Lower-bounded and grading-restricted twisted modules for affine vertex (operator) algebras
aa r X i v : . [ m a t h . QA ] O c t Lower-bounded and grading-restrictedtwisted modules for affine vertex (operator)algebras
Yi-Zhi Huang
Abstract
We apply the construction of the universal lower-bounded generalized twisted mod-ules by the author to construct universal lower-bounded and grading-restricted gen-eralized twisted modules for affine vertex (operator) algebras. We prove that theseuniversal twisted modules for affine vertex (operator) algebras are equivalent to suit-able induced modules of the corresponding twisted affine Lie algebra or quotients ofsuch induced modules by explicitly given submodules.
In [Hua5], the author constructed universal lower-bounded generalized twisted modules fora grading-restricted vertex algebra. In the present paper, we apply this construction toconstruct and identify explicitly universal lower-bounded and grading-restricted general-ized twisted modules for affine vertex (operator) algebras. In particular, general classes oflower-bounded and grading-restricted generalized twisted modules can be studied using theseuniversal ones.Let g be a finite-dimensional Lie algebra with a nondegenerate invariant symmetric bi-linear form ( · , · ) and g an automorphism of g . Then an induced module M ( ℓ,
0) of level ℓ ∈ C for the affine Lie algebra ˆ g generated by the trivial module C for g has a structureof grading-restricted vertex algebra. In the case that g is simple and ℓ = − h ∨ , where h ∨ is the dual Coxeter number of g , M ( ℓ,
0) has a conformal vector and is thus a vertex op-erator algebra. Let L ( ℓ,
0) be the irreducible quotient of M ( ℓ, L ( ℓ,
0) is also agraidng-restricted vertex algebra and, when ℓ + h ∨ = 0, is a vertex operator algebra. Anautomorphism g of g induces automorphisms, still denoted by g , of ˆ g , M ( ℓ,
0) and L ( ℓ, g [ g ] constructed using g , ( · , · ) and g . Note that g hasa rich automorphism group containing the Lie group corresponding to g . Automorphismsof g , M ( ℓ,
0) and L ( ℓ,
0) are mostly of infinite orders and many of them do not act on g , M ( ℓ,
0) and L ( ℓ,
0) semisimply.Twisted modules associated to automorphisms of finite orders of a vertex operator algebrawere introduced and studied first by Frenkel, Lepowsky and Meurman in [FLM1], [FLM2]1nd [FLM3] and by Lepowsky in [Le1] and [Le2]. In [Hua1], the author introduced twistedmodules associated to general automorphisms of a vertex operator algebra, including inparticular, automorphisms which do not act on the vertex operator algebra semisimply. Aparticular class of examples associated to such general automorphisms were also given in[Hua1]. In [Hua5], the author gave a construction of universal lower-bounded generalizedtwisted modules associated to such general automorphisms of a grading-restricted vertexalgebra.Applying the construction in [Hua5] to M ( ℓ, g -twisted M ( ℓ, g [ g ] I of ˆ g [ g ] ) with actions of g , its semisim-ple and unipotent parts, and some other operators and annihilated by the positive part ofˆ g [ g ] when M ( ℓ,
0) is viewed as a grading-restricted vertex algebra. When ℓ + h ∨ = 0 and M ( ℓ,
0) is viewed as a vertex operator algebra, we also construct universal lower-bounded(grading-restricted) generalized g -twisted M ( ℓ, g [ g ] I ) with additional structures as above. These universallower-bounded and grading-restricted generalized g -twisted M ( ℓ, g [ g ] -modulesare then proved to be equivalent to suitable induced modules for ˆ g [ g ] . The proofs of theseequivalences use the results in Section 2 of [Hua6] on a linearly independent set of gener-ators of the universal lower-bounded generalized twisted modules constructed in Section 5of [Hua5]. In the case that M ( ℓ,
0) is viewed as a vertex operator algebra, we also give ex-plicit formulas for the Virasoro operators on the universal lower-bounded generalized twisted M ( ℓ, g [ g ] .When g is simple and ℓ ∈ Z + and L ( ℓ,
0) is viewed as a vertex operator algebra, we con-struct universal lower-bounded (grading-restricted) generalized g -twisted M ( ℓ, g [ g ] I -module) with additional structures asabove. We also prove that these universal lower-bounded and grading-restricted generalized g -twisted L ( ℓ, g [ g ] -modules are equivalent to quotients by certain explicitlygiven submodules of the induced modules for ˆ g [ g ] equivalent to the universal lower-boundedand grading-restricted generalized g -twisted M ( ℓ, g -twisted M ( ℓ, L ( ℓ, g [ g ] I -submodules with addi-tional structures as above are quotients of these universal ones. Thus we can study thesetypes of twisted modules, including untwisted ones, using our results on the universal onesin the present paper.In the case that g is of finite order, Li gave the relationship between weak twisted modulesfor an affine vertex operator algebra and restricted modules for the corresponding twistedaffine Lie algebra in [Li]. In [B], Bakalov introduced twisted affine Lie algebras in the case2hat g is a general automorphsim of g and gave the relationship between weak twisted mod-ules for an affine vertex operator algebra and restricted modules for the corresponding twistedaffine Lie algebra. In the present paper, we do not study these most general weak twistedmodules and restricted modules. We study only lower-bounded and grading-restricted gener-alized twisted modules for affine vertex (operator) algebras and lower-bounded and grading-restricted modules for the twisted affine Lie algebras. We would like to emphasize that in therepresentation theory of vertex (operator) algebras, to obtain substantial results, we have torestrict ourselves to grading-restricted generalized (twisted) modules and we often have tofurther restrict ourselves to such modules of finite lengths. On the other hand, lower-boundedgeneralized (twisted) modules always appear in various constructions and proofs. One of thedifficult problems is to prove that these lower-bounded generalized (twisted) modules ap-pearing in our constructions and proofs are actually grading-restricted generalized (twisted)modules of finite lengths. So these two types of twisted modules are what we are mainlyinterested. Moreover, for such modules of finite lengths, we can reduce their study to thosemodules generated by subspaces annihilated by the positive part of the twisted affine Liealgebra. This is the reason why we choose to construct, identify and study these types oftwisted modules in this paper. Though weak twisted modules are more general, usually weneed them only in the formulations of certain notions in the representation theory of vertex(operator) algebras.As is mentioned in the preceding paragraph, one of the difficult problems in the represen-tation theory of vertex operator algebras is to prove that suitable lower-bounded generalizedtwisted modules are actually grading restricted. In fact, universal lower-bounded generalizedtwisted modules are in a certain sense analogous to Verma modules in the representationtheory of finite-dimensional Lie algebras. In the case of finite-dimensional Lie algebras, weknow that a Verma module generated from a highest weight vector has a finite-dimensionalquotient module if and only if the highest weight is dominant integral. For vertex operator al-gebra, we can ask an analogous question: Under what conditions, a universal lower-boundedgeneralized twisted module has a grading-restricted quotient. In this paper, our constructionand identification of grading-restricted generalized twisted modules for M ( ℓ,
0) and L ( ℓ, g of a finite-dimensional Lie algebra g , an automorphism g of g andthe twisted affine Lie algebra ˆ g [ g ] . In Section 3, we recall vertex (operator) algebras M ( ℓ, L ( ℓ,
0) associated to affine Lie algebras and their automorphisms induced from thoseof g . The construction, identification and basic properties of lower-bounded and grading-restricted generalized twisted modules for M ( ℓ,
0) are given in Section 4. In Subsection 4.1,we construct and identify explicitly lower-bounded and grading-restricted generalized twistedmodules for M ( ℓ,
0) viewed as a grading-restricted vertex algebra. In Subsection 4.2, we3onstruct and identify explicitly such twisted modules for M ( ℓ,
0) viewed as a vertex operatoralgebra. In Subsection 4.3, basic properties of these twisted modules for M ( ℓ, L ( ℓ,
0) are given in Section 5.
Acknowledgments
The author is grateful to Sven M¨oller for the argument (Lemma 8.3in [EMS]) needed in the last step of the proof of Proposition 5.3.
Let g be a finite-dimensional Lie algebra and ( · , · ) a nondegenerate invariant symmetricbilinear form on g . Recall that the affine Lie algebra ˆ g is the vector space g ⊗ C [ t, t − ] ⊕ C k equipped with the bracket operation[ a ⊗ t m , b ⊗ t n ] = [ a, b ] ⊗ t m + n + ( a, b ) mδ m + n, k , [ a ⊗ t m , k ] = 0 , for a, b ∈ g and m, n ∈ Z . Let ˆ g ± = g ⊗ t ± C [ t ± ]. Thenˆ g = ˆ g − ⊕ g ⊕ C k ⊕ ˆ g + . Let g be an automorphism of g . Assume also that ( · , · ) is invariant under g . This is truein the case that g is semisimple and ( · , · ) is proportional to the Killing form. Since g is finitedimensional, there exist operators L g , S g and N g on g such that g = e πi L g and S g and N g are the semisimple and nilpotent parts of L g , respectively. Then g , L g , S g and N g induceoperators, still denoted by g , L g , S g and N g , on the affine Lie algebra ˆ g such that g is alsoan automorphism of ˆ g .Let P g = { α ∈ C | ℜ ( α ) ∈ [0 , , e πiα is an eigenvalue of g } . Then g = a α ∈ P g g [ α ] , where for α ∈ P g , g [ α ] is the generalized eigenspace of g (or the eigenspace of e πi S g ) with theeigenvalue e πiα .For α, β ∈ [0 ,
1) + i R , let s ( α, β ) = (cid:26) α + β ℜ ( α + β ) < α + β − ℜ ( α + β ) ≥ . Then ℜ ( s ( α, β )) ∈ [0 ,
1) and for α, β ∈ P g , s ( α, β ) ∈ ( P g + P g ) ∪ ( P g + P g − emma 2.1 For α, β ∈ P g , [ g [ α ] , g [ β ] ] ⊂ g [ s ( α,β )] . In particular, in the case that [ g [ α ] , g [ β ] ] = 0 , s ( α, β ) ∈ P g ∩ (( P g + P g ) ∪ ( P g + P g − ⊂ P g and e πi ( α + β ) is an eigenvalue of g .Proof. For a ∈ g [ α ] and b ∈ g [ β ] , we have( g − e πi ( α + β ) )[ a, b ]= [ ga, gb ] − e πi ( α + β ) [ a, b ]= e πi ( α + β ) [ e πi N g a, e πi N g b ] − e πi ( α + β ) [ a, b ]= e πi ( α + β ) (cid:0) [(1 g + ( e πi N g − g )) a, (1 g + ( e πi N g − g )) b ] − [ a, b ] (cid:1) = e πi ( α + β ) (cid:0) [( e πi N g − g ) a, b ] + [ a, ( e πi N g − g ) b ] + [( e πi N g − g ) a, ( e πi N g − g ) b ] (cid:1) . Then there exists e K ∈ Z + such that( g − e πi ( α + β ) ) e K [ a, b ] = 0 . (Note that we can always take e K = dim g .) If [ a, b ] = 0, we have [ a, b ] ⊂ g [ s ( α,β )] . If [ a, b ] = 0,it is a generalized eigenvector of g with eigenvalue e πi ( α + β ) and thus is in g [ s ( α,β )] . In thiscase, g [ s ( α,β )] = 0. So s ( α, β ) ∈ P g . We also have either s ( α, β ) = α + β ∈ P g + P g or s ( α, β ) = α + β − ∈ P g + P g −
1. Thus s ( α, β ) ∈ P g ∩ (( P g + P g ) ∪ ( P g + P g − Corollary 2.2
The operators e πi S g and e πi N g are also automorphisms of g . The operator N g is a derivation of the Lie algebra g .Proof. Let a ∈ g [ α ] and b ∈ g [ β ] . By Lemma 2.1, e πi S g [ a, b ] = e πi ( α + β ) [ a, b ] = [ e πiα a, e πiα b ] = [ e πi S g a, e πi S g b ] . So e πi S g is an automorphism of g . Therefore e − πi S g is also an automorphism of g . Thus e πi N g = e − πi S g g is an automorphism of g .For a, b ∈ g , we have (ad e πi N g a ) b = [ e πi N g a, b ]= e πi N g [ a, e − πi N g b ]= e πi N g (ad a ) e − πi N g b = ((Ad 2 πi N g )(ad a )) b. Thus [ N g a, b ] = 12 πi (ad log e πi N g a ) b = 12 πi (log(Ad 2 πi N g )(ad a )) b = ((ad N g )(ad a )) b = N g [ a, b ] − [ a, N g b ] , proving that N g is a derivation of g . 5 emma 2.3 If α + β
6∈ { , } , then g [ α ] and g [ β ] are orthogonal. If α + β ∈ { , } , then ( · , · ) restricted to g [ α ] × g [ β ] is nondegenerate.Proof. For a ∈ g [ α ] , there exists p ∈ Z + such that ( g − e πiα ) p a = 0. On the other hand,since α + β
6∈ { , } , the restriction of ( g − − e πiα ) p to g [ β ] is a linear isomorphism from g [ β ] to itself. If there exist a ∈ g [ α ] and b ∈ g [ β ] such that ( a, b ) = 0. then there exists c ∈ g [ β ] suchthat b = ( g − − e πiα ) p c . Then ( a, ( g − − e πiα ) p c ) = ( a, b ) = 0. But since ( · , · ) is invariantunder g , we have ( a, ( g − − e πiα ) p c ) = (( g − e πiα ) p a, c ) = 0. Contradiction. So we musthave ( a, b ) = 0 for a ∈ g [ α ] and b ∈ g [ β ] .In the case that α + β ∈ { , } , if ( · , · ) restricted to g [ α ] × g [ β ] is degenerate, then thereexists a ∈ g [ α ] \ { } such that ( a, b ) = 0 for b ∈ g [ β ] . But for β ∈ P g such that α + β
6∈ { , } ,we just proved that ( a, b ) = 0 for b ∈ g [ β ] . Thus ( a, b ) = 0 for b ∈ g . Contradiction to thenondegeneracy of ( · , · ). Proposition 2.4
The nondegenerate invariant symmetric bilinear form ( · , · ) is also invari-ant under e πi S g and e πi N g .Proof. Let a ∈ g [ α ] and b ∈ g [ β ] . If α + β ∈ { , } , then ( e πi S g a, e πi S g b ) = e πi ( α + β ) ( a, b ) =( a, b ). If α + β
6∈ { , } , then by Lemma 2.3, ( e πi S g a, e πi S g b ) = 0 = ( a, b ). So ( · , · ) isinvariant under e πi S g .Since e πi N g = e − πi S g g and certainly ( · , · ) is also invariant under e − πi S g , ( · , · ) is invariantunder e πi N g . Corollary 2.5
For a, b ∈ g , we have ( N g a, b ) + ( a, N g b ) = 0 .Proof. For a, b ∈ g , we have( N g a, b ) = (cid:18) πi (1 g + (log e πi N g − g )) a, b (cid:19) = (cid:18) a, πi (1 g + (log e − πi N g − g ) b (cid:19) = − ( a, N g b ) . Remark 2.6
Note that if ℜ{ α } = ℜ{ β } = 0, then ℜ{ α + β } = ℜ{ s ( α, β ) } = 0. Inparticular, a ℜ{ α } =0 g [ α ] is a subalgebra of g . The fixed-point subalgebra g [0] is a subalgebra of this subalgebra.6he decomposition g = a α ∈ P g g [ α ] induces decompositions ˆ g = a α ∈ P g ˆ g [ α ] where ˆ g [ α ] for α ∈ P g are the generalized eigenspaces of g (or the eigenspaces of e πi S g ) on ˆ g with the eigenvalue e πiα .We now define the twisted affine Lie algebra associated to g , ( · , · ) and g (see, for example,[K] and [B]). Let ˆ g [ g ] = a α ∈ P g g [ α ] ⊗ t α C [ t, t − ] ⊕ C k . We define a bracket operation on ˆ g [ g ] by[ a ⊗ t m , b ⊗ t n ] = [ a, b ] ⊗ t m + n + m ( a, b ) δ m + n, k + ( N g a, b ) δ m + n, k , (2.1)[ k , a ⊗ t m ] = 0 , (2.2)for a ∈ g [ α ] , b ∈ g [ β ] , m ∈ α + Z , n ∈ β + Z , α, β ∈ P g . Then it is straightforward to verifythat the vector space ˆ g [ g ] equipped with the bracket operation defined above is a Lie algebra.Let ˆ g [ g ]+ = M α ∈ P g , ℜ{ α } > g [ α ] ⊗ t α C [ t ] ⊕ M α ∈ P g , ℜ{ α } =0 g [ α ] ⊗ t α +1 [ t ] , ˆ g [ g ] − = M α ∈ P g g [ α ] ⊗ t α − C [ t − ] , ˆ g [ g ] I = M α ∈ P g , ℜ{ α } =0 g [ α ] ⊗ C t α , ˆ g [ g ]0 = ˆ g [ g ] I ⊕ C k . Then ˆ g [ g ]+ , ˆ g [ g ] − , ˆ g [ g ] I and ˆ g [ g ]0 are subalgebras of ˆ g [ g ] and ˆ g [ g ] I is a subalgebra of ˆ g [ g ]0 . Moreover,we have a triangular decompositionˆ g [ g ] = ˆ g [ g ] − ⊕ ˆ g [ g ]0 ⊕ ˆ g [ g ]+ . In this paper, we are interested in only those ˆ g [ g ] -modules with lower-bounded C -gradingscompatible with the grading of ˆ g [ g ] and with actions of g . To be precise, we give the followingdefinition: 7 efinition 2.7 A graded ˆ g [ g ] -module is a ˆ g [ g ] -module W with a C -grading W = ` n ∈ C W [ n ] such that ˆ g [ g ][ m ] W [ n ] ⊂ W [ m + n ] for m ∈ P g + Z and n ∈ C , where ˆ g [ g ][ m ] = g [ α ] ⊗ t m for m ∈ ( P g + Z ) \ { } and ˆ g [ g ][0] = g ⊗ C t ⊕ C k . A graded ˆ g [ g ] -module of level ℓ is a gradedˆ g [ g ] -module such that k acts as ℓ ∈ C . A lower-bounded ˆ g [ g ] -module is a graded ˆ g [ g ] -module W = ` n ∈ C W [ n ] such that W [ n ] = 0 when ℜ ( n ) is sufficiently negative. A grading-restricted ˆ g [ g ] -module is a lower-bounded ˆ g [ g ] -module W = ` n ∈ C W [ n ] such that dim W [ n ] < ∞ for n ∈ C . A ˆ g [ g ] -module with a compatible action of g or simply a ˆ g [ g ] -module with an actionof g is a ˆ g [ g ] -module W with actions of g , S g and N g satisfying the following conditions:(i) W is a direct sum of generalized eigenspaces of g . (ii) g = e πi L g , where L g is theoperator on W such that S g and N g on W are the semisimple and nilpotent parts of L g . (iii) g ( uw ) = g ( u ) g ( w ) for u ∈ ˆ g [ g ] and w ∈ W .In this paper, ˆ g [ g ] -modules are always assumed to be graded and with compatible g ac-tions. So we shall call them simply ˆ g [ g ] -modules. In particular, in this paper, lower-boundedˆ g [ g ] -modules and grading-restricted ˆ g [ g ] -modules are always with compatible g actions. We recall the vertex operator algebras constructed from suitable modules for the affine Liealgebra ˆ g and their automorphsims in this section.Let M be a g -module and let ℓ ∈ C . Let ˆ g + act on M trivially and let k act as thescalar multiplication by ℓ . Then M becomes a g ⊕ C k ⊕ ˆ g + -module and we have an inducedˆ g -module c M ℓ = U (ˆ g ) ⊗ U ( g ⊕ C k ⊕ ˆ g + ) M, Let M = C and let g act on C trivially. The corresponding ˆ g -module b C ℓ is denoted by M ( ℓ, J ( ℓ,
0) be the maximal proper submodule of M ( ℓ,
0) and L ( ℓ,
0) = M ( ℓ, /J ( ℓ, L ( ℓ,
0) is the unique irreducible graded ˆ g -module such that k acts as ℓ and the spaceof all elements annihilated by ˆ g + is isomorphic to the trivial g -module C .Frenkel and Zhu [FZ] gave both M ( ℓ,
0) and L ( ℓ,
0) natural structures of vertex operatoralgebras (see also [LL]). In particular, M ( ℓ,
0) and L ( ℓ,
0) are grading-restricted vertex alge-bras. We shall apply the results in [Hua5] to construct lower-bounded and grading-restrictedgeneralized twisted M ( ℓ, L ( ℓ, M ( ℓ,
0) and L ( ℓ,
0) using the construction in Section 3 in [Hua2].We discuss M ( ℓ,
0) first. Note that U (ˆ g − ) is linearly isomorphic to M ( ℓ, Z + -grading on ˆ g − induces an N -grading on M ( ℓ, λ ). We denote the homogeneous subspace of M ( ℓ,
0) of degree (conformal weight) n by M ( n ) ( ℓ,
0) for n ∈ N . We denote the action of a ⊗ t n on M ( ℓ,
0) by a ( n ) for a ∈ g and n ∈ Z . We also denote 1 ∈ M ( ℓ,
0) by M ( ℓ, . Then M ( ℓ,
0) is spanned by elements of the form a ( − n ) · · · a k ( − n k ) M ( ℓ, for a , . . . , a k ∈ g and8 , . . . , n k ∈ − Z + . For a ∈ g , let a ( x ) = P n ∈ Z a ( n ) x − n − . In particular, z a ( z ) for z ∈ C × is an analytic map from C × to Hom( M ( ℓ, , M ( ℓ, L M ( ℓ, (0) be the operator on M ( ℓ,
0) giving the grading on M ( ℓ, L M ( ℓ, (0) v = nv for v ∈ ( M ( n ) ( ℓ, L M ( ℓ, ( −
1) on M ( ℓ,
0) by L M ( ℓ, ( − a ( − n ) · · · a k ( − n k ) M ( ℓ, = k X i =1 n i a ( − n ) · · · a i − ( − n i − ) a i ( − n i − a i +1 ( − n i +1 ) · · · a k ( − n k ) M ( ℓ, . It is easy to verify that the series a ( x ) for a ∈ g and the operators L M ( ℓ, (0) and L M ( ℓ, ( −
1) have the following properties:1. For a ∈ g , [ L M ( ℓ, (0) , a ( x )] = x ddx a ( x ) + a ( x ).2. L M ( ℓ, ( − = 0, [ L M ( ℓ, ( − , a ( x )] = ddz a ( x ) for a ∈ g .3. For a ∈ g , a ( x ) M ( ℓ, ∈ M ( ℓ, x ]]. Moreover, lim x → a ( x ) = a ( − M ( ℓ, .4. The vector space M ( ℓ,
0) is spanned by elements of the form a ( n ) · · · a k ( n k ) M ( ℓ, for a , . . . , a k ∈ g and n , . . . , n k ∈ Z .5. For a, b ∈ g , ( x − x ) a ( x ) b ( x ) = ( x − x ) b ( x ) a ( x ) . Then by Proposition 3.3 in [Hua2], h v ′ , a ( z ) · · · a k ( z k ) v i for a , . . . , a k ∈ g , v ∈ M ( ℓ, v ′ ∈ M ( ℓ, ′ is absolutely convergent in the region | z | > · · · > | z k | > R ( h v ′ , a ( z ) · · · a k ( z k ) v i ), in z , . . . , z k with the only possible poles at z i = 0 for i = 1 , . . . k and z i = z j for i < j , i, j = 1 , . . . , k .By Theorem 3.5 in [Hua2], the vector space M ( ℓ,
0) equipped with the vertex operatormap Y M ( ℓ, : M ( ℓ, ⊗ M ( ℓ, → M ( ℓ, x, x − ]]defined by h v ′ ,Y M ( ℓ, ( a ( n ) · · · a k ( n k ) M ( ℓ, , z ) v i = Res ξ =0 · · · Res ξ k =0 ξ n · · · ξ n k k R ( h v ′ , a ( ξ + z ) · · · a k ( ξ k + z ) v i )for z ∈ C × , a , . . . , a k ∈ g , n , . . . , n k ∈ Z , v ∈ M ( ℓ,
0) and v ′ ∈ M ( ℓ, ′ and the vacuum M ( ℓ, is a grading-restricted vertex algebra. Moreover, this is the unique grading-restrictedvertex algebra structure on M ( ℓ,
0) with the vacuum such that Y ( a ( − , x ) = a ( x ) for a ∈ g . In particular, this grading-restricted vertex algebra structure on M ( ℓ,
0) is the sameas the one constructed in [FZ] (see also [LL]), that is, the graded space M ( ℓ, Y M ( ℓ, , the operator L M ( ℓ, ( −
1) and the vacuum one are equal to the gradedspace, the vertex operator maps, the operator L ( −
1) and the vacuum in [FZ].9ince J ( ℓ,
0) is a ˆ g -module, we can define the action of a ( x ) for a ∈ g on L ( ℓ,
0) = M ( ℓ, /J ( ℓ, L M ( ℓ, (0) and L M ( ℓ, ( −
1) induce operators L L ( ℓ, (0) and L L ( ℓ, ( − L ( ℓ, = + J ( ℓ, ∈ L ( ℓ, L ( ℓ, L ( ℓ,
0) equipped with the vertex operator map Y L ( ℓ, defined by h v ′ ,Y L ( ℓ, ( a ( n ) · · · a k ( n k ) L ( ℓ, , z ) v i = Res ξ =0 · · · Res ξ k =0 ξ n · · · ξ n k k R ( h v ′ , a ( ξ + z ) · · · a k ( ξ k + z ) v i )for a , . . . , a k ∈ g , n , . . . , n k ∈ Z , v ∈ L ( ℓ,
0) and v ′ ∈ L ( ℓ, ′ and the vacuum L ( ℓ, isa grading-restricted vertex algebra. Moreover, this is the unique grading-restricted vertexalgebra structure on L ( ℓ,
0) with the vacuum such that Y ( a ( − , x ) = a ( x ) for a ∈ g . Inparticular, this grading-restricted vertex algebra structure on L ( ℓ,
0) is the same as the oneconstructed first in [FZ] (see also [LL]).Let g be an automorphism of g as is discussed in the preceding section. The actions of g , L g , S g and N g on ˆ g further induce their actions, still denoted by g , L g , S g and N g , on M ( ℓ, L ( ℓ, g , e πi S g and e πi N g are all automorphisms of the grading-restrictedvertex algebras M ( ℓ,
0) and L ( ℓ, g is simple, we shall always take ( · , · ) be the normalized Killing formsuch that ( α, α ) = 2 for a long root α . Let h ∨ be dual Coxeter number of g . In the case that ℓ + h ∨ = 0, the grading-restricted vertex algebra M ( ℓ,
0) has a conformal element ω M ( ℓ, = 12( ℓ + h ∨ ) dim g X i =1 ( a i ) ′ ( − a i ( − , where { a i } dim g i =1 is a basis of g and { ( a i ) ′ } dim g i =1 is its dual basis with respect to ( · , · ). See [FZ] andalso [LL]. The grading-restricted vertex algebra M ( ℓ,
0) with this conformal element is a ver-tex operator algebra (or a grading-restricted conformal vertex algebra). Moreover, L M ( ℓ, (0)and L M ( ℓ, ( −
1) above are in fact the coefficients of x − and x − in Y M ( ℓ, ( ω M ( ℓ, , x ). Since ω M ( ℓ, is not in J ( ℓ, L ( ℓ,
0) has a conformal element ω L ( ℓ, = ω M ( ℓ, + J ( ℓ, L ( ℓ,
0) with this conformal element is also a vertexoperator algebra and L L ( ℓ, (0) and L L ( ℓ, ( −
1) are in fact the coefficients of x − and x − in Y L ( ℓ, ( ω L ( ℓ, , x ). Again see [FZ] and also [LL] for details.Since the Killing form on g is invariant under the action of g , the conformal element ω M ( ℓ, and ω L ( ℓ, are also invariant under g . Thus g , e πi S g and e πi N g are in fact automorphismsof the vertex operator algebras M ( ℓ,
0) and L ( ℓ, N g is a nilpotent operator on g ,we must have N dim g g = 0 on M ( ℓ,
0) and L ( ℓ, g = a α ∈ P g ˆ g [ α ] induced from the decomposition g = a α ∈ P g g [ α ] M ( ℓ,
0) = a α ∈ P g M [ α ] ( ℓ, ,L ( ℓ,
0) = a α ∈ P g L [ α ] ( ℓ, , where M [ α ] ( ℓ,
0) and L [ α ] ( ℓ,
0) for α ∈ P g are the generalized eigenspaces of g (or theeigenspaces of e πi S g ) on M ( ℓ,
0) and L ( ℓ, e πiα .We now choose suitable generating fields a ( x ) such that Assumption 2.1 in [Hua5] issatisfied for the grading-restricted vertex algebra M ( ℓ, L g is an operator on afinite-dimensional vector space g , we can find a Jordan basis { a i } dim g i =1 for g , that is, a basis { a i } dim g i =1 of g such that under this basis, the matrix representation of L g is a Jordan canonicalform. We use I to denote the set { , . . . , dim g } . Then the Jordan basis can be written as { a i } i ∈ I . Since { a i } i ∈ I is a basis of g and M ( ℓ,
0) as a grading-restricted vertex algebra isgenerated by fields of the form a ( x ) for a ∈ g , M ( ℓ,
0) is also generated by the fields a i ( x )for i ∈ I . Since for i ∈ I , a i is an element of a Jordan basis, there exist an α i ∈ P g and n i ∈ Z such that a i is a generalized eigenvector of L g with the eigenvalue α i + n i , orequivalently, a generalized eigenvector of g with the eigenvalue e πiα i . Thus a i ( − is alsoa generalized eigenvector of L g on M ( ℓ,
0) with the eigenvalue α i + n i , or equivalently, ageneralized eigenvector of g on M ( ℓ,
0) with the eigenvalue e πiα i . Also since a i is an elementof a Jordan basis, either N g a i = 0 or N g a i is another element in the basis { a i } i ∈ I . Thereforethere exists N g ( i ) ∈ I such that N g a i = a N g ( i ) . Thus we also have N g a i ( − = a N g ( i ) ( − .These also hold for L ( ℓ, Proposition 3.1
Assumption 2.1 in [Hua5] is satisfied by M ( ℓ, and L ( ℓ, with the setof generating fields a i ( x ) for i ∈ I . M ( ℓ, -modules In this section, we first construct universal lower-bounded and grading-restricted generalizedtwisted modules for M ( ℓ,
0) viewed as a grading-restricted vertex algebra in Subsection 4.1.Then in the case that g is simple and ℓ + h ∨ = 0, we construct universal lower-boundedand grading-restricted generalized twisted modules for M ( ℓ,
0) viewed as a vertex operatoralgebra in Subsection 4.2. We also discuss their basic properties such as their universalproperties and so on in Subsection 4.3. M ( ℓ, is viewed as a grading-restrictedvertex algebra Before we give constructions of universal lower-bounded and grading-restricted generalizedtwisted M ( ℓ, g [ g ] . Let W be a lower-bounded generalized g -twisted M ( ℓ, α ∈ P g , a ∈ g [ α ] and n ∈ α + Z , we write Y gW ( a ( − , x ) = K X k =0 X n ∈ α + Z ( a W ) n,k x − n − (log x ) k . From (2.10) in [HY], Y gW ( a ( − , x ) = ( Y gW ) ( x N g a ( − , x ) = K X k =0 k ! ( Y gW ) (( N kg a )( − , x )(log x ) k , where ( Y gW ) ( x N g a ( − , x ) is the constant term when Y gW ( a ( − , x ) is viewed as a poly-nomial in log x . So in our notation,( Y gW ) ( x N g a ( − , x ) = X n ∈ α + Z ( a W ) n, x − n − ,Y gW ( a ( − , x ) = K X k =0 X n ∈ α + Z k ! (( N kg a ) W ) n, x − n − (log x ) k . We shall need the following version ((3.24) in [Hua4]) of the Jacobi identity for ( Y gW ) (ob-tained in [B] and proved to be equivalent to the duality property for ( Y gW ) in [HY]): x − δ (cid:18) x − x x (cid:19) Y gW ( u, x ) Y gW ( v, x ) − x − δ (cid:18) − x + x x (cid:19) Y gW ( v, x ) Y gW ( u, x )= x − δ (cid:18) x + x x (cid:19) Y gW Y V (cid:18) x + x x (cid:19) L g u, x ! v, x ! . (4.1)This Jacobi identity holds for lower-bounded generalized twisted modules and even moregeneral types of twisted modules for an arbitrary grading-restricted vertex algebra, including,in particular, M ( ℓ,
0) or L ( ℓ, Proposition 4.1
Let W be a lower-bounded generalized g -twisted M ( ℓ, -module. Then W ,with the action of ˆ g [ g ] given by a ⊗ t n ( a W ) n, and k ℓ W for α ∈ P g , a ∈ g [ α ] and n ∈ α + Z and with the existing action of g , S g and N g on W , is a lower-bounded ˆ g [ g ] -moduleof level ℓ .Proof. Taking Res x on both sides of (4.1) and taking u = a i ( − and v = a j ( − , weobtain the commutator formula Y gW ( a i ( − , x ) Y gW ( a j ( − , x ) − Y gW ( a j ( − , x ) Y gW ( a i ( − , x )= Res x x − δ (cid:18) x + x x (cid:19) Y gW Y M ( ℓ, (cid:18) x + x x (cid:19) L g a i ( − , x ! a j ( − , x ! . (4.2)12y definition, the left-hand side of (4.2) is equal to X m ∈ α i + Z X k ∈ N X n ∈ α j + Z X l ∈ N ( − k k ! ( − l l ! (( N kg a i ) W ) m, (( N lg a j ) W ) n, x − m − x − n − (log x ) k (log x ) l − X m ∈ α i + Z X k ∈ N X n ∈ α j + Z X l ∈ N ( − k k ! ( − l l ! (( N lg a j ) W ) n, (( N kg a i ) W ) m, x − m − x − n − (log x ) k (log x ) l . (4.3)On the other hand, by straightforward calculations, we see that the right-hand side of (4.2)is equal toRes x e x ∂∂x x − δ (cid:18) x x (cid:19) Y gW Y M ( ℓ, (cid:18) x x (cid:19) L g a i ( − , x ! a j ( − , x ! = X n ∈ Z X k ∈ N X l ∈ N X p ∈ α i + α j + Z ( − k k ! ( − l l ! (([ N kg a i , N lg a j ]) W ) p, x − n − α i − x n + α i − p − (log x ) k (log x ) l + X n ∈ Z X k ∈ N X l ∈ N ( − k k ! ( − l l ! ℓ ( N kg a i , N lg a j ) ∂∂x x − n − α i − x n + α i (log x ) k (log x ) l . (4.4)Taking coefficients of x − m − x − n − (log x ) k (log x ) l for m ∈ α i + Z , n ∈ α j + Z , k, l ∈ N inboth sides of (4.2), using (4.3) and (4.4), dividing the both results by ( − k k ! ( − l l ! and thenusing Corollary 2.5, we obtain[(( N kg a i ) W ) m, , (( N lg a j ) W ) n, ]= (([ N kg a i , N lg a j ]) W ) m + n, + m ( N kg a i , N lg a j ) δ m + n, ℓ + ( N k +1 g a i , N lg a j ) δ m + n, ℓ. (4.5)Let a = N kg a i and b = N lg a j . Also note that g is certainly spanned by such a and b . Then(4.5) is exactly what we can also obtain by replacing a ⊗ t n and k in (2.1) by ( a W ) n, and ℓ W for a ∈ g [ α ] and n ∈ α + Z . Thus (4.5) gives W a structure of a lower-bounded ˆ g [ g ] -moduleof level ℓ .We first construct and identify explicitly universal lower-bounded generalized g -twisted M ( ℓ, g [ g ]+ using the results in Section 5 of[Hua5] when M ( ℓ,
0) is viewed as a grading-restricted vertex algebra.Let M be a vector space. Assume that g acts on M and there is an operator L M (0)on M . If M is finite dimensional, then there exist operators L g , S g , N g such that on M , g = e πi L g and S g and N g are the semisimple and nilpotent, respectively, parts of L g . In thiscase, M is also a direct sum of generalized eigenspaces for the operator L M (0) and L M (0)can be decomposed as the sum of its semisimple part L M (0) S and nilpotent part L M (0) N .Moreover, the real parts of the eigenvalues of L M (0) has a lower bound. In the case that M is infinite dimensional, we assume that all of these properties for g and L M (0) hold. We callthe eigenvalue of a generalized eigenvector w ∈ M for L M (0) the (conformal) weight of w w . We first assume that M is itself a generalized eigenspace of L M (0)with eigenvalue h .Let { w a } a ∈ A be a basis of M consisting of vectors homogeneous in g -weights (eigenvaluesof g ) such that for a ∈ A , either L M (0) N w a = 0 or there exists L M (0) N ( a ) ∈ A such that L M (0) N w a = w L M (0) N ( a ) . For simplicity, when L M (0) N w a = 0, we shall use w L M (0) N ( a ) todenote 0. Then for a ∈ A , we always have L M (0) N w a = w L M (0) N ( a ) . For a ∈ A , let α a ∈ C such that ℜ ( α a ) ∈ [0 ,
1) and e πiα a is the eigenvalue of g for the generalized eigenvector w a .Taking the grading-restricted vertex algebra V , the space M and B ∈ R in Section 5of [Hua5] to be M ( ℓ, M above and ℜ ( h ), respectively, we obtain the universallower-bounded generalized g -twisted M ( ℓ, c M [ g ] h , which we shall denote by c M [ g ] ℓ,h toexhibit explicit the dependence on ℓ . The twisted generating fields and generator twist fieldsfor c M [ g ] ℓ,h are denoted by a i c M [ g ] ℓ,h ( x ) = X n ∈ α i + Z K i X k =0 ( a i c M [ g ] ℓ,h ) n,k x − n − (log x ) k for i ∈ I and ψ a c M [ g ] ℓ,h ( x ) = X n ∈ α i + Z K i X k =0 ( ψ a c M [ g ] ℓ,h ) n,k x − n − (log x ) k for a ∈ A . For simplicity, we shall denote a i c M [ g ] ℓ,h ( x ) and ( a i c M [ g ] ℓ,h ) n,k by a i [ g ] ,ℓ ( x ), ( a i [ g ] ,ℓ ) n,k ,respectively, since their commutators involve ℓ and denote ψ a c M [ g ] ℓ,h ( x ) and ( ψ a c M [ g ] ℓ,h ) n,k by ψ a [ g ] ( x )and ( ψ a [ g ] ) n,k , respectively. For a general element a ∈ g [ α ] and w ∈ M [ β ] , we shall use thesimilar notations to denote the twisted and twist fields associated to a and w , respectively,and similarly for their components.The construction above is based on the assumption that M is itself a generalized eigenspaceof L M (0) with eigenvalue h . In the general case, M = ` h ∈ Q M M [ h ] , where Q M is the setof all eigenvalues of L M (0) and M [ h ] is the generalized eigenspace of L M (0) with the eigen-value h . In this case, we have the lower-bounded generalized g -twisted M ( ℓ, ` h ∈ Q M \ ( M [ h ] ) [ g ] ℓ,h , which we shall denote by c M [ g ] ℓ . For h ∈ Q M , we have a basis { w a } a ∈ A h of M [ h ] satisfying the condition L M (0) N w a = w L M (0) N ( a ) for a ∈ A h . Let A = ⊔ h ∈ Q M A h . Thenwe have a basis { w a } a ∈ A of M satisfying the same condition for all a ∈ A .We now construct a lower-bounded ˆ g [ g ] -module that we will prove to be equivalent to c M [ g ] ℓ viewed as a lower-bounded ˆ g [ g ] -module. Let L − be a basis of a one-dimensional vector space C L − . Let T ( C L − ) be the tensor algebra of the one-dimensional space C L − . Considerthe vector space Λ( M ) = T ( C L − ) ⊗ M . We define actions of g , L g , S g and N g on Λ( M )by acting only on the second tensor factor M . We define an operator L Λ( M ) (0) on Λ( M )by L Λ( M ) (0)( L m − ⊗ w ) = m ( L m − ⊗ w ) + L m − ⊗ L M (0) w for m ∈ N and w ∈ M . We alsodefine operators L Λ( M ) (0) N and L Λ( M ) (0) S on Λ( M ) by L Λ( M ) (0) S ( L m − ⊗ w ) = m ( L m − ⊗ w ) + L m − ⊗ L M (0) S w and L Λ( M ) (0) N ( L m − ⊗ w ) = L m − ⊗ L M (0) N w , respectively, for m ∈ N and w ∈ M . Then L Λ( M ) (0) N and L Λ( M ) (0) S are the semisimple and nilpotent, respectively, parts14f L Λ( M ) (0). The space Λ( M ) is graded by the eigenvalues of L Λ( M ) (0). We define anotheroperator L Λ( M ) ( −
1) on Λ( M ) by L Λ( M ) ( − L m − ⊗ w ) = L m +1 − ⊗ w . Then Λ( M ) is spannedby elements of the form L Λ( M ) ( − m (1 ⊗ w ) for m ∈ N and w ∈ M . For simplicity, we shallidentify 1 ⊗ w with w ∈ M and hence embed M as a subspace of Λ( M ). Thus Λ( M ) isspanned by elements of the form L Λ( M ) ( − m w for w ∈ M .Let ˆ g [ g ]+ act on M as 0. We define an action of ˆ g [ g ]+ on Λ( M ) by the commutator formula[ a ( m ) , L Λ( M ) ( − ma ( m −
1) + ( N g a )( m −
1) (4.6)for a ∈ g [ α ] and m ∈ α + N when ℜ ( α ) > m ∈ α + Z + when ℜ ( α ) = 0. Let k act on Λ( M ) as ℓ . Then it is clear that Λ( M ) is a U (ˆ g [ g ]+ ⊕ C k )-module. and we havethe induced lower-bounded ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) (recalling that by a lower-bounded ˆ g [ g ] -module we mean a lower-bounded ˆ g [ g ] -module with a compatible g action).Using the commutator formula (4.6), we can extend the operator L Λ( M ) ( −
1) to an operatoron U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ). For simplicity, we shall still denote this extension of L Λ( M ) ( − L Λ( M ) ( − L Λ( M ) ( −
1) acts on U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M )now. Theorem 4.2
As a lower-bounded ˆ g [ g ] -module, c M [ g ] ℓ is equivalent to U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) .Proof. Consider the subspace b Λ( M ) of c M [ g ] ℓ spanned by elements of the form L c M [ g ] ℓ ( − k ( ψ b [ g ] ) − , (4.7)for k ∈ N and b ∈ A . Then we have a linear map ρ : Λ( M ) → b Λ( M ) defined by ρ ( L Λ( M ) ( − k w b ) = L c M [ g ] ℓ ( − k ( ψ b [ g ] ) − , for k ∈ N and b ∈ A . In particular, ρ ( w b ) = ( ψ b [ g ] ) − , for b ∈ A . So ρ ( M ) is the subspace of c M [ g ] ℓ spanned by ( ψ b [ g ] ) − , for b ∈ A . From the ˆ g [ g ] -module structure on c M [ g ] ℓ , we see thatˆ g [ g ]+ acts on ρ ( M ) as 0. From the commutator formula[ L c M [ g ] ℓ,h ( − , a [ g ] ,ℓ ( x )] = ddx a [ g ] ,ℓ ( x ) , we obtain [ L c M [ g ] ℓ,h ( − , ( a [ g ] ,ℓ ) m, ] = m ( a [ g ] ,ℓ ) m − , + (( N g a ) [ g ] ,ℓ ) m − , for a ∈ g [ α ] and m ∈ α + N when ℜ ( α ) > m ∈ α + Z + when ℜ ( α ) = 0. Thus wealso have an action of ˆ g [ g ]+ on b Λ( M ). From the ˆ g [ g ] -module structure on c M [ g ] ℓ again, we seethat k acts on c M [ g ] ℓ as ℓ . These actions give b Λ( M ) a ˆ g [ g ]+ ⊕ C k -module structure. Fromthe definitions of ρ and the ˆ g [ g ]+ ⊕ C k -module structures on Λ( M ) and b Λ( M ), we see that15 is in fact a ˆ g [ g ]+ ⊕ C k -module map. Moreover, by Theorem 2.4 in [Hua6], for h ∈ Q M , L c M [ g ] ℓ,h ( − k ( ψ b [ g ] ) − , for k ∈ N and b ∈ A h are linearly independent and thus form a basisof b Λ( M h ), which is the subspace of b Λ( M ) spanned by elements of the form (4.7) for k ∈ N and b ∈ A h . Then L c M [ g ] ℓ,h ( − k ( ψ b [ g ] ) − , for k ∈ N and b ∈ A form a basis of b Λ( M ). So ρ isin fact an equivalence of ˆ g [ g ]+ ⊕ C k -modules and commutes with the actions of L c M [ g ] ℓ ( −
1) and L c M [ g ] ℓ ( − U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ), thereexists a unique ˆ g [ g ] -module mapˆ ρ : U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) → c M [ g ] ℓ such that ˆ ρ | Λ( M ) = ρ . Since c M [ g ] ℓ as a ˆ g [ g ] -module is generated by Λ( M ), ˆ ρ is surjective. Weneed only prove that ˆ ρ is injective.For h ∈ Q M , by Theorem 2.3 in [Hua6], c M [ g ] ℓ,h is spanned by elements of the form( a i [ g ] ,ℓ ) n ,k · · · ( a i l [ g ] ,ℓ ) n l ,k l L c M [ g ] ℓ,h ( − k ( ψ b [ g ] ) − , for n j ∈ α i j + Z , 0 ≤ k j ≤ K j , k ∈ N and b ∈ A h . On the other hand, from a i [ g ] ,ℓ ( x ) = Y g c M [ g ] ℓ,h ( a i ( − , x ) = ( Y g c M [ g ] ℓ,h ) ( x −N g a i ( − , x ) , we obtain ( a i [ g ] ,ℓ ) n i ,k i = ( − k i k i ! ( N k i g a i [ g ] ,ℓ ) n i , . Moreover, N k i g a i [ g ] ,ℓ is a linear combination of a j for j ∈ I since a j for j ∈ I form a basis of g . Thus c M [ g ] ℓ,h is spanned by elements of the form( a i [ g ] ,ℓ ) n , · · · ( a i l [ g ] ,ℓ ) n l , L c M [ g ] ℓ,h ( − k ( ψ b [ g ] ) − , (4.8)for i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l , k ∈ N and b ∈ A h . Therefore c M g ] ℓ is spanned byelements of the form (4.8) with L c M [ g ] ℓ,h ( −
1) replaced by L c M [ g ] ℓ ( −
1) for i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l , k ∈ N and b ∈ A .On the other hand, U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) is spanned by elements of the form a i ( n ) · · · a i l ( n l ) L Λ( M ) ( − k w b (4.9)for i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l , k ∈ N , b ∈ A . Since ˆ ρ is a g [ g ] -module map, we haveˆ ρ ( a i ( n ) · · · a i l ( n l ) L Λ( M ) ( − w b ) = ( a i [ g ] ,ℓ ) n , · · · ( a i l [ g ] ,ℓ ) n l , L c M [ g ] ℓ ( − ψ b [ g ] ) − , i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l , k ∈ N and b ∈ A . To prove that ˆ ρ is injective,we prove that if we replace a i ( n ), L Λ( M ) ( −
1) and w b by ( a i [ g ] ,ℓ ) n, , L c M [ g ] ℓ ( −
1) and ( ψ b [ g ] ) − , ,respectively, for i ∈ I , n ∈ α i + Z and b ∈ A , the relations satisfied by elements of thespanning sets (4.9) of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) must be satisfied by elements of the spanningsets (4.8) of c M [ g ] ℓ .To prove this, we first list all the relations satisfied by (4.8). From the construction of c M [ g ] ℓ,h for h ∈ Q M given in Section 5 of [Hua5] and Theorem 2.4 in [Hua6], we see that theonly relations satisfied by elements of the form (4.8) for i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l and b ∈ A are generated by the following: (i) A homogeneous element of the form (4.8) with i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l , k ∈ N and b ∈ A h satisfying − n − · · · − n l < ℜ ( h )is equal to 0. (ii) The relations induced from the coefficients of the weak commutativity forthe generating g -twisted fields a i [ g ] ,ℓ ( x ) for i ∈ I . (iii) The commutator relations between( a i [ g ] ,ℓ ) n, and L c M [ g ] h ( −
1) for a ∈ g [ α ] and m ∈ α + N (when ℜ ( α ) >
0) and m ∈ α + Z + (when ℜ ( α ) = 0). The other relations given in Section 5 of [Hua5] involve elements that are not ofthe form (4.8).We need only prove that elements of the form (4.9) also satisfy the relations correspondingto the relations (i), (ii) and (iii). By the definitions of the actions of a i ( n ) on U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) and the fact that the weights of w b for b ∈ A h are h , elements of the form (4.9) satisfythe relations corresponding to (i). Since a i ( n ) and ( a i [ g ] ,ℓ ) n, for i ∈ I and n ∈ α i + Z satisfy thesame commutator formula, a i ( x ) = P n ∈ α i + Z a i ( n ) x − n − and a i [ g ] ,ℓ ( x ) for i ∈ I also satisfy thesame commutator formula. Since weak commutativity follows from the commutator formulafor generating twisted fields, we see that elements of the form (4.9) satisfy the relationscorresponding to (ii). Since ρ is in fact an equivalence of ˆ g [ g ]+ ⊕ C k -modules and commuteswith the actions of L c M [ g ] ℓ ( −
1) and L c M [ g ] ℓ ( − Remark 4.3
By the Poincar´e-Birkhoff-Witt theorem, the induced lower-bounded ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) is linearly isomorphic to U (ˆ g [ g ] − ) ⊗ U (ˆ g [ g ] I ) ⊗ Λ( M ). In particular, U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) is spanned by elements of the form a i ( n ) · · · a i l ( n l ) a j ( α j ) · · · a j m ( α j m ) L Λ( M ) ( − k w b for i p , j q ∈ I , n p ∈ α i p − Z + , ℜ ( α j q ) = 0 for p = 1 , . . . , l , q = 1 , . . . , m , k ∈ N and b ∈ A . Using the commutator formula between a j ( α j ) and L Λ( M ) ( −
1) for j ∈ I I , we see that U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) is also spanned by elements of the form a i ( n ) · · · a i l ( n l ) L Λ( M ) ( − k a j ( α j ) · · · a j m ( α j m ) w b (4.10)for i p , j q ∈ I , n p ∈ α i p − Z + , ℜ ( α j q ) = 0 for p = 1 , . . . , l , q = 1 , . . . , m , k ∈ N and b ∈ A . ByTheorem 4.2, we see that c M [ g ] ℓ,h is spanned by elements of the form( a i [ g ] ,ℓ ) n , · · · ( a i l [ g ] ,ℓ ) n l , ( a j [ g ] ,ℓ ) α j , · · · ( a j m [ g ] ,ℓ ) α jm , L c M [ g ] ℓ,h ( − k ( ψ b [ g ] ) − , i p , j q ∈ I , n p ∈ α i p − Z + , ℜ ( α j q ) = 0 for p = 1 , . . . , l , q = 1 , . . . , m , k ∈ N and b ∈ A .Using the commutator formula between ( a j [ g ] ,ℓ ) α j , and L c M [ g ] ℓ,h ( − c M [ g ] ℓ,h is alsospanned by elements of the form( a i [ g ] ,ℓ ) n , · · · ( a i l [ g ] ,ℓ ) n l , L c M [ g ] ℓ,h ( − k ( a j [ g ] ,ℓ ) α j , · · · ( a j m [ g ] ,ℓ ) α jm , ( ψ b [ g ] ) − , (4.11)for i p , j q ∈ I , n p ∈ α i p − Z + , ℜ ( α j q ) = 0 for p = 1 , . . . , l , q = 1 , . . . , m , k ∈ N and b ∈ A .Next we construct universal grading-restricted generalized g -twisted M ( ℓ, U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) to be finite dimensional since a i (0) for a i ∈ ˆ g [0] act on w b generate an infinite-dimensional homogeneous subspace. But if M is a finite-dimensional ˆ g I -module, a quotientof U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) might be grading restricted. Since U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) as alower-bounded ˆ g [ g ] -module is equivalent to c M [ g ] ℓ , the same discussion applies to c M [ g ] ℓ .Now we assume that M is in addition a finite-dimensional ˆ g I -module with a compatibleaction of g . Here by M has a compatible action of g we mean g ( a ( n ) w ) = ( g ( a ))( n ) g ( w ) for a ∈ g [ α ] such that ℜ ( α ) = 0, n ∈ α + Z and w ∈ M . We have a universal lower-boundedgeneralized g -twisted M ( ℓ, c M [ g ] ℓ . Since M in our case is a ˆ g [ g ] I -module but theconstruction of c M [ g ] ℓ above does not use such a structure on M , to incorporate such anaction on M , we need to take a further quotient. Let I I be the set of elements α i of I suchthat ℜ ( α i ) = 0. Then ˆ g [ g ] I is spanned by elements of the form a i ( α i ) for i ∈ I I . Since { w a } a ∈ A is a basis of M , there exist λ aic ∈ C for i ∈ I I and b, c ∈ A such that a i ( α i ) w b = X c ∈ A λ bic w c for i ∈ I I and b ∈ A .Consider the lower-bounded generalized g -twisted M ( ℓ, c M [ g ] ℓ,h generatedby elements of the form ( a i [ g ] ,ℓ ) α i , ( ψ b [ g ] ,h b ) − , − X c ∈ A λ bic ( ψ c [ g ] ,h c ) − , (4.12)for i ∈ I I and b ∈ A . We denote the quotient of c M [ g ] ℓ by this submodule by ( M [ g ] ℓ . Then ( M [ g ] ℓ is also a lower-bounded generalized g -twisted M ( ℓ, c M [ g ] ℓ to denotes the corresponding fields and their coefficients for ( M [ g ] ℓ .On the ˆ g [ g ] I -module M , we define an action of ˆ g [ g ]+ to be 0. Then we use the commutatorformula (4.6) for a ∈ g [ α ] and m ∈ α + N to define an action of ˆ g [ g ]+ ⊕ ˆ g [ g ] I on Λ( M ). Let k act on Λ( M ) as ℓ . Then Λ( M ) becomes a ˆ g [ g ]+ ⊕ ˆ g [ g ]0 -module and we have the inducedlower-bounded ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 Λ( M ). From the construction, we see that the18 g [ g ] -module U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 Λ( M ) is in fact the quotient of ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M )by the submodule generated by elements of the form a i ( α i ) ⊗ w b − X c ∈ A λ bic w c (4.13)for i ∈ I I and b ∈ A . Theorem 4.4
As a lower-bounded ˆ g [ g ] -module, ( M [ g ] ℓ is equivalent to the induced lower-bounded ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 Λ( M ) .Proof. By Theorem 4.2, c M [ g ] ℓ as a lower-bounded ˆ g [ g ] -module is equivalent to ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ C k Λ( M ). It is also clear that the submodule of c M [ g ] ℓ generated by elements ofthe form (4.12) for i ∈ I I and b ∈ A and the submodule of ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ C k Λ( M )generated by elements of the form (4.13) are equivalent under the equivalence from c M [ g ] ℓ toˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ C k Λ( M ). Thus their quotients ( M [ g ] ℓ and ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 Λ( M ) are equivalent. Remark 4.5
From the construction of U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 Λ( M ), it is spanned by elements ofthe form a i ( n ) · · · a i l ( n l ) L Λ( M ) ( − k w b (4.14)for i p ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l , k ∈ N and b ∈ A . Similarly, from the constructionof ( M [ g ] ℓ , it is spanned by elements of the form( a i [ g ] ,ℓ ) n , · · · ( a i l [ g ] ,ℓ ) n l , L c M [ g ] ℓ,h ( − k ( ψ b [ g ] ) − , (4.15)for i p , j q ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l , k ∈ N and b ∈ A .We are ready to prove that ( M [ g ] ℓ is in fact grading restricted now. Theorem 4.6
The lower-bounded generalized g -twisted M ( ℓ, -module ( M [ g ] ℓ is in fact grad-ing restricted.Proof. By Theorem 4.4, we need only prove that U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 Λ( M ) is grading restricted.By Remark 4.5, U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 Λ( M ) is spanned by elements of the form (4.14) for i p ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l , k ∈ N and b ∈ A . The weight of such an element is − n − · · · − n l + k + wt w a . For fixed n ∈ C , elements of weight n of the form (4.14) mustsatisfy n = − n − · · · − n l + k + wt w a . So we have n + · · · + n l − k = − n + wt w a . M is finite dimensional, there are only finitely many w a and thus finitely many wt w a .Let N ∈ R such that ℜ (wt w a ) ≥ N for a ∈ A . Then ℜ ( n ) + · · · + ℜ ( n l ) − k = −ℜ ( n ) + ℜ (wt w a ) ≥ −ℜ ( n ) + N. On the other hand, since n j ∈ α i j − Z + , ℜ ( n j ) < > ℜ ( n ) + · · · + ℜ ( n l ) − k ≥ −ℜ ( n ) + N. (4.16)Let P = max i ∈ I {ℜ ( α i ) − } . Then P ∈ [ − , n j = α i j − Z + = α i j − − N , we have ℜ ( n j ) ≤ ℜ ( α i j ) − ≤ P <
0. So ℜ ( n ) + · · · + ℜ ( n l ) ≤ lP . If lP < −ℜ ( n ) + N , we have ℜ ( n ) + · · · + ℜ ( n l ) − k < −ℜ ( n ) + N − k ≤ −ℜ ( n ) + N . Contradiction to (4.16). Thus wemust have lP ≥ −ℜ ( n ) + N or equivalently, l ≤ P ( −ℜ ( n ) + N ) (note that P < ℜ ( n j ) < j = 1 , . . . , l , from (4.16) and − k ≤
0, we obtain also ℜ ( n j ) ≥ −ℜ ( n ) + N and − k ≥ −ℜ ( n ) + N . From 0 > ℜ ( n j ) ≥ −ℜ ( n ) + N for j = 1 , . . . , l and 0 ≥ − k ≥ −ℜ ( n ) + N ,we see that for fixed l , there are only finitely many possible choices of a i j , n j and k . Thusfor fixed n ∈ C , there are only finitely many elements of weight n of the form (4.14). So U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ] I M , or equivalently, ( M [ g ] ℓ is grading restricted. M ( ℓ, is viewed as a vertex operatoralgebra Assume that g is simple and ℓ + h ∨ = 0. Then M ( ℓ,
0) has a conformal vector ω M ( ℓ, andthus is a vertex operator algebra. Now we want to construct and identify explicitly universallower-bounded generalized g -twisted modules for M ( ℓ,
0) viewed as a vertex operator alge-bra. Since in [Hua5], we give only the construction for a grading-restricted vertex algebraor a M¨obius vertex algebra, here we first give a construction of universal lower-boundedgeneralized twisted module for a general vertex operator algebra.Let V be a vertex operator algebra, that is, a grading-restricted vertex algebra V with aconformal element ω , and g an automorphism of V as a vertex operator algebra (meaning inparticular that g fixes ω ). Let M be a vector space with actions of g , S g , N g , L M (0), L M (0) S and L M (0) N and B a real number such that M is a direct sum of generalized eigenspaces of L M (0) and the real parts of the eigenvalues of L M (0) are larger than or equal to B . FromSection 5 of [Hua5], we have a universal lower-bounded generalized g -twisted V -module c M [ g ] B . Since g fix ω , the coefficients of Y g c M [ g ] B ( ω, x ) satisfy the Virasoro commutator relations.Note that for a lower-bounded generalized g -twisted V -module W , the operator L W (0) and L W ( −
1) must be equal to the coefficients of x − and x − , respectively, in the vertex operator Y W ( ω, x ). But L c M [ g ] B (0) and L c M [ g ] B ( −
1) for c M [ g ] B are not equal to the coefficients of x − and x − , respectively. To obtain a lower-bounded generalized g -twisted module for V viewed asa vertex operator algebra, we have to take the quotient by a submodule generated by thedifference of these operators acting on elements of c M [ g ] B .20onsider the lower-bounded generalized g -twisted V -submodule of c M [ g ] B generated byelements of the form L c M [ g ] B (0) w − Res x xY g c M [ g ] B ( ω, x ) w,L c M [ g ] B ( − w − Res x Y g c M [ g ] B ( ω, x ) w for w ∈ c M [ g ] B . We shall denote the quotient of c M [ g ] B by this submodule by > M [ g ] B and call this quotient module the lower-bounded generalized g -twisted V -module for thevertex operator algebra V , not the underlying grading-restricted vertex algebra V . ByTheorem 5.2 and the construction of c M [ g ] B in [Hua5], we immediately obtain the followingresult: Theorem 4.7
Let V be a vertex operator algebra and ( W, Y gW ) a lower-bounded generalized g -twisted V -module and M a subspace of W invariant under the actions of g , S g , N g , L W (0) = Res x xY gW ( ω, x ) , L W (0) S and L W (0) N . Let B ∈ R such that W [ n ] = 0 when ℜ ( n ) f : > M [ g ] B → W such that > f | M = f . If f is surjective and ( W, Y gW ) is generated by the coefficients of ( Y g ) WW V ( w, x ) v for w ∈ M and v ∈ V , where ( Y g ) WW V isthe twist vertex operator map obtained from Y gW , then > f is surjective. We now assume that g is simple and ℓ + h ∨ = 0. Then M ( ℓ,
0) is a vertex operatoralgebra. Take the vertex operator algebra V above to be M ( ℓ,
0) and g an automorphismof M ( ℓ,
0) induced from an automorphism of g as discussed above. Let M , as above, bea vector space with actions of g , L g , S g , N g , L M (0), L M (0) S and L M (0) N . We assumethat M = ` h ∈ Q M M [ h ] as above, where M [ h ] is the generalized eigenspace of L M (0) witheigenvalue h and Q M is the set of all eigenvalues of L M (0). For h ∈ Q h , take V , g , M and B in the construction above to be M ( ℓ, g , M [ h ] , ℜ ( h ). Then we have a universallower-bounded generalized g -twisted M ( ℓ, > ( M [ h ] ) [ g ] h . To exhibit its dependence on ℓ explicitly, we denote it by > ( M [ h ] ) [ g ] ℓ,h . Adding them together, we obtain a lower-boundedgeneralized g -twisted M ( ℓ, ` h ∈ Q M > ( M [ h ] ) [ g ] ℓ,h , which we shall denote by > M [ g ] ℓ . Weshall use the same notations a i [ g ] ,ℓ ( x ), ( a i [ g ] ,ℓ ) n,k , ψ b [ g ] ( x ) and ( ψ b [ g ] ) n,k and so on as those for c M [ g ] ℓ and ( M [ g ] ℓ to denote the generating twisted fields, their coefficients, the generator twistfields and their coefficients for > M [ g ] ℓ .We need to identify > M [ g ] ℓ with a suitable ˆ g [ g ] -module. We first need to identify L M (0)with the action of an element of U (ˆ g [ g ] ). Recall the Jordan basis { a i } i ∈ I of g that we havechosen in the end of the preceding section. Let { ( a i ) ′ } i ∈ I be the dual basis of { a i } i ∈ I with21espect to the nondegenerate bilinear form ( · , · ). For simplicity (but with an abuse of thenotation), we shall denote this dual basis by { a i ′ } i ∈ I . Then( e πi S g a i ′ , a j ) = ( a i ′ , e − πi S g a j ) = ( a i ′ , e − πiα j a j ) = e − πiα j δ ij = e − πiα i δ ij . This means that e πi S g a i ′ = e − πiα i a i ′ . So a i ′ ∈ g [1 − α i ] when ℜ ( α i ) > i ∈ I \ I I and a i ′ ∈ g [ − α i ] when ℜ ( α i ) = 0 or i ∈ I I . Byabuse of notation, let α i ′ = (cid:26) − α i i ∈ I \ I I , − α i i ∈ I I . By definition, the conformal element of M ( ℓ,
0) is ω M ( ℓ, = X i ∈ I a i ′ ( − a i ( − ∈ M [0] ( ℓ, , where M [0] ( ℓ,
0) is the fixed-point subalgebra of M ( ℓ, M ( ℓ, ω M ( ℓ, is in the fixed-pointsubalgebra of M ( ℓ, N g ω M ( ℓ, = 0. Hence Y g > M [ g ] ℓ ( ω M ( ℓ, , x ) = ( Y g > M [ g ] ℓ ) ( x −N g ω M ( ℓ, , x ) = ( Y g > M [ g ] ℓ ) ( ω M ( ℓ, , x )and from the equivariance property of the twisted vertex operators, Y g > ( M [ h ] ) [ g ] h ( ω, x ) or equiv-alently ( Y g > ( M [ h ] ) [ g ] h ) ( ω, x ) must have only integral powers of x . In particular, Y g > M [ g ] ℓ ( ω, x ) = ( Y g > M [ g ] ℓ ) ( ω, x ) = X n ∈ Z L > M [ g ] ℓ ( n ) x − n − . where L > M [ g ] ℓ ( n ) for n ∈ Z are the Virasoro operators on > M [ g ] ℓ satisfying the Virasoro commu-tator relations with central charge ℓ dim g ℓ + h ∨ . In particular, we have the operators L > M [ g ] ℓ (0) and L > M [ g ] ℓ ( − Proposition 4.8
For n ∈ Z , L > M [ g ] ℓ ( n )= X i ∈ I X p ∈ α i + Z + ℓ + h ∨ ) a i ′ [ g ] ,ℓ ( − p ) a i [ g ] ,ℓ ( p + n ) + X i ∈ I X p ∈ α i − N ℓ + h ∨ ) a i [ g ] ,ℓ ( p + n ) a i ′ [ g ] ,ℓ ( − p ) − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ] [ g ] ,ℓ ( n ) − X i ∈ I ℓδ n, ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i ) . (4.17)22 roof. We take W = > M [ g ] ℓ , u = x N g a i ′ ( − and v = x N g a i ( − in the Jacobi identity(4.1). Then Y g > M [ g ] ℓ ( u, x ) = ( Y g > M [ g ] ℓ ) ( a i ′ ( − , x ), Y g > M [ g ] ℓ ( v, x ) = ( Y g > M [ g ] ℓ ) ( a i ( − , x ) and S g a i ′ ( − = α i ′ a i ′ ( − . Then (4.1) becomes x − δ (cid:18) x − x x (cid:19) ( Y g > M [ g ] ℓ ) ( a i ′ ( − , x )( Y g > M [ g ] ℓ ) ( a i ( − , x ) − x − δ (cid:18) − x + x x (cid:19) ( Y g > M [ g ] ℓ ) ( a i ( − , x )( Y g > M [ g ] ℓ ) ( a i ′ ( − , x )= x − δ (cid:18) x + x x (cid:19) (cid:18) x + x x (cid:19) − α i ·· ( Y g > M [ g ] ℓ ) Y M ( ℓ, (cid:18) x x (cid:19) N g a i ′ ( − , x ! a i ( − , x ! , (4.18)where we have used x − δ (cid:18) x + x x (cid:19) (cid:18) x + x x (cid:19) α i ′ = x − δ (cid:18) x + x x (cid:19) (cid:18) x + x x (cid:19) − α i . Multiplying both sides of (4.18) by x − α i and then take Res x , rewriting ( x + x ) − α i as x − α i (cid:16) x x (cid:17) − α i and then multiplying both sides by x α i , we obtainRes x x − α i x α i x − δ (cid:18) x − x x (cid:19) ( Y g > M [ g ] ℓ ) ( a i ′ ( − , x )( Y g > M [ g ] ℓ ) ( a i ( − , x ) − Res x x − α i x α i x − δ (cid:18) − x + x x (cid:19) ( Y g > M [ g ] ℓ ) ( a i ( − , x )( Y g > M [ g ] ℓ ) ( a i ′ ( − , x )= (cid:18) x x (cid:19) − α i ( Y g > M [ g ] ℓ ) Y M ( ℓ, (cid:18) x x (cid:19) N g a i ′ ( − , x ! a i ( − , x ! . (4.19)Using the definition, we have( Y g > M [ g ] ℓ ) ( a ( − , x ) = ( a [ g ] ,ℓ ) ( x ) , where ( a [ g ] ,ℓ ) ( x ) = X n ∈ α + Z ( a [ g ] ,ℓ ) n, x − n − for a ∈ g [ α ] . Then the constant term in x (or equivalently, the result of applying Res x x − )23f the right-hand side of (4.19) is equal toRes x x − ( Y g > M [ g ] ℓ ) Y M ( ℓ, (cid:18) x x (cid:19) N g − α i a i ′ ( − , x ! a i ( − , x ! = X m ∈ N Res x x − (cid:18) x x (cid:19) m ( Y g > M [ g ] ℓ ) (cid:18) Y M ( ℓ, (cid:18)(cid:18) N g − α i m (cid:19) a i ′ ( − , x (cid:19) a i ( − , x (cid:19) = X m ∈ N Res x x − (cid:18) x x (cid:19) m ( Y g > M [ g ] ℓ ) (cid:18)(cid:18)(cid:18) N g − α i m (cid:19) a i ′ (cid:19) ( x ) a i ( − , x (cid:19) = X m ∈ N X n ∈ Z Res x x − (cid:18) x x (cid:19) m x − n − ( Y g > M [ g ] ℓ ) (cid:18)(cid:18)(cid:18) N g − α i m (cid:19) a i ′ (cid:19) ( n ) a i ( − , x (cid:19) = X m ∈ N x − m ( Y g > M [ g ] ℓ ) (cid:18)(cid:18)(cid:18) N g − α i m (cid:19) a i ′ (cid:19) ( m − a i ( − , x (cid:19) = ( Y g > M [ g ] ℓ ) ( a i ′ ( − a i ( − , x )+ x − ( Y g > M [ g ] ℓ ) ((( N g − α i ) a i ′ )(0) a i ( − , x )+ x − Y g > M [ g ] ℓ ) ((( N g − α i )( N g − α i − a i ′ )(1) a i ( − , x )= ( Y g > M [ g ] ℓ ) ( a i ′ ( − a i ( − , x ) + x − ([( N g − α i ) a i ′ , a i ] [ g ] ,ℓ ) ( x )+ ℓx − N g − α i )( N g − α i − a i ′ , a i ) . (4.20)Applying Res x x − to both sides of (4.19), using (4.20), taking sum over i ∈ I on both24ides and dividing both sides by 2( ℓ + h ∨ ), we obtain X n ∈ Z L > M [ g ] ℓ ( n ) x − n − = ( Y g > M [ g ] ℓ ) ( ω, x )= X i ∈ I ℓ + h ∨ ) ( Y g > M [ g ] ℓ ) ( a i ′ ( − a i ( − , x )= X i ∈ I ℓ + h ∨ ) Res x x − Res x x − α i x α i x − δ (cid:18) x − x x (cid:19) ( a i ′ [ g ] ,ℓ ) ( x )( a i [ g ] ,ℓ ) ( x ) − X i ∈ I ℓ + h ∨ ) Res x x − Res x x − α i x α i x − δ (cid:18) − x + x x (cid:19) ( a i [ g ] ,ℓ ) ( x )( a i ′ [ g ] ,ℓ ) ( x ) − X i ∈ I x − ℓ + h ∨ ) ([( N g − α i ) a i ′ , a i ] [ g ] ,ℓ ) ( x ) − X i ∈ I ℓx − ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i )= X i ∈ I ℓ + h ∨ ) Res x x − α i x α i ( x − x ) − ( a i ′ [ g ] ,ℓ ) ( x )( a i [ g ] ,ℓ ) ( x ) − X i ∈ I ℓ + h ∨ ) Res x x − α i x α i ( − x + x ) − ( a i [ g ] ,ℓ ) ( x )( a i ′ [ g ] ,ℓ ) ( x ) − X i ∈ I x − ℓ + h ∨ ) ([( N g − α i ) a i ′ , a i ] [ g ] ,ℓ ) ( x ) − X i ∈ I ℓx − ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i ) . (4.21)25aking the coefficients of x − n − of (4.21), we obtain L > M [ g ] ℓ ( n )= X i ∈ I X m ∈ Z + X k ∈− α i + Z X l ∈ α i + Z ℓ + h ∨ ) Res x Res x x − α i − m − k − x α i + m + n − l a i ′ [ g ] ,ℓ ( k ) a i [ g ] ,ℓ ( l )+ X i ∈ I X m ∈ Z + X k ∈− α i + Z X l ∈ α i + Z ℓ + h ∨ ) Res x Res x x − α i + m − k x α i + n − m − l − a i [ g ] ,ℓ ( l ) a i ′ [ g ] ,ℓ ( k ) − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ] [ g ] ,ℓ ( n ) − X i ∈ I ℓδ n, ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i )= X i ∈ I X p ∈ α i + Z + ℓ + h ∨ ) a i ′ [ g ] ,ℓ ( − p ) a i [ g ] ,ℓ ( p + n ) + X i ∈ I X p ∈ α i − N ℓ + h ∨ ) a i [ g ] ,ℓ ( p + n ) a i ′ [ g ] ,ℓ ( − p ) − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ] [ g ] ,ℓ ( n ) − X i ∈ I ℓδ n, ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i ) , proving (4.17).From (4.17), we obtain L > M [ g ] ℓ (0)= X i ∈ I X p ∈ α i + Z + ℓ + h ∨ ) a i ′ [ g ] ,ℓ ( − p ) a i [ g ] ,ℓ ( p ) + X i ∈ I X p ∈ α i − N ℓ + h ∨ ) a i [ g ] ,ℓ ( p ) a i ′ [ g ] ,ℓ ( − p ) − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ] [ g ] ,ℓ (0) − X i ∈ I ℓ ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i )(4.22)and L > M [ g ] ℓ ( − X i ∈ I X p ∈ α i + Z + ℓ + h ∨ ) a i ′ [ g ] ,ℓ ( − p ) a i [ g ] ,ℓ ( p −
1) + X i ∈ I X p ∈ α i − N ℓ + h ∨ ) a i [ g ] ,ℓ ( p − a i ′ [ g ] ,ℓ ( − p ) − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ] [ g ] ,ℓ ( − . (4.23)26ote that for b ∈ A , L > M [ g ] ℓ (0)( ψ b [ g ] ) − , = X i ∈ I I ℓ + h ∨ ) a i [ g ] ,ℓ ( α i ) a i ′ [ g ] ,ℓ ( − α i )( ψ b [ g ] ) − , − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ] [ g ] ,ℓ (0)( ψ b [ g ] ) − , − X i ∈ I ℓ ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i )( ψ b [ g ] ) − , , (4.24)that is, as an operator on the subspace of > M [ g ] ℓ spanned by ( ψ b [ g ] ) − , , L > M [ g ] ℓ (0) is equal to X i ∈ I I ℓ + h ∨ ) a i [ g ] ,ℓ ( α i ) a i ′ [ g ] ,ℓ ( − α i ) − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ] [ g ] ,ℓ (0) − X i ∈ I ℓ ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i ) . For h ∈ Q M and b ∈ A h , by definition, L > M [ g ] ℓ (0)( ψ b [ g ] ) − , = h ( ψ b [ g ] ) − , . Together with (4.24), we obtain the relation h ( ψ b [ g ] ) − , = X i ∈ I I ℓ + h ∨ ) a i [ g ] ,ℓ ( α i ) a i ′ [ g ] ,ℓ ( − α i )( ψ b [ g ] ) − , − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ] [ g ] ,ℓ (0)( ψ b [ g ] ) − , − X i ∈ I ℓ ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i )( ψ b [ g ] ) − , . (4.25)Let Ω [ g ] = X i ∈ I I a i ( α i ) a i ′ ( − α i ) − X i ∈ I [( N g − α i ) a i ′ , a i ](0) − X i ∈ I ℓ N g − α i )( N g − α i − a i ′ , a i ) ∈ U (ˆ g [ g ] ) [0] , (4.26)where U (ˆ g [ g ] ) [0] is the fixed-point subspace of U (ˆ g [ g ] ) under g . Let Ω [ g ] , ˆ g [ g ]+ and k act on M as L M (0), 0 and ℓ , respectively. Let G (Ω [ g ] , ˆ g [ g ]+ , k ) be the subalgebra of U (ˆ g [ g ] ) generated byΩ [ g ] , ˆ g [ g ]+ and k . Then we have the induced lower-bounded ˆ g [ g ]+ -module U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M .Note that this induced ˆ g [ g ]+ -module is a quotient of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) M .27 heorem 4.9 The universal lower-bounded generalized g -twisted M ( ℓ, -module > M [ g ] ℓ isequivalent as a lower-bounded ˆ g [ g ] -module to U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M .Proof. We know that > M [ g ] ℓ as a quotient of c M [ g ] ℓ is also spanned by elements of the form(4.8) for i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l , k ∈ N and b ∈ A . Using (4.23), we see thatelements of the form (4.8) in > M [ g ] ℓ can be written as linear combinations of elements of theform ( a i [ g ] ,ℓ ) n , · · · ( a i l [ g ] ,ℓ ) n l , ( ψ b [ g ] ) − , (4.27)for i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l and b ∈ A . On the other hand, by definition, U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M is spanned by elements of the form a i ( n ) · · · a i l ( n l ) w b (4.28)for i j ∈ I , n j ∈ α i j + Z for j = 1 , . . . , l and b ∈ A .By Theorem 4.2, we have an invertible ˆ g [ g ] -module map ˆ ρ : U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) → c M [ g ] ℓ such that ˆ ρ maps the element (4.8) to the element (4.9). In particular, ˆ ρ maps theelement of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) of the same form as (4.28) to the element (4.27). Wewant to use the map ˆ ρ restricted to elements of the same form as (4.28) to obtain an invertibleˆ g [ g ] -module map from U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M to > M [ g ] ℓ .To do this, we need only prove that the relations among elements of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) of the same form as (4.28) and the relations among elements of the form (4.27) in > M [ g ] ℓ are the same. The relations among elements of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) of the same formas (4.28) are generated by the following two types: The first type of relations are inducedfrom the relations in U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ). The second type is the additional relations ℓ + h ∨ ) Ω [ g ] w b = L M (0) w b for b ∈ A . For h ∈ Q M and b ∈ A h , this additional relations become hw b = X i ∈ I I ℓ + h ∨ ) a i ( α i ) a i ′ ( − α i ) w b − X i ∈ I ℓ + h ∨ ) [( N g − α i ) a i ′ , a i ](0) w b − X i ∈ I ℓ ℓ + h ∨ ) (( N g − α i )( N g − α i − a i ′ , a i ) w b . (4.29)The first type of relations are the same as the corresponding type of relations in > M [ g ] ℓ byTheorem 4.2. The second type of relations (4.29) correspond exactly to the relations (4.24)in > M [ g ] ℓ . The relations (4.24) are also the only relations in > M [ g ] ℓ in addition to the relationsinduced from the relations in c M [ g ] ℓ . Thus the theorem is proved.28 emark 4.10 We have seen in the proof of Theorem 4.9 that U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M isspanned by elements of the form (4.28). Using the commutator relations for a i ( n ) for i ∈ I and n ∈ α i + Z , we see that it is in fact spanned by elements of the form a i ( n ) · · · a i l ( n l ) a j ( α j ) · · · a j m ( α j m ) w b (4.30)for i p ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l , j q ∈ I I for q = 1 , . . . , m and b ∈ A . By Theorem4.2, we also see that > M [ g ] ℓ is spanned by elements of the form( a i [ g ] ,ℓ ) n , · · · ( a i l [ g ] ,ℓ ) n l , ( a j [ g ] ,ℓ ) α j , · · · ( a j m [ g ] ,ℓ ) α jm , ( ψ b [ g ] ,h ) − , (4.31)for i p ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l , j q ∈ I I for q = 1 , . . . , m and b ∈ A .We now construct and identify explicitly grading-restricted generalized g -twisted mod-ules for M ( ℓ,
0) viewed as a vertex operator algebra. We assume that M is in addition afinite-dimensional ˆ g I -module with a compatible action of g such that the action of Ω [ g ] , orequivalently, the operator L M (0) = ℓ + h ∨ ) Ω [ g ] , on M is induced from this ˆ g I -module struc-ture. In particular, M is a direct sum of generalized eigenspaces of L M (0) as above. We havea universal lower-bounded generalized g -twisted M ( ℓ, > M [ g ] ℓ . As in the precedingsubsection, since ˆ g [ g ] I is spanned by elements of the form a i ( α i ) for i ∈ I I and { w a } a ∈ A is abasis of M , there exist λ aic ∈ C for i ∈ I I and b, c ∈ A such that a i ( α i ) w b = X c ∈ A λ bic w c for i ∈ I I and b ∈ A . Consider the lower-bounded generalized g -twisted M ( ℓ, > M [ g ] ℓ generated by elements of the form( a i [ g ] ,ℓ ) α i , ( ψ b [ g ] ) − , − X c ∈ A λ bic ( ψ c [ g ] ) − , for i ∈ I I and b ∈ A . We denote the quotient of > M [ g ] ℓ by this submodule by f M [ g ] ℓ . Then f M [ g ] ℓ is also a lower-bounded generalized g -twisted M ( ℓ, > M [ g ] ℓ to denote the corresponding fields and coefficients for f M [ g ] ℓ .On the ˆ g [ g ] I -module M , we define actions of ˆ g [ g ]+ and k to be 0 and ℓ , respectively. Thenwe have the induced lower-bounded ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M . Theorem 4.11
As a lower-bounded ˆ g [ g ] -module, f M [ g ] ℓ is equivalent to the induced lower-bounded ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M .Proof. By Theorem 4.9, we have an invertible ˆ g [ g ] -module map > ρ from U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M to > M [ g ] ℓ which maps (4.30) to (4.31). Since f M [ g ] ℓ is a quotient of > M [ g ] ℓ , we have a surjectiveˆ g [ g ] -module map ˜ ̺ from U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M to f M [ g ] ℓ .29ote that by Poincar´e-Birkhoff-Witt theorem, U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M as a graded vectorspace is isomorphic to U (ˆ g [ g ] − ) ⊗ M . In particular, the ˆ g [ g ] -module U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M isspanned by elements of the form a i ( n ) · · · a i l ( n l ) w b (4.32)for i p ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l and b ∈ A .By Remark 4.10, > M [ g ] ℓ is spanned by elements of the form (4.31) for i p ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l , j q ∈ I I for q = 1 , . . . , m and b ∈ A . Then f M [ g ] ℓ is spanned by elements of theform ( a i [ g ] ,ℓ ) n , · · · ( a i l [ g ] ,ℓ ) n l , ( ψ b [ g ] ) − , (4.33)for i p ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l and b ∈ A .Since elements of U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M of the same form as (4.32) are sent under > ρ toelements of > M [ g ] ℓ of the same form as (4.33), ˜ ̺ maps elements of U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M of thesame form as (4.32) to elements of f M [ g ] ℓ of the form (4.33). But the only relations amongelements of U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M of the same form as (4.32) are generated by the commutatorrelations for a i ( n ) for i ∈ I and n ∈ α i − Z + . Since ( a i [ g ] ,ℓ ) n, for i ∈ I and n ∈ α i − Z + satisfy the same commutator relations as a i ( n ) and the only relations among elements of f M [ g ] ℓ of the form (4.33) are generated by these commutator relations, the surjective ˆ g [ g ] -module map ˜ ̺ induces a bijective ˆ g [ g ] -module map ˜ ρ from U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M to f M [ g ] ℓ .Thus U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M is equivalent to f M [ g ] ℓ . Remark 4.12
From the proof of Theorem 4.11, we see that U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ] I ) M and f M [ g ] ℓ are spanned by elements of the form (4.32) and (4.33), respectively, for i p ∈ I , n p ∈ α i p − Z + for p = 1 , . . . , l and b ∈ A . Theorem 4.13
The lower-bounded generalized g -twisted M ( ℓ, -module f M [ g ] ℓ is in fact grad-ing restricted.Proof. By Theorem 4.11, we need only prove that U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M is grading restricted.But U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M is a graded subspace of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) Λ( M ). By Theorem 4.6, U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) Λ( M ) is grading restricted, U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M is also grading restricted. The lower-bounded or grading-restricted g -twisted M ( ℓ, c M [ g ] ℓ , ( M [ g ] ℓ , > M [ g ] ℓ , f M [ g ] ℓ constructed above all have their own universal properties and other basic properties. Wefirst give the universal properties of c M [ g ] ℓ and > M [ g ] ℓ .30 heorem 4.14 Let ( W, Y gW ) be a lower-bounded generalized g -twisted module for M ( ℓ, viewed as a grading-restricted vertex algebra (vertex operator algebra when g is simple and ℓ + h ∨ = 0 ) and M a subspace of W invariant under the actions of g , S g , N g , L W (0) , L W (0) S and L W (0) N . Assume that ˆ g [ g ]+ acts on M as . If there is a linear map f : M → M commuting with the actions of g , S g , N g , L W (0) | M and L M (0) , L W (0) S | M and L M (0) S and L W (0) N | M and L M (0) N , then there exists a unique module map ˆ f : c M [ g ] ℓ → W ( > f : > M [ g ] ℓ → W ) such that ˆ f | M = f ( > f | M = f ) . If f is surjective and ( W, Y gW ) is generatedby the coefficients of ( Y g ) WW M ( ℓ, ( w, x ) for w ∈ M , where ( Y g ) WW M ( ℓ, is the twist vertexoperator map obtained from Y gW (see [Hua4]), then ˆ f ( > f ) is surjective.Proof. Since f commutes with the action of L W (0) | M and L c M [ g ] ℓ (0), we have f ( M [ h ] ) ⊂ M h ] for h ∈ Q M . Since ˆ g [ g ]+ acts on M h ] as 0, no nonzero elements of the submodule of W generated by M h ] have weights less than h . Then by the universal property of \ ( M [ h ] ) [ g ] ℓ,h (given by Theorem 5.2 in [Hua5]), there exists a unique module map ˆ f h : \ ( M [ h ] ) [ g ] ℓ,h → W suchthat ˆ f h | M [ h ] = f | M [ h ] . Let ˆ f : c M [ g ] ℓ → W be defined to be ˆ f h on \ ( M [ h ] ) [ g ] ℓ,h . Then ˆ f | M = f .The uniqueness of ˆ f follows from the uniqueness of ˆ f h for h ∈ Q M . The second conclusionalso follows from the property of ˆ f h (see Theorem 5.2 in [Hua5]) and the fact that \ ( M [ h ] ) [ g ] ℓ,h is generated by the subspace spanned by ( ψ b [ g ] ) − , (see Theorem 2.3 in [Hua6]).In the case that M ( ℓ,
0) is viewed as a vertex operator algebra, we have a module mapˆ f : c M [ g ] ℓ → W . Since on W , L W (0) = Res x xY gW ( ω M ( ℓ, , x ) and L W ( −
1) = Res x Y gW ( ω M ( ℓ, , x )and > f | M = f is obtained from c M [ g ] ℓ by taking the quotient by a submodule generated byexactly these relations, we see that ˆ f induces a module map > f : > M [ g ] ℓ → W . The otherconclusions follow from the properties of ˆ f which we have proved.We have the following immediate consequence whose proof is omitted: Corollary 4.15
Let W be a lower-bounded generalized g -twisted module for M ( ℓ, viewedas a grading-restricted vertex algebra (vertex operator algebra when g is simple and ℓ + h ∨ = 0 )generated by a subspace M invariant under g , L g , S g , N g , L W (0) , L W (0) S and L W (0) N andannihilated by ˆ g [ g ]+ . Then W as a lower-bounded ˆ g [ g ] -module is equivalent to a quotient of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) . Conversely, let M be a vector space withactions of g , L g , S g , N g , L W (0) , L W (0) S and L W (0) N satisfying the conditions given above.Then a quotient module of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M ) ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) has a naturalstructure of a lower-bounded generalized g -twisted M ( ℓ, -module when M ( ℓ, is viewed asa grading-restricted vertex algebra (vertex operator algebra when g is simple and ℓ + h ∨ = 0 ). Now we discuss the universal properties of ( M [ g ] ℓ and f M [ g ] ℓ .31 heorem 4.16 Let ( W, Y gW ) be a lower-bounded generalized g -twisted module for M ( ℓ, viewed as a grading-restricted vertex algebra (vertex operator algebra when g is simple and ℓ + h ∨ = 0 ). Let M a finite-dimensional ˆ g [ g ] I -submodule of W invariant also under theactions of g , S g , N g , L W (0) , L W (0) S and L W (0) N and annihilated by ˆ g [ g ]+ . Assume that thereis a ˆ g [ g ] I -module map f : M → M commuting with the actions of g , S g , N g , L W (0) | M and L M (0) , L W (0) S | M and L M (0) S and L W (0) N | M and L M (0) N . Then there exists a uniquemodule map ˘ f : ( M [ g ] ℓ → W ( ˜ f : f M [ g ] ℓ → W ) such that ˘ f | M = f ( ˜ f | M = f ) . If f issurjective and ( W, Y gW ) is generated by the coefficients of ( Y g ) WW M ( ℓ, ( w, x ) for w ∈ M ,where ( Y g ) WW M ( ℓ, is the twist vertex operator map obtained from Y gW , then ˘ f ( ˜ f ) is surjectiveand thus W is grading restricted.Proof. By Theorem 4.14, we have a unique module map ˆ f : c M [ g ] ℓ → W such that ˆ f | M = f .Since f is a ˆ g [ g ] I -module map, we have a iW ( α i ) f ( w b ) − X c ∈ A λ bic f ( w c ) = f a i ( α i ) w b − X c ∈ A λ bic w c ! = f (0) = 0for i ∈ I I , b ∈ A . Since ˆ f is a module map, we haveˆ f (( ψ b [ g ] ,h b ) − , ) = ˆ f ((( Y g ) c M [ g ] ℓ c M [ g ] ℓ M ( ℓ, ) − , ( w b ) ) = (( Y g ) WW M ( ℓ, ) − , ( f ( w b )) = f ( w b )for b ∈ A . Thus we obtainˆ f ( a i [ g ] ,ℓ ) α i , ( ψ b [ g ] ,h b ) − , − X c ∈ A λ bic ( ψ c [ g ] ,h c ) − , ! = a iW ( α i ) f ( w b ) − X c ∈ A λ bic f ( w c ) = 0 . So ( a i [ g ] ,ℓ ) α i , ( ψ b [ g ] ,h b ) − , − X c ∈ A λ bic ( ψ c [ g ] ,h c ) − , is in the kernel of ˆ f . In particular, we have a module map ˘ f : ( M [ g ] ℓ,h → W . The uniquenessof ˘ f and the surjectivity of ˘ f when f is surjective follow from the uniqueness of ˆ f andthe surjectivity of ˆ f . Since ( M [ g ] ℓ is grading restricted, W is grading restricted when ˘ f issurjective.The proof for f M [ g ] ℓ is the same except that we use > M [ g ] ℓ instead of c M [ g ] ℓ .We also have the following immediate consequence whose proof is also omitted: Corollary 4.17
Let W be a lower-bounded generalized g -twisted module for M ( ℓ, viewedas a grading-restricted vertex algebra (vertex operator algebra when g is simple and ℓ + h ∨ = 0 )generated by a finite-dimensional ˆ g I I -submodule M invariant under g , L g , S g , N g , L W (0) , L W (0) S and L W (0) N and annihilated by ˆ g [ g ]+ . Then W as a lower-bounded ˆ g [ g ] -module is quivalent to a quotient of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) Λ( M ) ( U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M ) and, in particular, W is grading restricted. Conversely, let M be a finite-dimensional ˆ g I I -module with compatibleactions of g , L g , S g , N g , L W (0) , L W (0) N and L W (0) N satisfying the conditions we discussedabove. Then a quotient module of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) Λ( M ) ( U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M ) has anatural structure of a grading-restricted generalized g -twisted M ( ℓ, -module when M ( ℓ, is viewed as a grading-restricted vertex algebra (vertex operator algebra when g is simple and ℓ + h ∨ = 0 ). Remark 4.18
In [Hua6], only the existence of a grading-restricted generalized g -twisted V -module for a grading-restricted vertex algebra V and an automorphism g of V is provedunder suitable conditions. But no construction is given in that paper. Theorems 4.6, 4.13and Corollary 4.17 give explicit constructions of grading-restricted generalized g -twisted M ( ℓ, L ( ℓ, -modules In this section, we construct lower-bounded and grading-restricted generalized g -twisted L ( ℓ, g is simple, ℓ ∈ Z + and L ( ℓ,
0) isviewed as a vertex operator algebra.We first give some straightforward general results.
Proposition 5.1
Let W be a lower-bounded generalized g -twisted module for L ( ℓ, viewedas a grading-restricted vertex algebra (vertex operator algebra when g is simple and ℓ + h ∨ = 0 ) generated by a subspace M invariant under g , L g , S g , N g , L W (0) , L W (0) N and L W (0) N and annihilated by ˆ g [ g ]+ . Then W is a lower-bounded generalized g -twisted mod-ule for M ( ℓ, viewed as a grading-restricted vertex algebra (vertex operator algebra). Inparticular, W is a lower bounded ˆ g [ g ] -module and is a quotient of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M )( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) . If M is in addition a finite-dimensional ˆ g I -module, then W is aquotient of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) Λ( M ) ( U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M ) and in particular, W is gradingrestricted.Proof. Since L ( ℓ,
0) is a quotient of M ( ℓ, W must be a lower-bounded generalized g -twisted M ( ℓ, W is a lower-bounded ˆ g [ g ] -module. ByCorollary 4.15, W as a lower-bounded ˆ g [ g ] -module is a quotient of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ C k ) Λ( M )( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ).If M is in addition a finite-dimensional ˆ g I -module, then by Corollary 4.17, W as a lower-bounded ˆ g [ g ] -module is a quotient of U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) Λ( M ) ( U (ˆ g [ g ] ) ⊗ U (ˆ g [ g ]+ ⊕ ˆ g [ g ]0 ) M ) and, inparticular, W is grading restricted. 33sing the structure of L ( ℓ,
0) as a ˆ g -module and properties of lower-bounded or grading-restricted generalized g -twisted M ( ℓ, Proposition 5.2
Assume that g is simple with a given Cartan subalgebra and a given setof simple roots and ℓ ∈ Z + . Let ( W, Y gW ) be a lower-bounded (grading-restricted) generalized g -twisted M ( ℓ, -module. Then ( W, Y gW ) is a lower-bounded (grading-restricted) generalized g -twisted L ( ℓ, -module if and only if Y gW ( e θ ( − ℓ +1 , x ) = 0 , where θ is the highest root of g and e θ ∈ g θ \ { } .Proof. From [K] and Proposition 6.6.17 in [LL], we know that L ( ℓ,
0) = M ( ℓ, /I ( ℓ, I ( ℓ,
0) = U (ˆ g ) e θ ( − ℓ +1 . Then W is a lower-bounded (grading-restricted) general-ized g -twisted L ( ℓ, Y gW ( I ( ℓ, , x ) = 0. We need only prove that Y gW ( I ( ℓ, , x ) = 0 if and only if Y gW ( e θ ( − ℓ +1 , x ) = 0.If Y gW ( I ( ℓ, , x ) = 0, then certainly Y gW ( e θ ( − ℓ +1 , x ) = 0. If Y gW ( e θ ( − ℓ +1 , x ) = 0,then Y gW ( e θ ( − ℓ +1 , x ) w = 0 for w ∈ W . Therefore we have ( Y g ) WW M ( ℓ, ( w, x ) e θ ( − ℓ +1 =0 for w ∈ W , where ( Y g ) WW M ( ℓ, is the twist vertex operator map introduced and studiedin [Hua4]. But by Corollary 4.3 in [Hua4] (commutativity for twisted and twist vertexoperators), for v ∈ M ( ℓ,
0) and w ′ ∈ W ′ , F p ( h w ′ , ( Y g ) WW M ( ℓ, ( w, z ) Y M ( ℓ, ( v, z ) e θ ( − ℓ +1 i )= F p ( h w ′ , Y gW ( v, z )( Y g ) WW M ( ℓ, ( w, z ) e θ ( − ℓ +1 i )= 0 , where the first two lines are different expressions of the p -th branch of a multivalued analyticfunction that h w ′ , Y gW ( v, z )( Y g ) WW M ( ℓ, ( w, z ) e θ ( − ℓ +1 i and h w ′ , ( Y g ) WW M ( ℓ, ( w, z ) Y M ( ℓ, ( v, z ) e θ ( − ℓ +1 i converge to in the regions | z | > | z | > | z | > | z | >
0, respectively. Thus we have( Y g ) WW M ( ℓ, ( w, x ) Y M ( ℓ, ( v, x ) e θ ( − ℓ +1 = 0for w ∈ W . From the definition of the twist vertex operator map ( Y g ) WW V , we obtain Y gW ( Y M ( ℓ, ( v, x ) e θ ( − ℓ +1 , x ) = 0 . But the coefficients of Y M ( ℓ, ( v, x ) span U (ˆ g ). So we obtain Y gW ( I ( ℓ, , x ) = 0.Our goal is to give and identify explicitly universal lower-bounded and grading-restrictedgeneralized g -twisted L ( ℓ, g [ g ] -modules. To do this, we first prove someresults on S g and N g .For S g , or equivalently, the semisimple automorphism σ = e πi S g , we have the followinggeneralization of Proposition 8.1 in [K] on automorphisms of finite orders of a simple Liealgebra: 34 roposition 5.3 Assume that g is simple with a given Cartan subalgebra h and a givenset ∆ of simple roots. Let σ = e πi S g be the semisimple part of g . Then there exists anautomorphism τ σ of g such that σ = τ σ e ad h µτ − σ , where µ is a diagram automorphism of g preserving h and ∆ and h is an element of the fixed-point subspace h [0] of h .Proof. Let ˜ h [0] ⊂ g [0] be a maximal toral subalgebra (that is, maximal ad-diagonalizablesubalgebra) of g [0] and C g (˜ h [0] ) the centralizer of ˜ h [0] in g . Extend ˜ h [0] to a maximal toralsubalgebra ˜ h of g . Then ˜ h is a Cartan subalgebra of g and C g (˜ h [0] ) = ˜ h + P ξ ˜ g ξ , wherethe sum is over the roots ξ in ˜ h such that ξ restricted to ˜ h [0] is 0 and ˜ g ξ is the root spaceassociated to ξ . We first prove that C g (˜ h [0] ) = ˜ h .Let s = P ξ ˜ g ξ . By definition, s is a subalgebra of g invariant under g such that s ∩ g [0] = 0.Moreover, the restriction of the bilinear form ( · , · ) to s is nondegenerate because ˜ g ξ and ˜ g ξ are orthogonal if ξ + ξ = 0 and the restriction of ( · , · ) to ˜ g ξ × ˜ g − ξ is nondegenerate. Since s isinvariant under g , we have s = ` α ∈ P s s [ α ] where P s is the set of α ∈ [0 ,
1) + i R such that e πiα is an eigenvalue of σ (or equivalently, of g ) and for α ∈ s , s [ α ] = s ∩ g [ α ] is the eigenspaceof σ (or equivalently, the generalized eigenspace of g ) in s with the eigenvalue e πiα . For α ∈ ([0 ,
1) + i R ) \ P s , let s [ α ] = 0. Then we have s = ` α ∈ [0 , i R s [ α ] . Moreover, by Lemma2.1 and the fact that s is a subalgebra of g , we have [ s [ α ] , s [ β ] ] ⊂ s [ s ( α,β )] for α, β ∈ [0 ,
1) + i R (recall s ( α, β ) defined before Lemma 2.1). We need only prove that s = 0.Since g is finite dimensional, s is finite dimensional and hence P s is a finite set. Weuse induction on the finitely many real parts of the elements of P s . First, we know that s [0] = s ∩ g [0] = 0. For α ∈ ( P s \ { } ) ∩ ( i R ) and a ∈ s [ α ] , we know that (ad a ) r s [ β ] ∈ s [ rα + β ] for β ∈ P g . Since P s is a finite set, s [ rα + β ] must be 0 when r is sufficiently large. So ad a is nilpotent on g . Applying Lemma 2.3 to s and the restriction of σ to s , we see that ( · , · )restricted to s [ α ] × s [ − α ] is nondegenerate. In particular, if s [ α ] = 0, then s [ − α ] = 0. If s [ α ] = 0, let a ∈ s [ α ] \ { } and b ∈ s [ − α ] \ { } . Then both ad a and ad b are nilpotent on s . Therefore the eigenvalues of ad a and ad b are all 0. Since [ s [ α ] , s [ − α ] ] ⊂ s [0] = 0, ad a and ad b commute and hence can be diagonalized simultaneously. In particular, the traceof (ad a )(ad b ) is 0. But this contradicts the nondegneracy of ( · , · ) restricted to s [ α ] × s [ − α ] because ( a, b ) is proportional to this trace. Thus s [ α ] = 0.Note that for α ∈ P s such that ℜ ( α ) >
0, if s [ α ] = 0, then s [1 − α ] = 0 since by Lemma2.3, the restriction of ( · , · ) to s is nondegenerate and s [ α ] is orthogonal to s [ β ] for β = 1 − α .We now assume that for α ∈ P s with ℜ ( α ) > s [ α ′ ] = 0 for α ′ ∈ P g and ℜ ( α ′ ) < ℜ ( α )and for α ′ ∈ P g and ℜ ( α ′ ) > ℜ (1 − α ). Then for a ∈ s [ α ] , (ad a ) r s [ β ] ∈ s [ s ( α,r,β )] , where s ( α, r, β ) ≡ rα + β mod Z satisfying 0 ≤ ℜ ( s ( α, r, β )) <
1. Since 0 ≤ β <
1, there exists r ∈ Z + such that ℜ (( r − α + β ) < ℜ ( rα + β ) ≥
1. From ℜ (( r − α + β ) <
1, weobtain 0 ≤ s ( α, r, β ) = ℜ ( rα + β ) − < ℜ ( α ). By the induction assumption, s [ s ( α,r,β )] = 0.So we obtain (ad a ) r s [ β ] = 0 for β ∈ P s . Thus ad a is nilpotent on s . Similarly, for b ∈ s [1 − α ] , (ad b ) r s [ β ] ∈ s [ s (1 − α,r,β )] , where s (1 − α, r, β ) ≡ r (1 − α ) + β mod Z satisfying0 ≤ ℜ ( s (1 − α, r, β )) <
1. When ℜ ( β ) = 0, since we have proved s [ β ] = 0, (ad b ) r s [ β ] = 0for r ∈ Z + . When ℜ ( β ) = 0, there exists r ∈ Z + such that ℜ ( − rα + β ) ≥ − ℜ ( − ( r + 1) α + β ) < −
1. Then ℜ ( r (1 − α ) + β ) ≥ r − ℜ (( r − − α ) + β ) < r .Since 0 < ℜ ( β ) <
1, we obtain 0 < ℜ (1 − α ) < ℜ (1 − α + β ) = ℜ ( s (1 − α, r, β )) <
1. By35he induction assumption, s [ s (1 − α,r,β )] = 0. So we obtain (ad b ) r s [ β ] = 0. Thus ad b is alsonilpotent on s . If s [ α ] = 0, let a ∈ s [ α ] \ { } and b ∈ s [1 − α ] \ { } . Then we have provedthat both ad a and ad b are nilpotent on s . Therefore the eigenvalues of ad a and ad b areall 0. Since [ s [ α ] , s [1 − α ] ] ⊂ s [0] = 0, ad a and ad b commute and hence can be diagonalizedsimultaneously. In particular, the trace of (ad a )(ad b ) is 0. Contradiction. Thus s [ α ] = 0.This proves s = 0.We have proved that C g (˜ h [0] ) = ˜ h . Now choose a ∈ ˜ h [0] such that the centralizer C g ( a ) of a in g is minimal among the collection of all centralizer C g ( b ) of b in g for b ∈ ˜ h [0] . Note thatsince elements of ˜ h [0] are all semisimple or ad-diagonalizable, C g ( b ) for b ∈ ˜ h [0] is equal to thespace of all elements of g on which ad b acts nilpotently. Since ˜ h [0] ⊂ C g ( a ), by Lemma Ain Subsection 15.2 in [Hum], we have C g ( a ) ⊂ C g ( b ) for b ∈ ˜ h [0] . But C g (˜ h [0] ) = ∩ b ∈ ˜ h [0] C g ( b ).So C g ( a ) ⊂ C g (˜ h [0] ) = ˜ h . But ˜ h ⊂ C g ( a ). So we must have ˜ h = C g ( a ), that is, a ∈ ˜ h [0] is aregular semisimple element. As the centralizer of a fixed point of g , ˜ h is a Cartan subalgebraof g invariant under σ . In particular, we have a root system e Φ obtained from ˜ h .The regular semisimple element a cannot be orthogonal to any root ξ . Otherwise [ a, e ξ ] =( ξ, a ) e ξ = 0 for e ξ ∈ g ξ so that e ξ ∈ C g ( a ) = ˜ h , which is impossible. Thus if we let e Φ + = { ξ ∈ e Φ | ( ξ, a ) > } , then e Φ = e Φ + − e Φ + . By Theorem ′ in Subsection 10.1 in [Hum],the set e ∆ of all indecomposable roots in e Φ + is a set of simple roots of e Φ and e Φ + is the set ofpositive roots. Since a is fixed by σ , σ induces an automorphism of e Φ + . Choose e ξ ∈ g ξ \ { } for ξ ∈ e ∆. Let ˜ µ be the diagram automorphism of g corresponding to this automorphismof e Φ + and e ξ for ξ ∈ e ∆. Then σ ˜ µ − fix every element of ˜ h . In particular, σ ˜ µ − commuteswith ad ˜ a for all ˜ a ∈ ˜ h and thus can be diagonalized simultaneously together with ad ˜ a .The root space decomposition g = ˜ h ⊕ ` ξ ∈ e Φ ˜ g ξ gives a diagonalization of ad ˜ a . Also ˜ g ξ for ξ ∈ e Φ are all one dimensional and σ ˜ µ − acts as the identity on ˜ h , we see the this root spacedecomposition also give a diagonalization of σ ˜ µ − . So on each root space ˜ g ξ , it must act as ascalar multiplication by λ ξ ∈ C × . Let l ( λ ξ ) = log | λ ξ | + i arg λ ξ , where 0 ≤ arg λ ξ < π . Let˜ h ∈ ˜ h be defined by ( ξ, ˜ h ) = πi l ( λ ξ ) for ξ ∈ e ∆. Then λ ξ e ξ = e πi ( ξ, ˜ h ) e ξ = e πi (ad ˜ h ) e ξ . Thuswe obtain σ ˜ µ − = e πi (ad ˜ h ) , or equivalently, σ = e πi (ad ˜ h ) ˜ µ . Since σ ˜ µ − fix every element of˜ h , ˜ h must be in ˜ h .Since any two Cartan subalgebras are conjugate to each other, there exists an automor-phism ν of g such that ν ( h ) = ˜ h and ν (∆) = e ∆. Let µ = ν ˜ µν − and ˘ h = ν − (˜ h ) ∈ h . Thenit is clear that µ is a diagram automorphism of g preserving h and ∆ and we have σ = e πi (ad ν (˘ h )) νµν − = νe πi (ad ˘ h ) µν − . But ˘ h might not be fixed by µ . We need to find another automorphism such that after theconjugation by this automorphism, we have h ∈ h [0] (that is, fixed by µ ). This argument wasin fact given by the proof of Lemma 8.3 in [EMS]: Let r be the order of µ (in fact r = 1 , h = m P r − k =1 µ k ˘ h and η = e πir P r − k =0 k (ad µ k ˘ h ) . Then µh = h and h ∈ h since h is invariant36nder µ . Moreover, ηe πi (ad ˘ h ) µη − = e πir P r − k =1 k (ad µ k ˘ h ) e πi (ad ˘ h ) µe − πir P r − k =1 k (ad µ k ˘ h ) = e πir P r − k =1 k (ad µ k ˘ h ) e πi (ad ˘ h ) e − πir P r − k =1 k (ad µ k +1 ˘ h ) µ = e πi r P r − k =1 (ad µ k ˘ h ) µ = e πi (ad h ) µ. Let τ σ = ην − . Then σ = νe πi (ad ˘ h ) µν − = νη − e πi (ad h ) µην − = τ σ e πi (ad h ) µτ − σ . For N g , we have the following result: Proposition 5.4
Assume that g is semisimple. Then we have the following:1. There exists a N g ∈ g [0] such that N g b = [ a N g , b ] for b ∈ g , that is, N g = ad a N g .2. On ˆ g , N g ( b ⊗ t m ) = [ a N g ⊗ t , b ⊗ t m ] = [ a N g , b ] ⊗ t m for b ∈ g , m ∈ Z and N g k =[ a N g ⊗ t , k ] = 0 .3. On ˆ g [ g ] , N g ( b ⊗ t m ) = [ a N g ⊗ t , b ⊗ t m ] = [ a N g , b ] ⊗ t m for b ∈ g [ β ] and m ∈ β + Z and N g k = [ a N g ⊗ t , k ] = 0 .4. On M ( ℓ, or L ( ℓ, , N g = a N g (0) .Proof. By Corollary 2.2, N g is a derivation of g . Since g is semisimple, we know that everyderivation of g is inner. So there exists a N g ∈ g such that N g b = [ a N g , b ] for b ∈ g . Since S g commutes with N g , for b ∈ g [ β ] , S g [ a N g , b ] = S g N g b = N g S g b = β N g b = β [ a N g , b ]. So[ a N g , b ] ∈ g [ β ] . Thus a N g ∈ g [0] . This finishes the proof of Conclusion 1.Conclusions 2, 3 and 4 follow immediately from the definitions of the actions of N g on ˆ g ,ˆ g [ g ] , M ( ℓ,
0) and L ( ℓ, g is simple with a Cartan subalgebra h and aset ∆ of simple roots which gives a root system Φ. For a ∈ g , a = P α ∈ P g a α , where a α ∈ g [ α ] .Given a ˆ g -module W , we have introduced a α ( x ) = P n ∈ α + Z a ( n ) x − n − for α ∈ P g above.We shall need the following result later: Proposition 5.5
Assume that g is simple with a given Cartan subalgebra h and a given set ∆ of simple roots. Let g , µ , h and τ σ be the same as in Proposition 5.3. Let θ be the highestroot of g and W a ˆ g [ g ] -module. Then there exists r θ ∈ Z such that e θ ∈ g [( θ,h )+ r θ ] and [( τ σ e θ )( x ) , ( τ σ e θ )( x )] = 0 . (5.1)37 roof. Since (ad h ) e θ = [ h, θ ] = ( θ, h ) e θ , e πi (ad h ) e θ = e πi ( θ,h ) e θ . Then e πi (ad τ σ h ) τ σ e θ = τ σ e πi (ad h ) e θ = e πi ( θ,h ) τ σ e θ . Since θ is the highest root and µ is an automorphism of Φ + , θ is fixed under µ by thedefinition and by the uniqueness of the highest root. Thus e θ is also fixed under µ . So τ σ µτ − σ fixes τ σ e θ . Then by Proposition 5.3, στ σ e θ = τ σ e πi (ad h ) µe θ = e πi ( θ,h ) τ σ e θ . (5.2)From σ = e πi S g and (5.2), there exists r θ ∈ Z such that ( θ, h ) + r θ ∈ P g and S g τ σ e θ = (( θ, h ) + r θ ) τ σ e θ . (5.3)Thus e θ ∈ g [( θ,h )+ r θ ] .To prove (5.1), first we have [ τ σ e θ , τ σ e θ ] = 0 . (5.4)Since θ + θ = 0, ( e θ , e θ ) = 0. Then by the invariance of the bilinear form ( · , · ),( τ σ e θ , τ σ e θ ) = 0 . (5.5)Using the invariance of the bilinear form ( · , · ) and N g = ad a N g (Part 1 of Proposition 5.4),we obtain ( N g τ σ e θ , τ σ e θ ) = ([ a N g , τ σ e θ ] , τ σ e θ ) = ( a N g , [ τ σ e θ , τ σ e θ ]) = 0 . (5.6)Let γ = ( θ, h ) + r θ . Then e θ ∈ g [ γ ] . Using (5.4), (5.5) and (5.6), we have[( τ σ e θ )( x ) , ( τ σ e θ )( x )]= X m ∈ γ + Z X n ∈ γ + Z [( τ σ e θ )( m ) , ( τ σ e θ )( n )]= X m ∈ γ + Z X n ∈ γ + Z (cid:16) [( τ σ e θ ) , ( τ σ e θ )]( m + n )+ m (( τ σ e θ ) , ( τ σ e θ )) δ m + n, ℓ + ( N g ( τ σ e θ ) , ( τ σ e θ )) δ m + n, ℓ (cid:17) = 0 . (5.7)We also need the following general lemma: Lemma 5.6
Let V be a grading-restricted vertex algebra (or a vertex operator algebra), g anautomorphism of V and W a lower-bounded generalized g -twisted V -module. Assume thatfor some u, v ∈ V , ( Y gW ) ( u, x )( Y gW ) ( v, x ) = ( Y gW ) ( v, x )( Y gW ) ( u, x ) , here ( Y gW ) ( v, x ) for v ∈ V is the constant term of Y gW ( v, x ) viewed as a power series in log x . Then ( Y gW ) ( u, x )( Y gW ) ( v, x ) is well defined and ( Y gW ) (( Y V ) − ( u ) v, x )) = ( Y gW ) ( u, x )( Y gW ) ( v, x ) . (5.8) Proof.
For w ∈ W , Y gW ( u, x ) Y gW ( v, x ) w = Y gW ( v, x ) Y gW ( u, x ) w has only finitely many negative power terms in both x and x . In particular, we can let x = x = x to obtain a well defined formal series ( Y gW ) ( u, x )( Y gW ) ( v, x ).To prove (5.8), we use the the Jacobi identity (4.1). Using Y gW ( u, x ) = ( Y gW ) ( x −N g u, x )((2.10) in [HY]) for u ∈ V and x N g Y V ( u, x ) = Y V ( x N g u, x ) x N g ((2.5) in [Hua4]), and replac-ing u and v in (4.1) by x N g u and x N g v , respectively, we see that that (4.1) becomes x − δ (cid:18) x − x x (cid:19) ( Y gW ) ( u, x )( Y gW ) ( v, x ) − x − δ (cid:18) − x + x x (cid:19) ( Y gW ) ( v, x )( Y gW ) ( u, x )= x − δ (cid:18) x + x x (cid:19) ( Y gW ) Y V (cid:18) x x (cid:19) S g (cid:18) x x (cid:19) L g u, x ! v, x ! . (5.9)By the assumption, the left-hand side of (5.9) is equal to (cid:18) x − δ (cid:18) x − x x (cid:19) − x − δ (cid:18) − x + x x (cid:19)(cid:19) ( Y gW ) ( u, x )( Y gW ) ( v, x )= x − δ (cid:18) x + x x (cid:19) ( Y gW ) ( u, x )( Y gW ) ( v, x ) . Thus from (5.9), we obtain x − δ (cid:18) x + x x (cid:19) ( Y gW ) Y V (cid:18) x x (cid:19) S g (cid:18) x x (cid:19) L g u, x ! v, x ! = x − δ (cid:18) x + x x (cid:19) Y gW ( u, x ) Y gW ( v, x ) . (5.10)Replacing u in (5.10) by (cid:0) x x (cid:1) −L g (cid:0) x x (cid:1) −S g u , we obtain x − δ (cid:18) x + x x (cid:19) ( Y gW ) ( Y V ( u, x ) v, x )= x − δ (cid:18) x + x x (cid:19) ( Y gW ) (cid:18) x x (cid:19) −L g (cid:18) x x (cid:19) −S g u, x ! ( Y gW ) ( v, x ) . (5.11)39ince V = ` α ∈ P V V [ α ] , we have u = P α ∈ P V u α , where u α ∈ V [ α ] for α ∈ P V . Also note that( Y gW ) ( u α , x ) ∈ x − α (End W )[[ x, x − ]]. Then we have x − δ (cid:18) x + x x (cid:19) ( Y gW ) (cid:18) x x (cid:19) −L g (cid:18) x x (cid:19) −S g u, x ! = X α ∈ P V x − δ (cid:18) x + x x (cid:19) x α ( Y gW ) (cid:18) x x (cid:19) −L g x −S g u α , x ! = X α ∈ P V x − δ (cid:18) x + x x (cid:19) ( x + x ) α ( Y gW ) (cid:18) x x (cid:19) −L g x −S g u α , x + x ! = x − δ (cid:18) x + x x (cid:19) ( Y gW ) (cid:18) x x (cid:19) −L g (cid:18) x x + x (cid:19) −S g u, x + x ! = x − δ (cid:18) x + x x (cid:19) ( Y gW ) (cid:18) x x (cid:19) −N g u, x + x ! . (5.12)Using (5.12), we see that the right-hand side of (5.11) is equal to x − δ (cid:18) x + x x (cid:19) ( Y gW ) (cid:18) x x (cid:19) −N g u, x + x ! ( Y gW ) ( v, x ) . (5.13)Then Res x of the left-hand side of (5.11) and (5.13) are also equal, that is,( Y gW ) ( Y V ( u, x ) v, x ) = ( Y gW ) (cid:18) x x (cid:19) −N g u, x + x ! Y gW ( v, x ) . (5.14)Taking the constant terms in x in both sides of (5.14) and then replacing x by x , we obtain(5.8).Applying Proposition 5.5 and Lemma 5.6 to the lower-bounded (grading-restricted) gen-eralized g -twisted M ( ℓ, > M [ g ] ℓ ( f M [ g ] ℓ ) and using Theorems 4.9 and 4.11, we have thefollowing consequence: Corollary 5.7
On the lower-bounded ˆ g [ g ] -module > M [ g ] ℓ (the grading-restricted ˆ g [ g ] -module f M [ g ] ℓ ), ( τ σ e θ )( x ) m for m ∈ N are well defined and Y g > M [ g ] ℓ (( τ σ e θ )( − m , x ) = ( τ σ e θ )( x ) m (5.15) (cid:18) Y g f M [ g ] ℓ (( τ σ e θ )( − m , x ) = ( τ σ e θ )( x ) m (cid:19) . (5.16) In particular, on U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M and U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M , ( τ σ e θ )( x ) m for m ∈ N arewell defined. roof. By the definition of ˆ g [ g ] -module structure on > M [ g ] ℓ (see Proposition 4.1), we have( Y g > M [ g ] ℓ ) (( τ σ e θ )( − , x ) = ( τ σ e θ )( x ) . Then from (5.1), we have( Y g > M [ g ] ℓ ) (( τ σ e θ )( − , x )( Y g > M [ g ] ℓ ) (( τ σ e θ )( − , x )= ( Y g > M [ g ] ℓ ) (( τ σ e θ )( − , x )( Y g > M [ g ] ℓ ) (( τ σ e θ )( − , x ) . (5.17)By (5.17), we can use Lemma 5.6 for u = v = ( τ σ e θ )( − Y g > M [ g ] ℓ (( τ σ e θ )( − , x ) = ( τ σ e θ )( x ) , where the right-hand side is well defined. From (5.1), we obtain[( τ σ e θ )( x ) , ( τ σ e θ )( x ) m ] = m X i =1 ( τ σ e θ )( x ) i − [( τ σ e θ )( x ) , ( τ σ e θ )( x )]( τ σ e θ )( x ) m − i = 0 . Using induction and Lemma 5.6, we see that for m ∈ N , ( τ σ e θ )( x ) m is well defined and (5.15)holds. The proof for f M [ g ] L ( ℓ, is completely the same.By Theorems 4.9 and 4.11, we see that ( τ σ e θ )( x ) m for m ∈ N are also well defined on U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M and U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M .Let V be a grading-restricted vertex algebra (or a vertex operator algebra), g and h auto-morphisms of V and ( W, Y gW ) a lower-bounded (grading-restricted) generalized g -twisted V -module. Recall the lower-bounded (grading-restricted) hgh − -twisted V -module ( W, φ h ( Y g ))(see Proposition 3.2 in [Hua3]), where φ h ( Y gW ) : V × W → W { x } [log x ] v ⊗ w φ h ( Y gW )( v, x ) w is the linear map defined by φ h ( Y gW )( v, x ) w = Y gW ( h − v, x ) w. The hgh − -twisted V -module( W, φ h ( Y gW )) is also denoted by φ h ( W ).Let M , as in Subsection 4.2, be a vector space with actions of g , L g , S g , N g , L M (0), L M (0) S and L M (0) N such that M = ` h ∈ Q M M [ h ] , where Q M is the set of all eigenvaluesof L M (0) and M [ h ] is the generalized eigenspace of L M (0) with eigenvalue h ∈ Q M . Thenjust as in the construction of > M [ g ] ℓ in Subsection 4.2 for the vertex operator algebra M ( ℓ, h ∈ Q M , we have a universal lower-bounded generalized g -twisted L ( ℓ, M h (see Theorem 4.7 for its universal property). We shall denote them by > M [ g ] L ( ℓ, ,h for h ∈ Q M . Let > M [ g ] L ( ℓ, = ` h ∈ Q M > M [ g ] L ( ℓ, ,h . This is the universal lower-boundedgeneralized g -twisted L ( ℓ, M annihilated by ˆ g + . If M is in addition a finite-dimensional ˆ g I -module such that the action of g is compatible, then41lso as in Subsection 4.2, we have a quotient of > M [ g ] L ( ℓ, ,h . We shall denote this quotient by f M [ g ] L ( ℓ, .From the definitions of > M [ g ] L ( ℓ, and f M [ g ] L ( ℓ, , we have the following universal properties forthem: Theorem 5.8
Let ( W, Y gW ) be a lower-bounded generalized g -twisted L ( ℓ, -module (when L ( ℓ, is viewed as a vertex operator algebra). Let M a subspace (finite-dimensional ˆ g [ g ] I -submodule) of W invariant under the actions of g , S g , N g , L W (0) , L W (0) S and L W (0) N andannihilated by ˆ g [ g ]+ . Assume that there is a linear ( ˆ g [ g ] I -module map) f : M → M commutingwith the actions of g , S g , N g , L W (0) | M and L M (0) , L W (0) S | M and L M (0) S and L W (0) N | M and L M (0) N . Then there exists a unique module map > f : > M [ g ] L ( ℓ, → W ( ˜ f : f M [ g ] L ( ℓ, → W ) such that > f | M = f ( ˜ f | M = f ) . If f is surjective and ( W, Y gW ) is generated by the coefficientsof ( Y g ) WW L ( ℓ, ( w, x ) for w ∈ M , where ( Y g ) WW L ( ℓ, is the twist vertex operator map obtainedfrom Y gW , then > f ( ˜ f ) is surjective.Proof. For h ∈ Q M , by the universal property of > M [ g ] L ( ℓ, ,h (see the construction of > M [ g ] L ( ℓ, ,h and Theorem 4.7, ), there is a unique L ( ℓ, > f h from > M [ g ] L ( ℓ, ,h to the submoduleof W generated by f ( M ). Let > f : > M [ g ] L ( ℓ, → W be defined by > f ( w ) = > f h ( w ) for w ∈ > M [ g ] L ( ℓ, ,h . Then > f is clearly a module map. The uniqueness of > f follows from the uniquenessof > f h for h ∈ Q M . It is also clear that the second conclusion holds.In the case that M is a finite-dimensional ˆ g [ g ] I -submodule of W , the proof is the same asthat of Theorem 4.16 except that we should use > M [ g ] L ( ℓ, instead of c M [ g ] ℓ .Let > I [ g ] L ( ℓ, ( e I [ g ] L ( ℓ, ) be the submodules of U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ( U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M ) gener-ated by the coefficients of ( τ σ e θ )( x ) ℓ +1 w for w ∈ U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ( w ∈ U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M ). Then on ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) / > I [ g ] L ( ℓ, , ( U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M ) / e I [ g ] L ( ℓ, and their quotients,( τ σ e θ )( x ) ℓ +1 = 0 . (5.18) Theorem 5.9
Assume that g is simple and ℓ ∈ Z + . The universal lower-bounded (grading-restricted) generalized g -twisted L ( ℓ, -module > M [ g ] L ( ℓ, ( f M [ g ] L ( ℓ, ) is equivalent as a lower-bounded (grading-restricted) ˆ g [ g ] -module to ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) / > I [ g ] L ( ℓ, (( U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M ) / e I [ g ] L ( ℓ, ) . In particular, the lower-bounded generalized g -twisted L ( ℓ, -module f M [ g ] L ( ℓ, isin fact grading restricted.Proof. Note that the automorphism τ σ of g induces automorphisms, denoted still by τ σ ,of the vertex operator algebras M ( ℓ,
0) and L ( ℓ, τ σ ( > M [ g ] L ( ℓ, ) is a lower-bounded42eneralized τ − σ gτ σ -twisted L ( ℓ, τ σ ( > M [ g ] L ( ℓ, ) is also a lower-bounded generalized τ − σ gτ σ -twisted M ( ℓ, φ τ σ (cid:18) Y g > M [ g ] L ( ℓ, (cid:19) ( e θ ( − ℓ +1 , x ) = 0 , or equivalently, Y g > M [ g ] L ( ℓ, ( τ σ e θ ( − ℓ +1 , x ) = 0 . Since τ σ is an automorphism of L ( ℓ,
0) induced from the automorphism τ σ of g , we have τ σ ( e θ ( − ℓ +1 ) = ( τ σ e θ )( − ℓ +1 . Hence we have Y g > M [ g ] L ( ℓ, (( τ σ e θ )( − ℓ +1 , x ) = 0 . (5.19)From (5.19) and (5.15), we see that (5.18) holds on the ˆ g [ g ] -module > M [ g ] L ( ℓ, . By Proposition5.1, > M [ g ] L ( ℓ, is also a lower-bounded generalized g -twisted M ( ℓ, M .Then by Corollary 4.15, > M [ g ] L ( ℓ, is equivalent to a quotient of U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M . Sinceon > M [ g ] L ( ℓ, (5.18) holds, we see that > M [ g ] L ( ℓ, is equivalent to a quotient of ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) / > I [ g ] L ( ℓ, .On the other hand, by Theorem 4.9, ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) / > I [ g ] L ( ℓ, is equivalent as a ˆ g -module to the lower-bounded generalized g -twisted M ( ℓ, > M [ g ] ℓ . Then by Corollary4.15, W = ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) / > I [ g ] L ( ℓ, is also a lower-bounded generalized g -twisted M ( ℓ, W . So we have φ τ σ ( Y gW )( e θ ( − ℓ +1 , x ) = Y gW ( τ σ e θ ( − ℓ +1 , x )= Y gW (( τ σ e θ )( − ℓ +1 , x )= ( τ σ e θ )( x ) ℓ +1 = 0 . By Proposition 5.2, φ τ σ ( W ) is a lower-bounded generalized τ − σ gτ σ -twisted L ( ℓ, W is a lower-bounded generalized g -twisted L ( ℓ, M can be viewedas a subspace of W invariant under the actions of g , S g , N g , L W (0), L W (0) S and L W (0) N and with ˆ g [ g ]+ acting on M as 0 and we have the identity map from M to itself. Thus byTheorem 5.8, there exists a unique surjective L ( ℓ, > M [ g ] L ( ℓ, to W . Inparticular, this surjective L ( ℓ, g -module map. Thus we have asurjective ˆ g -module map from > M [ g ] L ( ℓ, to W . Since we have proved that > M [ g ] L ( ℓ, is a quotientof W , the existence of such a surjective ˆ g -module map means that > M [ g ] L ( ℓ, is equivalent to W = ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) / > I [ g ] L ( ℓ, . 43he proof for f M [ g ] L ( ℓ, is completely the same except that we use the results in Subsections4.2 and 4.3 on f M [ g ] ℓ instead of > M [ g ] ℓ . Since ( U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M ) / e I [ g ] L ( ℓ, is grading-restricted,we see that f M [ g ] L ( ℓ, is grading restricted.We also have the following immediate consequence whose proof is also omitted: Corollary 5.10
Let W be a lower-bounded generalized g -twisted L ( ℓ, -module (when L ( ℓ, is viewed as a vertex operator algebra) generated by a subspace (finite-dimensional ˆ g I -submodule) M invariant under g , L g , S g , N g , L W (0) , L W (0) S and L W (0) N and annihilated by ˆ g [ g ]+ . Then W as a lower-bounded ˆ g [ g ] -module is equivalent to a quotient of ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) / > I [ g ] L ( ℓ, (( U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M ) / e I [ g ] L ( ℓ, and, in particular, W is grading restricted). Conversely, let M be a vector space (finite-dimensional ˆ g I I -module) with (compatible) actions of g , L g , S g , N g , L W (0) , L W (0) S and L W (0) N satisfying the conditions discussed in Section 4. Then a quo-tient module of ( U (ˆ g [ g ] ) ⊗ G (Ω [ g ] , ˆ g [ g ]+ , k ) M ) / > I [ g ] L ( ℓ, (( U (ˆ g [ g ] ) ⊗ ˆ g [ g ]+ ⊕ ˆ g [ g ]0 M ) / e I [ g ] L ( ℓ, ) has a naturalstructure of a grading-restricted generalized g -twisted L ( ℓ, -module (when L ( ℓ, is viewedas a vertex operator algebra). References [B] B. Bakalov, Twisted logarithmic modules of vertex algebras,
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Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscat-away, NJ 08854-8019