), no. 1, 252–269.[6] Y. Billig, M. Lau, Irreducible modules for extended affine Lie algebras, J. Algebra 327(2011), 208–235.[7] Y. Billig, V. Futorny, Classification of irreducible representations of Lie algebra ofvector fields on a torus, J. Reine Angew. Math. 720 (2016), 199–216.[8] A. Borel, G. Mostow, On semi-simple automorphisms of Lie algebras, Ann. of Math.(2)61 (1955), 389–405.[9] V. Chari, Integrable representations of affine Lie-algebras, Invent. Math. 85 (1986),no. 2, 317–335.[10] V. Chari, A. Pressley, New unitary representations of loop groups, Math. Ann. 275(1986), no. 1, 87–104.[11] V. Chari, A. Pressley, A new family of irreducible, integrable modules for affine Liealgebras, Math. Ann. 277 (1987), no. 3, 543–562. [12] V. Chari, A. Pressley, Integrable representations of twisted affine Lie algebras, J.Algebra 113 (1988), no. 2, 438–464.[13] A. Elduque, M. Kochetov, Graded modules over classical simple Lie algebras with agrading, Israel J. Math. 207 (2015), no. 1, 229–280.[14] P. Di Francesco, P. Mathieu, D. S´en´echal, Conformal field theory, Graduate Texts inContemporary Physics, Springer-Verlag, New York, 1997.[15] J. Fu, C. Jiang, Integrable representations for the twisted full toroidal Lie algebras, J.Algebra 307 (2007), no. 2, 769–794.[16] V. Futorny, A. Tsylke, Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras, J. Algebra 238 (2001), no. 2, 426–441.[17] C. Jiang, D. Meng, Integrable representations for generalized Virasoro-toroidal Liealgebras, J. Algebra 270 (2003), no. 1, 307–334.[18] V.G. Kac, Highest weight representations of conformal current algebras, Topologicaland geometrical methods in field theory (Espoo, 1986), 3–15, World Sci. Publ., Teaneck,NJ, 1986.[19] V.G. Kac, Infinite-dimensional Lie algebras, Third edition. Cambridge UniversityPress, Cambridge, 1990.[20] C. Kassel, K¨ahler differentials and coverings of complex simple Lie algebras extendedover a commutative algebra, in: Proceedings of the Luminy Conference on AlgebraicK-Theory, Luminy, 1983, vol. 34, 1984, pp. 265–275.[21] G. Kuroki, Fock space representations of affine Lie algebras and integral representa-tions in the Wess-Zumino-Witten models, Comm. Math. Phys. 142 (1991), no. 3, 511–542.[22] M. Lau, Representations of multiloop algebras, Pacific J. Math. 245 (2010), no. 1,167–184.[23] M. Lau, Classification of Harish-Chandra modules for current algebras, Proc. Amer.Math. Soc. 146 (2018), no. 3, 1015–1029.[24] H. Li, Classification of irreducible weight modules, Math. Z. 248 (2004), no. 3, 635–664.[25] O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math. 107 (1992), no. 2, 225–234.[26] O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Greno-ble) 50 (2000), no. 2, 537–592.[27] R.V. Moody, S. Eswara Rao, T. Yokonuma, Toroidal Lie algebras and vertex repre-sentations, Geom. Dedicata 35 (1990), no. 1-3, 283–307.[28] K. Naoi, Multiloop Lie algebras and the construction of extended affine Lie algebras, J. Algebra 323 (2010), no. 8, 2103–2129.[29] E. Neher, Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004),no. 3, 90–96.[30] E. Neher, A. Savage, P. Senesi, Irreducible finite-dimensional representations of equi-variant map algebras, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2619–2646.[31] S. Pal, Integrable modules for graded Lie tori with finite dimensional weight spaces, https://arxiv.org/abs/2005.07381.[32] E. Ramos, C.H. Sah, R.E. Shrock, Algebras of diffeomorphisms of the N-torus, J.Math. Phys. 31 (1990), no. 8, 1805–1816.[33] S. Eswara Rao, R.V. Moody, Vertex representations for n-toroidal Lie algebras and ageneralization of the Virasoro algebra, Comm. Math. Phys. 159 (1994), no. 2, 239–264.[34] S. Eswara Rao, Classification of irreducible integrable modules for toroidal Lie algebraswith finite dimensional weight spaces, J. Algebra 277 (2004), no. 1, 318–348. NTEGRABLE MODULES FOR TWISTED FULL TOROIDAL LIE ALGEBRAS 27 [35] S. Eswara Rao, Partial classification of modules for Lie algebra of diffeomorphismsof d-dimensional torus, J. Math. Phys. 45 (2004), no. 8, 3322–3333.[36] S. Eswara Rao, C. Jiang, Classification of irreducible integrable representations forthe full toroidal Lie algebras, J. Pure Appl. Algebra 200 (2005), no. 1-2, 71–85.[37] S. Eswara Rao, S. Sharma, P. Batra Integrable modules for twisted toroidal extendedaffine Lie algebras, J. Algebra 556 (2020), 1057–1072.[38] S. Eswara Rao, Classification of bounded modules for the twisted full toroidal Liealgebras, Communications in Algebra (2020).[39] J. Sun, Universal central extensions of twisted forms of split simple Lie algebras overrings, J. Algebra 322 (2009), no. 5, 1819–1829.[40] Y. Yoshii, Root systems extended by an abelian group and their Lie algebras, J. LieTheory 14 (2004), no. 2, 371–394.[41] Y. Yoshii, Lie tori–a simple characterization of extended affine Lie algebras, Publ.Res. Inst. Math. Sci. 42 (2006), no. 3, 739–762. Souvik Pal, Harish-Chandra Research Institute (HBNI), Chhatnag Road,Jhunsi, Prayagraj(Allahabad) 211019, Uttar Pradesh, India. Email address : [email protected], [email protected] S. Eswara Rao, School of Mathematics, Tata Institute of FundamentalResearch, Homi Bhabha Road, Colaba, Mumbai 400005, India. Email address ::