Associative algebras and the representation theory of grading-restricted vertex algebras
aa r X i v : . [ m a t h . QA ] S e p Associative algebras and the representationtheory of grading-restricted vertex algebras
Yi-Zhi Huang
Abstract
We introduce an associative algebra A ∞ ( V ) using infinite matrices with entries ina grading-restricted vertex algebra V such that the associated graded space Gr ( W ) = ` n ∈ N Gr n ( W ) of a filtration of a lower-bounded generalized V -module W is an A ∞ ( V )-module satisfying additional properties (called a graded A ∞ ( V )-module). We provethat a lower-bounded generalized V -module W is irreducible or completely reducibleif and only if the graded A ∞ ( V )-module Gr ( W ) is irreducible or completely reducible,respectively. We also prove that the set of equivalence classes of the lower-boundedgeneralized V -modules are in bijection with the set of the equivalence classes of graded A ∞ ( V )-modules. For N ∈ N , there is a subalgebra A N ( V ) of A ∞ ( V ) such that thesubspace Gr N ( W ) = ` Nn =0 Gr n ( W ) of Gr ( W ) is an A N ( V )-module satisfying addi-tional properties (called a graded A N ( V )-module). We prove that A N ( V ) are finitedimensional when V is of positive energy (CFT type) and C -cofinite. We prove thatthe set of the equivalence classes of lower-bounded generalized V -modules is in bijec-tion with the set of the equivalence classes of graded A N ( V )-modules. In the case that V is a M¨obius vertex algebra and the differences between the real parts of the lowestweights of the irreducible lower-bounded generalized V -modules are less than or equalto N ∈ N , we prove that a lower-bounded generalized V -module W of finite length isirreducible or completely reducible if and only if the graded A N ( V )-module Gr N ( W )is irreducible or completely reducible, respectively. In the representation theory of Lie algebras, the universal enveloping algebra of a Lie algebraplays a crucial role because the module categories for a Lie algebra and for its universalenveloping algebra are isomorphic. For a vertex operator algebra, there is also a universalenveloping algebra introduced by Frenkel and Zhu [FZ] such that the module categoriesfor these algebras are isomorphic. Unfortunately, the universal enveloping algebra of avertex operator algebra is not very useful since it involves certain infinite sums in a suitabletopological completion of the tensor algebra of the algebra. On the other hand, the classesof modules that we are interested in the representation theory of vertex operator algebrasand two-dimensional conformal field theory do not involve such infinite sums since the vertexoperators on these modules are lower truncated when acting on elements of these modules.1nstead, in the representation theory of vertex operator algebras, we have the Zhu algebra A ( V ) introduced by Zhu in [Z] and it’s generalizations A n ( V ) for n ∈ N by Dong, Li andMason in [DLM] for a vertex operator algebra V . These algebras can be used to classifyirreducible modules for the vertex operator algebra and to study problems related to differenttypes of modules. But compared with the universal enveloping algebra of a Lie algebra, therole of these associative algebras played in the representation theory of vertex operatoralgebras is quite limited. For example, the module for one of these associative algebrasobtained from a suitable V -module in general do not tell us whether the original V -moduleis irreducible or completely reducible.In the present paper, we introduce an associative algebra A ∞ ( V ) using infinite matriceswith entries in a grading-restricted vertex algebra V . The associated graded space Gr ( W ) = ` n ∈ N Gr n ( W ) of a filtration of a lower-bounded generalized V -module W is an A ∞ ( V )-module with an N -grading and some operators having suitable properties (called a graded A ∞ ( V )-module). In fact, the algebra A ∞ ( V ) is defined using the associated graded spacesof all lower-bounded generalized V -modules. We prove that a lower-bounded generalized V -module W is irreducible or completely reducible if and only if the graded A ∞ ( V )-module Gr ( W ) is irreducible or completely reducible, respectively. We also prove that the set of theequivalence classes of irreducible lower-bounded generalized V -modules is in bijection withthe set of the equivalence classes of irreducible graded A ∞ ( V )-modules.We show that A ( V ) in [Z] and A n ( V ) [DLM] mentioned above are isomorphic to veryspecial subalgebras of A ∞ ( V ). This fact gives a conceptual explanation of the role that theseassociative algebras played in the representation theory of vertex operator algebras.We then introduce new subalgebras A N ( V ) of A ∞ ( V ) for N ∈ N . These subalgebras canalso be obtained using finite matrices with entries in V . In the case that V is of positiveenergy (or CFT type) and C -cofinite, we prove that A N ( V ) are finite dimensional. Thesubspace Gr N ( W ) = ` Nn =0 Gr n ( W ) of Gr ( W ) of a lower-bounded generalized V -module W is an A N ( V )-module with some operators having suitable properties (called a graded A N ( V )-module). Wer prove that if a lower-bounded generalized V -module W is irreducibleor completely reducible, then the graded A N ( V )-module Gr N ( W ) is irreducible or completelyreducible, respectively. We also prove that the set of the equivalence classes of irreduciblelower-bounded generalized V -modules is in bijection with the set of the equivalence classesof irreducible graded A N ( V )-modules.In the case that V is a M¨obius vertex algebra so that a lowest weight of a lower-boundedgeneralized V -module is defined, under the assumption that the differences between thereal parts of the lowest weights of the irreducible lower-bounded generalized V -modules areless than or equal to N ∈ N , we prove that a lower-bounded generalized V -module W offinite length is irreducible or completely reducible if and only if the graded A N ( V )-module Gr N ( W ) is irreducible or completely reducible, respectively. When A N ( V ) for all N ∈ N arefinite dimensional (for example, when V is of positive energy (or CFT type) and C -cofinite),we prove that an irreducible lower-bounded generalized V -module is an ordinary V -moduleand thus every lower-bounded generalized V -module of finite length is grading-restricted.In this case, under the assumptions above on V , lowest weights and N , a lower-bounded2eneralized V -module W of finite length or a grading-restricted generalized V -module W is a direct sum of irreducible ordinary V -modules if and only if the graded A N ( V )-module Gr N ( W ) is completely reducible.Many of the main results mentioned above need the construction of universal lower-bounded generalized V -modules in [H3] and some results from [H4].The category of lower-bounded generalized V -modules and the category of graded A ∞ ( V )-modules are not equivalent because of morphisms, but they are “almost” equivalent. We shallstudy the relations between these categories, the category of lower-bounded generalized V -modules of finite lengths and the categories of graded A N ( V )-modules for N ∈ N in anotherpaper.This paper is organized as follows: In the next section, we introduce the associave algebra A ∞ ( V ) associated to a grading-restricted vertex algebra V and prove that the associatedgraded space Gr ( W ) of a filtration of a lower-bounded generalized V -module W is an A ∞ ( V )-module. In Section 3, we introduce graded A ∞ ( V )-modules and prove the results mentionedabove on the relations between lower-bounded generalized V -modules and graded A ∞ ( V )-modules. We show that the Zhu algebra and their generalizations by Dong, Li and Masonare isomorphic to subalgebras of A ∞ ( V ) in Subsection 4.1 and introduce the new subalgebras A N ( V ) of A ∞ ( V ) for N ∈ N in Subsection 4.2. We also prove in Subsection 4.2 that when V is of positive energy and C -cofinite, A N ( V ) for N ∈ N are finite dimensional. In Section 5,we introduce graded A N ( V )-modules and prove the results mentioned above on the relationsbetween lower-bounded generalized V -modules, lower-bounded generalized V -modules offinite lengths and graded A N ( V )-modules. A ∞ ( V ) and modules In this paper, we fix a grading-restricted vertex algebra V . Most of the constructions andresults work and hold for more general vertex algebras, for example, lower-bounded vertexalgebras or superalgebras. The constructions and results certainly work and hold for aM¨obius vertex algebra or a vertex operator algebra. For some results in Section 5, we shallassume that V is a M¨obius vertex algebra.Let U ∞ ( C ) be the space of all column-finite infinite matrices with entries in C , but doublyindexed by N instead of Z + . In other words, U ∞ ( C ) is the space of all infinite matrices ofthe form [ a kl ] for a kl ∈ C , k, l ∈ N such that for each fixed l ∈ N , there are only finitelymany nonzero a kl . Let I ∞ = [ δ kl ] be the infinite identity matrix. Then U ∞ ( C ) is in fact anassociative algebra with the identity I ∞ . The space U ∞ ( C ) has a set of linearly independentelements of the form E kl for k, l ∈ N with the only nonzero entry being the one in the k -throw and l -th column. These infinite matrices do not form a basis of U ∞ ( C ) but form a basisof the subspace U ∞ ( C ) of U ∞ ( C ) consisting of finitary matrices (matrices with only finitelymany nonzero entries). In particular, U ∞ ( C ) = a k,l ∈ N C E kl . U ∞ ( C ) ⊂ Y k,l ∈ N C E kl , where Q k,l ∈ N C E kl is the algebraic completion of U ∞ ( C ) viewed as a graded space. Thoughelements of U ∞ ( C ) are infinite linear combinations of E kl for k, l ∈ N , any binary producton U ∞ ( C ) satisfying the distribution axioms is still determined completely by the product of E kl for k, l ∈ N . For example, for the usual matrix product, we know that E kl E mn = δ lm E kn for k, l, m, n ∈ N . Let A = P k,l ∈ N a kl E kl and B = P k,l ∈ N b kl E kl , where a kl , b kl ∈ C for k, l ∈ N . Then AB = X k,l ∈ N a kl E kl ! X m,n ∈ N b mn E mn ! = X k,n ∈ N X m ∈ N a km b mn ! E kn . So even though E kl for k, l ∈ N do not form a basis of U ∞ ( C ), all the properties of the asso-ciative algebra structure on U ∞ ( C ) can still be derived from the properties these matrices.Thus we can study U ∞ ( C ) using E kl for k, l ∈ N , k ≤ l . Also what we are mainly interestedis the subalgebra C I ∞ ⊕ U ∞ ( C ) of U ∞ ( C ). This subalgebra has a basis { I ∞ } ∪ { E kl } k,l ∈ N .Let U ∞ ( V ) = V ⊗ U ∞ ( C ). Then U ∞ ( V ) is the space of column-finite infinite matriceswith entries in V , but doubly indexed by N instead of Z + . Elements of U ∞ ( V ) are of theform v = [ v kl ] for v kl ∈ V , k, l ∈ N such that for each fixed l ∈ N , there are only finitelymany nonzero v kl . Let U ∞ ( V ) be the subspace of U ∞ ( V ) spanned by elements of the form v ⊗ E kl for v ∈ V and k, l ∈ N . Then U ∞ ( V ) = a k,l ∈ N V ⊗ C E kl and U ∞ ( V ) ⊂ Y k,l ∈ N V ⊗ C E kl . We shall denote v ⊗ E kl simply by [ v ] kl . Then elements of U ∞ ( C ) can all be written as X k,l ∈ N [ v kl ] kl for v kl ∈ V , k, l ∈ N . As in the case of U ∞ ( C ), we can study any binary product on U ∞ ( V )satisfying the distribution axioms using [ v ] kl for v ∈ V and k, l ∈ N . We are also mainlyinterested in the subspace V ⊗ I ∞ ⊕ U ∞ ( V ) of U ∞ ( V ). This subspace is spanned by elementsof the form v ⊗ I ∞ and [ v ] kl for v ∈ V and k, l ∈ N . Because of this reason, though we mightgive definitions of products and related notions using general elements of U ∞ ( V ), we shallstudy them using only [ v ] kl for v ∈ V and k, l ∈ N .We also need some particular formal series and polynomials. In this paper, we shall usethe convention that a complex power or the integral power of the logarithm of an orderedlinear combination of formal variables and a complex number, always means its expansion4n nonnegative powers of the formal variables or the complex number that are not the firstone in the ordered linear combination. For example, ( x + 1) − k − for k ∈ N and (1 + x ) n for n ∈ N mean the expansions in nonnegative powers of 1 and in nonnegative powers of x ,respectively. For k, n, l ∈ N , we have( x + 1) − k + n − l − = X m ∈ N (cid:18) − k + n − l − m (cid:19) x − k + n − l − m − = T k + l +1 (( x + 1) − k + n − l − ) + R k + l +1 (( x + 1) − k + n − l − ) , (2.1)where T k + l +1 (( x + 1) − k + n − l − ) = n X m =0 (cid:18) − k + n − l − m (cid:19) x − k + n − l − m − is the Taylor polynomial in x − of order k + l + 1 of ( x + 1) − k + n − l − and R k + l +1 (( x + 1) − k + n − l − ) = X m ∈ n +1+ N (cid:18) − k + n − l − m (cid:19) x − k + n − l − m − is the remainder of order k + l + 1.We define a product ⋄ on U ∞ ( V ) by u ⋄ v = [( u ⋄ v ) kl ]for u = [ u kl ] , v = [ v kl ] ∈ U ∞ ( V ), where( u ⋄ v ) kl = l X n = k Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l Y V ((1 + x ) L V (0) u kn , x ) v nl = n X n = k l X m =0 (cid:18) − k + n − l − m (cid:19) Res x x − k + n − l − m − (1 + x ) l Y V ((1 + x ) L V (0) u kn , x ) v nl (2.2)for k, l ∈ N . Then U ∞ ( V ) equipped with ⋄ is an algebra but in general is not even associative.Let O ∞ ( V ) be the subspace of U ∞ ( V ) spanned by elements of the form l X n = k Res x x − k − l − p − (1 + x ) l (cid:2) Y V ((1 + x ) L V (0) u kn , x ) v nl (cid:3) for u = [ u kl ] , v = [ v kl ] ∈ U ∞ ( V ), p ∈ N and elements of the form[( L V ( −
1) + L V (0) + l − k ) v kl ]for v = [ v kl ] ∈ U ∞ ( V ).The product ⋄ on U ∞ ( V ) looks complicated. But as we mentioned above, though [ v ] kl for v ∈ V and k, l ∈ N does span U ∞ ( V ), their infinite linear combinations give all the elements5f U ∞ ( V ) and U ∞ ( V ) can be studied using these elements. In particular, the product ⋄ canbe studied using these elements. So instead of working with arbitrary matrices in U ∞ ( V ),we use [ v ] kl for v ∈ V and k, l ∈ N to write down ⋄ . For u, v ∈ V and k, m, n, l ∈ N , bydefinition, [ u ] km ⋄ [ v ] nl = 0when m = n and[ u ] kn ⋄ [ v ] nl = Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l (cid:2) Y V ((1 + x ) L V (0) u, x ) v (cid:3) kl = n X m =0 (cid:18) − k + n − l − m (cid:19) Res x x − k + n − l − m − (1 + x ) l (cid:2) Y V ((1 + x ) L V (0) u, x ) v (cid:3) kl . (2.3)Since [ u ] km ⋄ [ v ] nl = 0 when m = n , we need only consider [ u ] kn ⋄ [ v ] nl for u, v ∈ V and k, n, l ∈ N . By taking u = [ u ] kn and v = [ v ] nl , we see also that the subspace O ∞ ( V ) isspanned by infinite linear combinations of elements of the formRes x x − k − l − p − (1 + x ) l [ Y V ((1 + x ) L V (0) u, x ) v ] kl for u, v ∈ V , k, l, p ∈ N and elements of the form[( L V ( −
1) + L V (0) + l − k ) v ] kl for v ∈ V and k, l ∈ N , with each pair ( k, l ) appearing in the linear combinations only finitelymany times.Let ∞ be the element of U ∞ ( V ) with diagonal entries being ∈ V and all the otherentries being 0. Then ∞ = ⊗ I ∞ .We shall take a quotient of U ∞ ( V ) such that the quotient with the product induced from ⋄ is an associative algebra and such that the associated graded space of a filtration of everylower-bounded generalized V -module is a module for this associaive algebra. To do this, weneed to first give an action of the (nonassociative) algebra U ∞ ( V ) with the product ⋄ on alower-bounded generalized V -module.We briefly recall the notion of lower-bounded generalized V -module. We refer the readerto Definition 1.2 in [H1], where a lower-bounded generalized V -module is called a lower-truncated generalized V -module. Definition 1.2 in [H1] is for a vertex operator algebra V but the definition applies also to a grading-restricted vertex algebra except that we haveto require the existence of operators L W (0) and L W ( −
1) satisfying the same axioms forthe corresponding operators coming from the vertex operator of the conformal element of avertex operator algebra. We also refer the reader to Definition 3.1 in [H2] for this notionin the special case that V is a grading-restricted vertex algebra (not a superalgebra) andthe automorphism of V is 1 V . Roughly speaking, a lower-bounded generalized V -moduleis a C -graded vector space W = ` n ∈ C W [ n ] equipped with a vertex operator map Y W : V ⊗ W → W [[ x, x − ]] and operators L W (0) and L W ( −
1) on W satisfying all the axioms foran (ordinary) V -module except that for n ∈ C , W [ n ] does not have to be finite dimensional6nd is the generalized eigenspace with the eigenvalue n of L W (0) instead of the eigenspacewith the eigenvalue n of L W (0). Module maps between lower-bounded generalized V -modulesare defined in the obvious way as in Definition 1.1 in [H1], not those defined in Definition4.2 in [H4]. On the other hand, if we replace V -module maps in the results below by thosein Definition 4.2 in [H4], these results still hold. The notion of generalized V -submoduleof a lower-bounded generalized V -module is defined in the obvious way. A generalized V -submodule of a lower-bounded generalized V -module is certainly also lower bounded.Let W be a lower-bounded generalized V -module. For n ∈ N , letΩ n ( W ) = { w ∈ W | ( Y W ) k ( v ) w = 0 for homogeneous v ∈ V, wt v − k − < − n } . Then Ω n ( W ) ⊂ Ω n ( W )for n ≤ n and W = [ n ∈ N Ω n ( W ) . So { Ω n ( w ) } n ∈ N is an ascending filtration of W . Let Gr ( W ) = X n ∈ N Gr n ( W )be the associated graded space, where Gr n ( W ) = Ω n ( W ) / Ω n − ( W ) . Sometimes we shall use [ w ] n to denote the element w + Ω n − ( W ) of Gr n ( W ), where w ∈ Ω n ( W ). Lemma 2.1
For w ∈ Ω n ( W ) and l ∈ N , Res x x l − Y W ( x L V (0) v, x ) w ∈ Ω n − l ( W ) .Proof. The operator Res x x l − Y W ( x L V (0)2 v, x ) has weight − l . Then for homogeneous u ∈ V , ( Y W ) p ( u )Res x x l − Y W ( x L V (0)2 v, x ) has weight wt u − p − − l . Consider the gen-eralized V -submodule of W generated by w . Then ( Y W ) p ( u )Res x x l − Y W ( x L V (0)2 v, x ) w is in this generalized V -submodule. Using the associativity for Y W , we know that thegeneralized V -submodule generated by w is spanned by elements of the form ( Y W ) m (˜ u ) w for ˜ u ∈ V . So ( Y W ) p ( u )Res x x l − Y W ( x L V (0)2 v, x ) w is a linear combination of such ele-ments. But for homogeneous w , the weight of ( Y W ) p ( u )Res x x l − Y W ( x L V (0)2 v, x ) w is wt u − p − − l + wt w . So the elements of the form ( Y W ) m (˜ u ) w whose linear combination is( Y W ) p ( u )Res x x l − Y W ( x L V (0)2 v, x ) w can also be chosen to be of weight wt u − p − − l +wt w ,that is, the weight wt ˜ u − m − Y W ) m (˜ u ) is equal to wt u − p − − l . Since w ∈ Ω n ( W ),( Y W ) m (˜ u ) w = 0 when wt ˜ u − m − < − n , or equivalently, wt u − p − < − ( n − l ). So wehave proved that ( Y W ) p ( u )Res x x l − Y W ( x L V (0)2 v, x ) w = 0 when wt u − p − < n − l . Thismeans that Res x x l − Y W ( x L V (0) v, x ) w ∈ Ω n − l ( W ).7y Lemma 2.1, the operator Res x x l − Y W ( x L V (0) v, x ) in fact induces an operator, stilldenoted by the same notation, on Gr ( W ), which maps Gr n ( W ) to Gr n − l ( W ).For v = [ v kl ] ∈ U ∞ ( V ), where v kl ∈ V and k, l ∈ N , we define an operator ϑ Gr ( W ) ( v ) on Gr ( W ) as follows: For w ∈ Gr ( W ), we define ϑ Gr ( W ) ( v ) w = X k,l ∈ N Res x x l − k − Y W ( x L V (0) v kl , x ) π Gr l ( W ) w , where π Gr l ( W ) is the projection from Gr ( W ) to Gr l ( W ). Note that since w is a sum ofelements of Gr l ( W ) for finitely many l ∈ N and for each l , there are only finitely manynonzero v kl , the sum over k and l is finite. So ϑ Gr ( W ) ( v ) w is indeed a well defined elementof Gr ( W ). In the case v = [ v ] kl and w = [ w ] n for v ∈ V , w ∈ W and k, l, n ∈ N , we have ϑ Gr ( W ) ([ v ] kl )[ w ] n = δ ln [Res x x l − k − Y W ( x L V (0) v, x ) w ] k . (2.4)In the case that v is homogeneous and w ∈ Gr l ( W ), we have ϑ Gr ( W ) ([ v ] kl )[ w ] l = [( Y W ) wt v + l − k − ( v ) w ] k . (2.5)We now have a linear map ϑ Gr ( W ) : U ∞ ( V ) → End Gr ( W ) v ϑ Gr ( W ) ( v ) . Let Q ∞ ( V ) be the intersection of ker ϑ Gr ( W ) for all lower-bounded generalized V -modules W and A ∞ ( V ) = U ∞ ( V ) /Q ∞ ( V ).We shall need the following lemma: Lemma 2.2
For l ∈ Z , k ∈ N and m ∈ Z + and v ∈ V , Res x x l − k − Y W (cid:18) x L V (0) (cid:18) L V ( −
1) + L V (0) + lk + m (cid:19) v, x (cid:19) = 0 . (2.6) In particular, when k = 0 and m = 1 , we have Res x x l − Y W (cid:0) x L V (0) ( L V ( −
1) + L V (0) + l ) v, x (cid:1) = 0 . (2.7) For l ∈ Z and v ∈ V , Res x x l − k − Y W (cid:18) x L V (0) (cid:18) L V ( −
1) + L V (0) + lk (cid:19) v, x (cid:19) = Res x x l − k − Y W ( x L V (0)+ l v, x ) . (2.8) Proof.
For l ∈ Z , n ∈ N and v ∈ V , using the L ( − Y W repeatedly, we have1 n ! d n dx n Y W ( x L V (0)+ l v, x ) = Y W (cid:18) x L V (0)+ l − n (cid:18) L V ( −
1) + L V (0) + ln (cid:19) v, x (cid:19) . (2.9)8ultiplying x p to both sides and then taking Res x , we obtainRes x x p n ! d n dx n Y W ( x L V (0)+ l v, x ) = Res x x l − n + p Y W (cid:18) x L V (0) (cid:18) L V ( −
1) + L V (0) + ln (cid:19) v, x (cid:19) . (2.10)When 0 ≤ p ≤ n −
1, the left-hand side of (2.10) is 0. Thus we obtainRes x x l − n + p Y W (cid:18) x L V (0) (cid:18) L V ( −
1) + L V (0) + ln (cid:19) v, x (cid:19) = 0 . (2.11)Let n = k + m and p = m − k ∈ N and m ∈ Z + . Then we obtain (2.6).Let p = − n = k in (2.10), we obtainRes x x − k ! d k dx k Y W ( x L V (0)+ l v, x ) = Res x x l − k − Y W (cid:18) x L V (0) (cid:18) L V ( −
1) + L V (0) + lk (cid:19) v, x (cid:19) . (2.12)Since left-hand side of (2.12) is equal toRes x x l − k − Y W ( x L V (0)+ l v, x ) , we obtain (2.8). Proposition 2.3
We have O ∞ ( V ) ⊂ Q ∞ ( V ) .Proof. We need to prove ϑ Gr ( W ) ( O ∞ ( V )) = 0 for every lower-bounded generalized V -module W . For Res x x − k − l − p − (1 + x ) l [ Y V ((1 + x ) L (0) v , x ) v ] kl ∈ O ∞ ( V ) , v , v ∈ V , ≤ k ≤ n ≤ l ≤ N and p ∈ N , and w ∈ Ω l ( W ), we have ϑ Gr ( W ) (Res x x − k − l − p − (1 + x ) l [ Y V ((1 + x ) L (0) v , x ) v ] kl )[ w ] l = Res x x l − k − Res x x − k − l − p − (1 + x ) l [ Y W ( x L V (0)2 Y V ((1 + x ) L (0) v , x ) v , x ) π G l ( W ) w ] k = Res x Res x x − k − l − p − (1 + x ) l x l − k − ·· [ Y W ( Y V ( x L V (0)2 (1 + x ) L (0) v , x x ) x L V (0)2 v , x ) w ] k = Res x Res x x − k − l − p − x l x − k − Res x x − δ (cid:18) x + x x x (cid:19) ·· [ Y W ( Y V ( x L V (0)1 v , x x ) x L V (0)2 v , x ) w ] k = Res x Res x x − k − l − p − x l x − k − Res x x − x − δ (cid:18) x − x x x (cid:19) ·· [ Y W ( x L (0)1 v , x ) Y W ( x L V (0)2 v , x ) w ] k − Res x Res x x − k − l − p − x l x − k − Res x x − x − δ (cid:18) x − x − x x (cid:19) ·· [ Y W ( x L V (0)2 v , x ) Y W ( x L (0)1 v , x ) w ] k = Res x Res x x − k − p − (1 − x − x ) − k − l − p − x l + p [ Y W ( x L (0)1 v , x ) Y W ( x L V (0)2 v , x ) w ] k − Res x Res x ( − x x − ) − k − p − x l x − k − [ Y W ( x L V (0)2 v , x ) Y W ( x L (0)1 v , x ) w ] k . (2.13)Since w ∈ Ω l ( W ) and the series (1 − x − x ) − k − l − p − contains only nonneative powers of x ,Res x (1 − x − x ) − k − l − p − x l + p Y W ( x L V (0)2 v , x ) w = 0 . So the first term in the right-hand side of (2.13) is 0. Since w ∈ Ω l ( W ) and the series( − x x − ) − k − p − contains only nonnegative powers of x ,Res x ( − x x − ) − k − p − x l Y W ( x L (0)1 v , x ) w = 0 . So the second term in the right-hand side of (2.13) is also 0.Taking l in (2.7) to be l − k , we obtain ϑ Gr ( W ) ([( L V ( −
1) + L V (0) + l − k ) v ] kl )[ w ] l = [Res x x l − k − Y W (cid:0) x L V (0) ( L V ( −
1) + L V (0) + l − k ) v, x (cid:1) w ] k = 0 (2.14)for v ∈ V , k, l ∈ N and w ∈ Ω l ( W ). Thus we have ϑ Gr ( W ) ( O ∞ ( V )) = 0. Theorem 2.4
Let W be a lower-bounded generalized V -module. Then the linear map ϑ Gr ( W ) : U ∞ ( V ) → End Gr ( W )10 ives a U ∞ ( V ) -module structure on Gr ( W ) (that is, ϑ Gr ( W ) is a homomorphism of (nonas-sociative) algebras from U ∞ ( V ) to End Gr ( W ) ). In particular, U ∞ ( V ) / ker ϑ Gr ( W ) is anassociative algebra isomorphic to a subalgebra of End Gr ( W ) .Proof. For u, v ∈ V , k, n, l ∈ N and w ∈ Ω l ( W ), using (2.3), we have ϑ Gr ( W ) ([ u ] kn ⋄ [ v ] nl )[ w ] l = Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l Res x x l − k − ·· [ Y W ( x L V (0)2 Y V ((1 + x ) L V (0) u, x ) v, x ) w ] k = Res x Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l x l − k − ·· [ Y W ( Y V (( x + x x ) L V (0) u, x x ) x L V (0)2 v, x ) w ] k = Res x Res x Res x x − δ (cid:18) x + x x x (cid:19) T k + l +1 (( x + 1) − k + n − l − ) x l x − k − ·· [ Y W ( Y V ( x L V (0)1 u, x x ) x L V (0)2 v, x ) w ] k = Res x Res x Res x x − x − δ (cid:18) x − x x x (cid:19) T k + l +1 (( x + 1) − k + n − l − ) x l x − k − ·· [ Y W ( x L V (0)1 u, x ) Y W ( x L V (0)2 v, x ) w ] k − Res x Res x Res x x − x − δ (cid:18) x − x − x x (cid:19) T k + l +1 (( x + 1) − k + n − l − ) x l x − k − ·· [ Y W ( x L V (0)2 v, x ) Y W ( x L V (0)1 u, x ) w ] k = Res x Res x T k + l +1 (( x + 1) − k + n − l − ) (cid:12)(cid:12)(cid:12)(cid:12) x =( x − x ) x − x l x − k − ·· [ Y W ( x L V (0)1 u, x ) Y W ( x L V (0)2 v, x ) w ] k − Res x Res x T k + l +1 (( x + 1) − k + n − l − ) (cid:12)(cid:12)(cid:12)(cid:12) x =( − x + x ) x − x l x − k − ·· [ Y W ( x L V (0)2 v, x ) Y W ( x L V (0)1 u, x ) w ] k . (2.15)Since w ∈ Ω l ( W ), the second term in the right-hand side of (2.15) is 0. Expanding T k + l +1 (( x + 1) − k + n − l − ) explicitly, we see that the first term in the right-hand side of (2.15)is equal to n X m =0 (cid:18) − k + n − l − m (cid:19) Res x Res x ( x − x ) − k + n − l − m − x k − n + l + m +12 x l x − k − ·· [ Y W ( x L V (0)1 u, x ) Y W ( x L V (0)2 v, x ) w ] k = n X m =0 X j ∈ N (cid:18) − k + n − l − m (cid:19)(cid:18) − k + n − l − m − j (cid:19) ( − j Res x Res x x − k + n − m − − j ·· x − n + l + m − j [ Y W ( x L V (0)1 u, x ) Y W ( x L V (0)2 v, x ) w ] k . (2.16)11n the case j > n − m , since w ∈ Ω l ( W ),wt v − (wt v − n + l + m − j ) − < − l in the case that v is homogeneous and hence we haveRes x x − n + l + m − j Y W ( x L V (0)2 v, x ) w = 0 . Hence those terms in the right-hand side of (2.16) with j > n − m is 0 So the right-handside of (2.16) is equal to n X m =0 n − m X j =0 (cid:18) − k + n − l − m (cid:19)(cid:18) − k + n − l − m − j (cid:19) ( − j ·· Res x Res x x − k + n − m − − j x − n + l + m − j [ Y W ( x L V (0)1 u, x ) Y W ( x L V (0)2 v, x ) w ] k = n X m =0 n X p = m (cid:18) − k + n − l − m (cid:19)(cid:18) − k + n − l − m − p − m (cid:19) ( − p − m ·· Res x Res x x − k + n − − p x − n + l − p [ Y W ( x L V (0)1 u, x ) Y W ( x L V (0)2 v, x ) w ] k = n X p =0 p X m =0 (cid:18) − k + n − l − m (cid:19)(cid:18) − k + n − l − m − p − m (cid:19) ( − p − m ! ·· Res x Res x x − k + n − − p x − n + l − p [ Y W ( x L V (0)1 u, x ) Y W ( x L V (0)2 v, x ) w ] k . (2.17)For p = 0 , . . . , n , p X m =0 (cid:18) − k + n − l − m (cid:19)(cid:18) − k + n − l − m − p − m (cid:19) ( − p − m = p X m =0 ( − k + n − l − · · · ( − k + n − l − m ) m ! ·· ( − k + n − l − m − · · · ( − k + n − l − p )( p − m )! ( − p − m = p X m =0 ( − k + n − l − · · · ( − k + n − l − p ) p ! p ! m !( p − m )! ( − p − m = (cid:18) − k + n − l − p (cid:19) p X m =0 (cid:18) pm (cid:19) ( − p − m = (cid:18) − k + n − l − p (cid:19) ( − p = (cid:18) − k + n − l − p (cid:19) δ p, . (2.18)12sing (2.18), we see that the right-hand side of (2.17) is equal toRes x Res x x − k + n − x − n + l − [ Y W ( x L V (0)1 u, x ) Y W ( x L V (0)2 v, x ) w ] k = ϑ Gr ( W ) ([ u ] kn )[Res x x l − n − Y W ( x L V (0)2 v, x ) w ] n = ϑ Gr ( W ) ([ u ] kn ) ϑ Gr ( W ) ([ v ] nl )[ w ] l . (2.19)From (2.15), (2.16), (2.17) and (2.19), we obtain ϑ Gr ( W ) ([ u ] kn ⋄ [ v ] nl ) = ϑ Gr ( W ) ([ u ] kn ) ϑ Gr ( W ) ([ v ] nl )for u, v ∈ V and k, n, l ∈ N . Thus ϑ Gr ( W ) gives an U ∞ ( V )-module structure on W . Lemma 2.5
Let L U ( − and L U (0) be operators on a vector space U satisfying [ L U (0) , L U ( − L U ( − . We have e xL U ( − (1 + x ) L U (0) = (1 + x ) L U ( − L U (0) . (2.20) Proof.
This can be proved easily by showing ddx e xL U ( − (1 + x ) L U (0) (1 + x ) − ( L U ( − L U (0)) = 0so that it must be independent of x and then setting x = 0 to obtain e xL U ( − (1 + x ) L U (0) (1 + x ) − ( L U ( − L U (0)) = 1 U . We can now write down explicitly the expressions of elements of the form [ v ] kl ⋄ ∞ for v ∈ V and k, l ∈ N satisfying k ≤ l . Lemma 2.6
For v ∈ V and k, l ∈ N , [ v ] kl ⋄ ∞ = l X m =0 (cid:18) − k − m (cid:19) (cid:20)(cid:18) L V ( −
1) + L V (0) + lk + m (cid:19) v (cid:21) kl . (2.21) Proof.
By the definition (2.2) of ⋄ and the skew-symmetry of Y V ,([ v ] kl ⋄ N ∞ ) mn = δ km δ ln Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) n Y V ((1 + x ) L (0) v, x ) = δ km δ ln Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) n e xL V ( − (1 + x ) L V (0) v. (2.22)Thus we obtain[ v ] kl ⋄ ∞ = Res x T k + l +1 (( x + 1) − k − )(1 + x ) l [ e xL V ( − (1 + x ) L V (0) v ] kl . (2.23)13sing (2.20) with U = V , expanding the formal series explicitly and then evaluating theformal residue, we see that the right-hand side of (2.23) is equal toRes x T k + l +1 (( x + 1) − k − )(1 + x ) l [(1 + x ) L V (0)+ L V (0) v ] kl = l X m =0 (cid:18) − k − m (cid:19) Res x x − k − m − [(1 + x ) L V ( − L V (0)+ l v ] kl = l X m =0 X j ∈ N (cid:18) − k − m (cid:19) Res x x − k − m − j (cid:18) L V ( −
1) + L V (0) + lj (cid:19) v = l X m =0 (cid:18) − k − m (cid:19)(cid:18) L V ( −
1) + L V (0) + kk + m (cid:19) v, (2.24)proving (2.21). Proposition 2.7
For v ∈ V and k, l ∈ N , [ v ] kl ⋄ ∞ − [ v ] kl ∈ O ∞ ( V ) . For u, v ∈ V and k, l, n ∈ N , ([ v ] kl ⋄ ∞ − [ v ] kl ) ⋄ [ u ] ln ∈ O ∞ ( V ) .Proof. For m ∈ N , (cid:18) L V ( −
1) + L V (0) + lk + m (cid:19) v = (cid:18) ( L V ( −
1) + L V (0) + l − k ) + kk + m (cid:19) v = (cid:18) kk + m (cid:19) v + ( L V ( −
1) + L V (0) + l − k )˜ v m ≡ (cid:26) m ∈ Z + v m = 0 mod O ∞ ( V ) , (2.25)where ˜ v m is an element of V depending on m . Thus by (2.21),[ v ] kl ⋄ ∞ ≡ v mod O ∞ ( V ) . By (2.21), (2.25) and (2.30),([ v ] kl ⋄ ∞ ) ⋄ [ u ] ln = l X m =0 (cid:18) − k − m (cid:19) (cid:20)(cid:18) L V ( −
1) + L V (0) + lk + m (cid:19) v (cid:21) kl ⋄ [ u ] ln = l X m =0 (cid:18) − k − m (cid:19)(cid:18) kk + m (cid:19) [ v ] kl ⋄ [ u ] ln + l X m =0 (cid:18) − k − m (cid:19) [( L V ( −
1) + L V (0) + l − k )˜ v m ] kl ⋄ [ u ] ln ≡ [ v ] kl ⋄ [ u ] ln mod O ∞ ( V ) . heorem 2.8 The product ⋄ on U ∞ ( V ) induces a product, denoted still by ⋄ , on A ∞ ( V ) = U ∞ ( V ) /Q ∞ ( V ) such that A ∞ ( V ) equipped with ⋄ is an associative algebra with ∞ as iden-tity. Moreover, the associated graded space Gr ( W ) of the ascendant filtration { Ω n ( W ) } n ∈ N of a lower-bounded generalized V -module W is an A ∞ ( V ) -module.Proof. Since ker ϑ Gr ( W ) for a lower-bounded generalized V -module W is a two-sided idealof U ∞ ( V ), Q ∞ ( V ) as the intersection of such two-sided ideals is still a two-sided ideal of U ∞ ( V ). Thus ⋄ on U ∞ ( V ) induces a product on A ∞ ( V ). Since for each lower-boundedgeneralized V -module W , the quotient algebra U ∞ ( V ) / ker ϑ Gr ( W ) is associative, A ∞ ( V ) asa quotient of the associative algebra U ∞ ( V ) / ker ϑ Gr ( W ) is also associative.By definition, we have ∞ ⋄ [ v ] kl = [ v ] kl . So ∞ is in fact a left identity of the algebra U ∞ ( V ). By Proposition 2.3, O ∞ ( V ) ⊂ Q ∞ ( V ). Then by Proposition 2.7, we have([ v ] kl + Q ∞ ( V )) ⋄ ( ∞ + Q ∞ ( V )) = [ v ] kl + Q ∞ ( V ) . So ∞ + Q ∞ ( V ) is an identity of A ∞ ( V ).For a lower-bounded generalized V -module W , by Theorem 2.4, Gr ( W ) is a modulefor U ∞ ( V ) / ker ϑ Gr ( W ) . Since Q ∞ ( V ) is a two-sided subideal of ker ϑ Gr ( W ) , Gr ( W ) is an A ∞ ( V )-module.The ideal Q ∞ ( V ) of U ∞ ( V ) is defined using all lower-bounded generalized V -modules. Itis not easy to find an explicit description of Q ∞ ( V ). We even do not know whether Q ∞ ( V )is generated by O ∞ ( V ). One research problem is to find an explicit description of Q ∞ ( V ),or at least prove some structure theorems about A ∞ ( V ).The only result above on Q ∞ ( V ) is Proposition 2.3. Below we give another result (Propo-sition 2.10). To prove this result, we need the following commutator formula: Lemma 2.9
For v ∈ V , [ L V ( −
1) + L V (0) , Y V ((1 + x ) L V (0) v, x )] = Y V ((1 + x ) L V (0) ( L V ( −
1) + L V (0)) v, x ) . (2.26) Proof.
By the L ( −
1) and L (0)-commutator formula with the vertex operator map Y V andthe fact that the weight of L V ( −
1) is 1,[ L V ( −
1) + L V (0) , Y V ((1 + x ) L V (0) v, x )]= Y V (((1 + x ) L V ( −
1) + L V (0))(1 + x ) L V (0) v, x )= Y V ((1 + x ) L V (0) ( L V ( −
1) + L V (0)) v, x ) . By Theorem 2.4, every lower-bounded generalized V -module is an A ∞ ( V )-module. Proposition 2.10
For u, v ∈ V and k, n, l ∈ V , both [( L V ( −
1) + L V (0) + n − k ) u ] kn ⋄ [ v ] nl and [ v ] kn ⋄ [( L V ( −
1) + L V (0) + l − n ) u ] nl are in O ∞ ( V ) . roof. For u, v ∈ V , k, l, m ∈ N , by definition,[( L V ( −
1) + L V (0) + n − k ) u ] kn ⋄ [ v ] nl = Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l [ Y V ((1 + x ) L V (0) ( L V ( −
1) + L V (0) + n − k ) u, x ) v ] kl = Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) k − n + l +1 ddx [ Y V ((1 + x ) L V (0)+ n − k u, x ) v ] kl = − Res x (cid:18) ddx T k + l +1 (( x + 1) − k + n − l − )(1 + x ) k − n + l +1 (cid:19) [ Y V ((1 + x ) L V (0)+ n − k u, x ) v ] kl = − Res x (cid:18) ddx T k + l +1 (( x + 1) − k + n − l − ) (cid:19) (1 + x ) k − n + l +1 + ( k − n + l + 1) T k + l +1 (( x + 1) − k + n − l − )(1 + x ) k − n + l ! ·· (1 + x ) − k + n [ Y V ((1 + x ) L V (0) u, x ) v ] kl . (2.27)Applying − xk − n + l +1 ddx to bother sides of (2.1), we obtain( x + 1) − k + n − l − = − xk − n + l + 1 ddx T k + l +1 (( x + 1) − k + n − l − ) − xk − n + l + 1 ddx R k + l +1 (( x + 1) − k + n − l − ) . (2.28)Since the first and second terms in the right-hand side of (2.28) contain only the terms withpowers in x − less than or equal to and larger than, respectively, k + l + 2, we must have − xk − n + l + 1 ddx T k + l +1 (( x + 1) − k + n − l − ) = T k + l +2 (( x + 1) − k + n − l − ) , or equivalently,(1 + x ) ddx T k + l +1 (( x + 1) − k − )= − ( k − n + l + 1) T k + l +1 (( x + 1) − k + n − l − ) − ( k − n + l + 1) (cid:18) − k + n − l − n + 1 (cid:19) x − k − l − , (2.29)Using (2.29), the right-hand side of (2.27) becomes( k − n + l + 1) (cid:18) − k + n − l − n + 1 (cid:19) Res x x − k − l − (1 + x ) l [ Y V ((1 + x ) L V (0) u, x ) v ] kl ∈ O ∞ ( V ) . Thus we obtain[( L V ( −
1) + L V (0) + n − k ) u ] kn ⋄ [ v ] nl = ( k − n + l + 1) (cid:18) − k + n − l − n + 1 (cid:19) Res x x − k − l − (1 + x ) l [ Y V ((1 + x ) L V (0) u, x ) v ] kl ∈ O ∞ ( V ) . (2.30)16or u, v ∈ V , k, l, n ∈ N satisfying n ≤ k ≤ l , by the definition, (2.26) and l − n =( l − k ) − ( n − k )[ v ] kn ⋄ [( L V ( −
1) + L V (0) + l − n ) u ] nl = Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l ·· [ Y V ((1 + x ) L V (0) v, x )( L V ( −
1) + L V (0) + l − n ) u ] kl = Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l ·· [( L V ( −
1) + L V (0) + l − k ) Y V ((1 + x ) L V (0) v, x ) u ] kl − Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l ·· [ Y V ((1 + x ) L V (0) ( L V ( −
1) + L V (0) + n − k ) v, x ) u ] kl . (2.31)The first term in the right-hand side of (2.31) is by definition in O ∞ ( V ). The second termin the right-hand side of (2.31) is equal to [( L V ( −
1) + L V (0) + n − k ) v ] kn ⋄ [ u ] nl , which isalso in O ∞ ( V ) (2.30). So[ v ] kn ⋄ [( L V ( −
1) + L V (0) + l − n ) u ] nl ∈ O ∞ ( V ) . V -modules and graded A ∞ -modules We study the relations between lower-bounded generalized V -modules and suitable A ∞ ( V )-modules in this section.Note that W is graded by the generalized eigenspaces of L W (0). Since Ω n ( W ) for n ∈ N isinvariant under L W (0), L W (0) induces an operator on Gr n ( W ) = Ω n ( W ) / Ω n − ( W ) such that Gr n ( W ) is also graded by the generalized eigenspaces of this operator. These operators on Gr n ( W ) for n ∈ N together define an operator, denoted by L Gr ( W ) (0), on Gr ( W ) preservingthe N -grading on Gr ( W ). Then Gr ( W ) is also graded by the generalized eigenspaces of L Gr ( W ) (0).For v ∈ V , k ∈ Z and w ∈ Ω n ( W ), by the L ( − Y W ) k ( v ) L W ( − w = L W ( − Y W ) k ( v ) w + k ( Y W ) k − ( v ) w. When wt v − k − < − ( n + 1), we have wt v − k − < − n and wt v − ( k − − < − n . Soin this case, ( Y W ) k ( v ) w = ( Y W ) k − ( v ) w = 0 since w ∈ Ω n ( W ). Thus ( Y W ) k ( v ) L W ( − w = 0when wt v − k − < − ( n + 1). This means that L W ( − w ∈ Ω n +1 ( W ). In particular, L W ( −
1) induces a linear map from Gr n ( W ) to Gr n +1 ( W ) for n ∈ N . These maps for n ∈ N together define an operator, denoted by L Gr ( W ) ( − Gr ( W ).The operators L Gr ( W ) (0), L Gr ( W ) ( −
1) and ϑ Gr ( W ) ([ v ] kl ) satisfy the same commutatorformulas as those between L W (0), L W ( −
1) and Res x x l − k − Y W ( x L V (0) v, x ) for v ∈ V and k, l ∈ N . These structures on Gr ( W ) motivates the following definition:17 efinition 3.1 Let G be an A ∞ ( V )-module with the A ∞ ( V )-module structure on G givenby a homomorphism ϑ G : A ∞ ( V ) → End G of associative algebras. We say that G is a graded A ∞ ( V ) -module if the following conditions are satisfied:1. G is graded by N , that is, G = ` n ∈ N G n , and for v ∈ V , k, l ∈ N , ϑ G ([ v ] kl + Q ∞ ( V ))maps G n to 0 when n = l and to G k when n = l .2. For g ∈ G l , if ϑ G ([ v ] l + Q ∞ ( V )) g = 0 for all v ∈ V , then g = 0.3. G is a direct sum of generalized eigenspaces of an operator L G (0) on G and the realparts of the eigenvalues of L G (0) has a lower bound.4. There is an operator L G ( −
1) on G mapping G n to G n +1 for n ∈ N .5. The commutator relations[ L G (0) , L G ( − L G ( − , [ L G (0) , ϑ G ([ v ] kl + Q ∞ ( V ))] = ( k − l ) ϑ G ([ v ] kl + Q ∞ ( V )) , [ L G ( − , ϑ G ([ v ] kl + Q ∞ ( V ))] = ϑ G ([ L V ( − v ] ( k +1) l + Q ∞ ( V ))hold for v ∈ V and k, l ∈ N Let G and G be graded A ∞ ( V )-modules. A graded A ∞ ( V ) -module map from G to G isan A N ( V )-module map f : G → G such that f (( G ) n ) ⊂ ( G ) n , f ◦ L G (0) = L G (0) ◦ f and f ◦ L G ( −
1) = L G ( − ◦ f . A graded A ∞ ( V ) -submodule of a graded A ∞ ( V )-module G is an A ∞ ( V )-submodule of G that is also an N -graded subspace of G and invariant under theoperators L G (0) and L G ( − A ∞ ( V )-module G is said to be generated by a subset S if G is equal to the smallest graded A ∞ ( V )-submodule containing S , or equivalently, G is spanned by homogeneous elements with respect to the N -grading and the grading givenby L G (0) obtained by applying elements of A ∞ ( V ), L G (0) and L G ( −
1) to homogeneoussummands of elements of S . A graded A ∞ ( V )-module is said to be irreducible if it has nononzero proper graded A ∞ ( V )-submodules. A graded A ∞ ( V )-module is said to be completelyreducible if it is a direct sum of irreducible graded A ∞ ( V )-modules.From Theorem 2.8 and the properties of a lower-bounded generalized V -module W andits associated graded space Gr ( W ), we obtain immediately: Theorem 3.2
For a lower-bounded generalized V -module W , Gr ( W ) is a graded A ∞ -module.Let W and W be lower-bounded generalized V -modules and f : W → W a V -module map.Then f induces a graded A ∞ ( V ) -module map Gr ( f ) : Gr ( W ) → Gr ( W ) . We now give a direct and explicit description of Gr ( W ) for a completely reducible lower-bounded generalized V -module W . In this case, W = a µ ∈M W µ , M is an index set and W µ for µ ∈ M are irreducible lower-bounded generalized V -modules. For µ ∈ M , since W µ is irreducible, there exists h µ ∈ C such that W µ = a n ∈ N W µ [ h µ + n ] , where as usual, W µ [ h µ + n ] for n ∈ N is the subspace of W µ of weight h µ + n , and W µ [ h µ ] = 0.For n ∈ N , let G n ( W ) = a µ ∈M W [ h µ + n ] . Then W = a n ∈ N G n ( W ) . For n ∈ N , let T n ( W ) = n a m =0 G m ( W ) . It is clear that T n ( W ) ⊂ Ω n ( W ). In particular, G n ( W ) ⊂ Ω N ( W ) for n ≤ N . Let e W : W → Gr ( W ) be defined by e W ( w ) = w + Ω n − ( W ) for w ∈ G n ( W ) and n ∈ N . Then e W preservesthe N -grading. We also define a map ϑ W : U ∞ ( V ) → End W by ϑ W ( v ) w = X k,l ∈ N Res x x l − k − Y W ( x L V (0) v kl , x ) π G l ( W ) w for v ∈ U ∞ and w ∈ W , where π G l ( W ) is the projection from W to G l ( W ). In the case v = [ v ] kl and w ∈ G n ( W ) for v ∈ V and k, l, n ∈ N , we have ϑ W ([ v ] kl ) w = δ ln Res x x l − k − Y W ( x L V (0) v, x ) w. (3.1) Proposition 3.3
Let W be a completely reducible lower-bounded generalized V -module.Then Ω n ( W ) = T n ( W ) for n ∈ N . Moreover, W equipped with ϑ W is a graded A ∞ ( V ) -module and e W : W → Gr ( W ) is an isomorphism of graded A ∞ ( V ) -modules.Proof. If T n ( W ) = Ω n ( W ), then there exists homogeneous w ∈ Ω n ( W ) but w T n ( W ).Then w = P µ ∈M w µ , where w µ ∈ W µ for µ ∈ M and only finitely many w µ is not 0. Since w is homogeneous, we can assume that w µ for µ ∈ M are homogeneous. Since w ∈ Ω n ( W )but w T n ( W ), there is at least one w µ such that w µ ∈ Ω n ( W µ ) but w µ T n ( W µ ) = ` nm =0 W µ [ h µ + m ] . Let W µ be the generalized V -submodule of W µ generated by such a w µ .Since w µ T n ( W µ ), w µ = 0 and hence W µ = 0. But W µ is irreducible. So W µ = W µ .Since w µ is homogeneous, there is m ∈ N such that wt w µ ∈ W µ [ h µ + m ] . Since w µ T n ( W µ ),we must have m > n . Since W µ = W µ , W µ is spanned by elements of the form ( Y W ) k ( v ) w µ for v ∈ V and k ∈ Z . Since w µ ∈ Ω n ( W µ ), ( Y W ) k ( v ) w µ = 0 for homogeneous v ∈ V and k ∈ Z satisfying wt v − k − < − n . Thus the homogeneous subspaces of W µ [ h µ + m − n − p ] = 019or p ∈ Z + . But for p = m − n ∈ Z + , W µ [ h µ + m − n − p ] = W µ [ h µ ] = 0. Contradiction. Thus T n ( W ) = Ω n ( W ).For n ∈ N , we have Gr n ( W ) = Ω n ( W ) / Ω n − ( W ) = T n ( W ) /T n − ( W ). Then e W (cid:12)(cid:12) G n ( W ) isclearly a linear isomorphism from G n ( W ) to T n ( W ) /T n − ( W ) = Gr n ( W ). This shows that e W is an isomorphism of graded spaces. For v ∈ V , k, l ∈ N and w ∈ G l ( W ), e W ( ϑ W ([ v ] kl ) w ) = e W (Res x x l − k − Y W ( x L V (0) v, x ) w )= Res x x l − k − Y W ( x L V (0) v, x ) w + T k − ( W )= Res x x l − k − Y W ( x L V (0) v, x )( w + T l − ( W )= ϑ Gr ( W ) ([ v ] kl ) e W ( w ) . Thus we have e W ◦ ϑ W = ϑ Gr ( W ) ◦ e W . In particular, the A ∞ ( V )-module structure on Gr ( W )given by ϑ Gr ( W ) is transported to W by e W so that W equipped with ϑ W is an A ∞ ( V )-moduleand e W : W → Gr ( W ) is an isomorphism of A ∞ ( V )-modules. Theorem 3.4
A lower-bounded generalized V -module W is irreducible or completely re-ducible if and only if the graded A ∞ ( V ) -module Gr ( W ) is irreducible or completely reducible,respectively.Proof. Let W be an irreducible lower-bounded generalized V -module. By Theorem 3.3, W is a graded A ∞ ( V )-module isomorphic to Gr ( W ). Let W be a nonzero graded A ∞ ( V )-submodule of the graded A ∞ ( V )-module W . For a homogeneous element v ∈ V , n ∈ Z and w ∈ W , Res x x n Y W ( v, x ) w = X l ∈ N ϑ W ([ v ] (wt v − n − l ) l ) π G l ( W ) w ∈ W . This means that W is invariant under the action of the vertex operators on W . By the defi-nition of graded A ∞ ( V )-submodule, W is invariant under the actions of L W (0) and L W ( − L W (0) (cid:12)(cid:12) W . Thus W is also a nonzerolower-bounded generalized V -submodule of W . Since W is an irreducible lower-bounded gen-eralized V -module, W = W . So as a graded A ∞ ( V )-module, W is also irreducible. Since asa graded A ∞ ( V )-module, Gr ( W ) is equivalent to W , we see that Gr ( W ) is irreducible.Conversely, assume that for a lower-bounded generalized V -module W , the graded A ∞ ( V )-module Gr ( W ) is irreducible. Let W be a nonzero generalized V -submodule of W . ThenΩ n − ( W ) ⊂ Ω n − ( W ) for n ∈ N (when n = 0, Ω − ( W ) = 0). We have a map from Gr ( W )to Gr ( W ) given by w +Ω n − ( W ) w +Ω n − ( W ) for n ∈ N and w ∈ Ω n ( W ). This map isan injective graded A ∞ ( V )-module map. So the image of Gr ( W ) under this map is a graded A ∞ ( V )-submodule of Gr ( W ). Since W is nonzero, Gr ( W ) is nonzero. Since Gr ( W ) is ir-reducible, the image of Gr ( W ) under this map is equal to Gr ( W ). Now it is easy to derive W = W . In fact, for n ∈ N , the image of Gr n ( W ) under the map from Gr ( W ) to Gr ( W )above is { w + Ω n ( W ) | w ∈ Ω n ( W ) } . So Gr n ( W ) = { w + Ω n − ( W ) | w ∈ Ω n ( W ) } .For n = 0, we obtain Ω ( W ) = Gr ( W ) = Gr ( W ) = Ω ( W ). Assume that Ω n − ( W ) =20 n − ( W ). Given w ∈ Ω n ( W ), w + Ω n − ( W ) ∈ Gr n ( W ). By Gr n ( W ) = { w + Ω n − ( W ) | w ∈ Ω n ( W ) } , there exists w ∈ Ω n ( W ) such that w + Ω n − ( W ) = w + Ω n − ( W ), or equiv-alently, w − w ∈ Ω n − ( W ) = Ω n − ( W ). Thus w ∈ Ω n ( W ). This shows Ω n ( W ) = Ω n ( W )for n ∈ N . Then we have W = ∪ n ∈ N Ω n ( W ) = ∪ n ∈ N Ω n ( W ) = W . So W is irreducible.Assume that a lower-bounded generalized V -module W is completely reducible. Then W = ` µ ∈M W µ , where W µ for µ ∈ µ are irreducible generalized V -modules. From whatwe have proved above, W µ for µ ∈ M as graded A ∞ ( V )-modules are also irreducible. So W as a graded A ∞ ( V )-module is completely reducible. But Gr ( W ) is equivalent to W as a graded A ∞ ( V )-module by Proposition 3.3. So Gr ( W ) is also completely reducible.Conversely, assume that for a lower-bounded generalized V -module W , the graded A ∞ ( V )-module Gr ( W ) is completely reducible. Then Gr ( W ) = ` µ ∈M G µ , where G µ for µ ∈ M areirreducible graded A ∞ ( V )-submodules of Gr ( W ). For µ ∈ M , since G µ is a graded A ∞ ( V )-submodule of Gr ( W ), we have G µn ⊂ Gr n ( W ) = Ω n ( W ) / Ω n − ( W ). Let W µ be the subspaceof W consisting of elements of the form w µ ∈ Ω n ( W ) such that w µ +Ω n − ( W ) ∈ G µn for n ∈ N .Since G µ is a graded A ∞ ( V )-submodule of Gr ( W ), for v ∈ V , k, l ∈ N and w µ ∈ Ω l ( W ) suchthat w µ + Ω l − ( W ) ∈ G µl ,Res x x l − k − Y W ( x L V (0) v, x ) w µ + Ω k − ( W ) ∈ G µk . By the definition of W µ , we obtain Res x x l − k − Y W ( x L V (0) v, x ) w µ ∈ W µ . Since w µ ∈ Ω l ( W ),Res x x l − k − Y W ( x L V (0) v, x ) w µ = 0 for k ∈ − Z + . Thus Res x x l − k − Y W ( x L V (0) v, x ) w µ ∈ W µ for k ∈ N are all the nonzero coefficients of Y W ( v, x ) w µ . This means that W µ is closed under theaction of the vertex operators on W . Since G µ is invariant under the actions of L Gr ( W ) (0)and L Gr ( W ) ( −
1) and is a direct sum of generalized eigenspaces of L Gr ( W ) (0), W µ is invariantunder the actions of L W (0) and L W ( −
1) and is a direct sum of generalized eigenspaces of L W (0). Thus W µ is a generalized V -submodule of W .Let w µ + Ω n − ( W µ ) ∈ Gr n ( W µ ), where n ∈ N and w µ ∈ Ω n ( W µ ) ⊂ Ω n ( W ). By thedefinition of W µ , we see that since w µ is an element of W µ , w µ + Ω n − ( W ) ∈ G µn . Sowe obtain a linear map from Gr n ( W µ ) to G µn given by w µ + Ω n − ( W µ ) w µ + Ω n − ( W )for w µ + Ω n − ( W µ ) ∈ Gr n ( W µ ). These maps for n ∈ N give a map from Gr ( W µ ) to G µ .It is clear that this map is a graded A ∞ ( V )-module map. If the image w µ + Ω n − ( W )of w µ + Ω n − ( W µ ) ∈ Gr n ( W µ ) under this map is 0 in G µ , then w µ ∈ Ω n − ( W ). But w µ ∈ Ω n ( W µ ) ⊂ W µ . So w µ ∈ Ω n − ( W µ ) and w µ + Ω n − ( W µ ) is 0 in Gr ( W µ ). This meansthat this graded A ∞ ( V )-module map is injective. In particular, the image of Gr ( W µ ) underthis map is a nonzero graded A ∞ ( V )-submodule of G µ . But G µ is irreducible. So Gr ( W µ )must be equivalent to G µ and is therefore also irreducible. From what we have proved above,since Gr ( W µ ) is irreducible, W µ is irreducible. This shows that W is complete reducible.Theorem 3.4 implies that there is a map from the set of the equivalence classes of lower-bounded generalized V -modules to the set of equivalence classes of graded A ∞ ( V )-modules.This map is in fact a bijection. To prove this, we need to construct a lower-bounded general-ized V -module S ( G ) from a graded A ∞ ( V )-module G . We use the construction in Section 5of [H3]. Take the generating fields for the grading-restricted vertex algebra V to be Y V ( v, x )for v ∈ V . By definition, G is a direct sum of generalized eigenspaces of L G (0) and the real21arts of the eigenvalues of L G (0) has a lower bound B ∈ R . We take M and B in Section 5of [H3] to be G and the lower bound B above. Using the construction in Section 5 of [H3],we obtain a universal lower-bounded generalized V -module b G [1 V ] B . For simplicity, we shalldenote it simply by b G .By Theorem 3.3 in [H4] and the construction in Section 5 of [H3] and by identifyingelements of the form ( ψ a b G ) − , with basis elements g a ∈ G for a ∈ A for a basis { g a } a ∈ A of G , we see that b G is generated by G (in the sense of Definition 3.1 in [H4]). Moreover, afteridentifying ( ψ a b G ) − , with basis elements w a ∈ G for a ∈ A , Theorems 3.3 and 3.4 in [H4] infact say that elements of the form L b G ( − p w a for p ∈ N and a ∈ A are linearly independentand b G is spanned by elements obtained by applying the components of the vertex operatorsto these elements. In particular, G can be embedded into b G as a subspace. So from now on,we shall view G as a subspace of b G . Let J G be the generalized V -submodule of b G generatedby elements of the forms Res x x l − k − Y b G ( x L V (0) v, x ) g (3.2)for l ∈ N , k ∈ − Z + and g ∈ G l ,Res x x l − k − Y c M ( x L V (0) v, x ) g − ϑ G ([ v ] kl + Q ∞ ( V )) g (3.3)for v ∈ V , k, l ∈ N , g ∈ G l and L b G ( − g − L G ( − g (3.4)for l ∈ N , g ∈ G l .Let S ( G ) = b G/J G . Then S ( G ) is a lower-bounded generalized V -module. Let π S ( G ) bethe projection from b G to S ( G ). Since b G is generated by G (in the sense of Definition 3.1 in[H4]), S ( G ) is generated by π S ( G ) ( G ) (in the same sense). In particular, S ( G ) is spanned byelements of the formRes x x ( l + p ) − n − Y S ( G ) ( x L S ( G ) (0) v, x ) L S ( G ) ( − p π S ( G ) ( g ) (3.5)for v ∈ V , n, l, p ∈ N and g ∈ G l . For n ∈ N , let G n ( S ( G )) be the subspace of S ( G ) spannedby elements of the form (3.5) for v ∈ V , l, p ∈ N and g ∈ G l . Proposition 3.5
Let G be a graded A ∞ ( V ) -module.1. For n ∈ N , G n ( S ( G )) = π S ( G ) ( G n ) and for n = n , G n ( S ( G )) ∩ G n ( S ( G )) = 0 .Moreover, S ( G ) = π S ( G ) ( G ) = a n ∈ N G n ( S ( G )) .2. For n ∈ N , Ω n ( S ( G )) = n a j =0 π S ( G ) ( G j ) = n a j =0 G j ( S ( G )) .3. Gr ( S ( G )) is equivalent to G as a graded A ∞ ( V ) -module. roof. Since elements of the forms (3.3) and (3.4) are in J G , for n ∈ N , the element (3.5)for v ∈ V for l, p ∈ N and g ∈ G l is in fact equal to π S ( G ) ( ϑ G ([ v ] n ( l + p ) + Q ∞ ( V )) L G ( − p g ) . (3.6)Since ϑ G ([ v ] n ( l + p ) + Q ∞ ( V )) L G ( − p g for l, p ∈ N and g ∈ G l certainly span G n and elementsof the form (3.5) for v ∈ V for l, p ∈ N and g ∈ G l span G n ( S ( G )), elements of the form (3.6)for v ∈ V for l, p ∈ N and g ∈ G l also span G n ( S ( G )). Thus G n ( S ( G )) = π S ( G ) ( G n ). When n = n , we know G n ∩ G n = 0. Then G n ( S ( G )) ∩ G n ( S ( G )) = π S ( G ) ( G n ∩ G n ) = 0.As is mentioned above, S ( G ) is spanned by elements of the form (3.5) for v ∈ V , k, l, p ∈ N and g ∈ G l . But we already see that (3.5) is in fact equal to (3.6). Thus S ( G ) = π S ( G ) ( G ).Since G n ( S ( G )) = π S ( G ) ( G n ) and G n ( S ( G )) ∩ G n ( S ( G )) = 0, we have S ( G ) = π S ( G ) ( G ) = ` n ∈ N G n ( S ( G )).By definition, for j ≤ n , G j ( S ( G )) ⊂ Ω n ( S ( G )). Then for j = 0 , . . . , n , π S ( G ) ( G j ) ⊂ Ω j ( S ( G )) ⊂ Ω n ( S ( G )). So we obtain π S ( G ) ( ` nj =0 G j ) ⊂ Ω n ( S ( G )). By Condition 2 in thedefinition of graded A ∞ ( V )-module, nonzero elements of G j for j > n are not in Ω n ( b G ).From the construction of b G , nonzero elements of the form (3.2), (3.3) or (3.4) are not in G ⊂ b G . In particular, the intersection of J ( G ) with G is 0. So π S ( G ) (cid:12)(cid:12) G : G → S ( G ) isinjective. Since π S ( G ) (cid:12)(cid:12) G is injective, we conclude that nonzero elements of π S ( G ) ( G j ) for j > n are not in Ω n ( S ( G )). So we haveΩ n ( S ( G )) = π S ( G ) n a j =0 G j ! = n a j =0 π S ( G ) ( G j ) = n a j =0 G j ( S ( G )) . Since Ω n ( S ( G )) = ` nj =0 G j ( S ( G )) for n ∈ N , we see that as a N -graded space, Gr ( S ( G ))is isomorphic to ` n ∈ N G n ( S ( G )) = π S ( G ) ( G ). We use f G to denote the isomorphism from Gr ( S ( G )) to π S ( G ) ( G ). Then we have f G ◦ ϑ Gr ( S ( G )) ([ v ] kl + Q ∞ ( V )) = Res x x l − k − Y S ( G ) ( x L S ( G ) (0) v, x ) ◦ f G for v ∈ V , k, l ∈ N , f G ◦ L Gr ( S ( G )) (0) = L S ( G ) (0) ◦ f G and f G ◦ L Gr ( S ( G )) ( −
1) = L S ( G ) ( − ◦ f G .We have proved that π S ( G ) (cid:12)(cid:12) G is injective and surjective and preserves the N -gradings. So itis an isomorphism of graded spaces from G to S ( G ). From the fact that π S ( G ) is a V -modulemap and on G ⊂ b G , L b G (0) = L G (0) and L b G ( −
1) = L G ( − π S ( G ) (cid:12)(cid:12) G ◦ ϑ G ([ v ] kl + Q ∞ ( V )) = Res x x l − k − Y S ( G ) ( x L S ( G ) (0) v, x ) ◦ π S ( G ) (cid:12)(cid:12) G for v ∈ V , k, l ∈ N , π S ( G ) (cid:12)(cid:12) G ◦ L G (0) = L S ( G ) (0) ◦ π S ( G ) (cid:12)(cid:12) G and π S ( G ) (cid:12)(cid:12) G ◦ L G ( −
1) = L S ( G ) ( − ◦ π S ( G ) (cid:12)(cid:12) G . Then by the properties of f G and π S ( G ) (cid:12)(cid:12) G above, we see that ( π S ( G ) (cid:12)(cid:12) G ) − ◦ f G is anequivalence of graded A ∞ ( V )-modules from Gr ( S ( G )) to G . Remark 3.6
Note that our construction of the lower-bounded generalized V -module b G seems to depend on the lower bound B of the real parts of the eigenvalues of L G (0). But byProposition 3.5, S ( G ) depends only on G , not on B .23 heorem 3.7 The set of the equivalence classes of irreducible lower-bounded generalized V -modules is in bijection with the set of the equivalence classes of irreducible graded A ∞ ( V ) -modules.Proof. Let [ W ] irr be the set of the equivalence classes of irreducible lower-bounded gener-alized V -modules and [ G ] irr the set of the equivalence classes of irreducible graded A ∞ ( V )-modules. Given an irreducible lower-bounded generalized V -module W , by Theorem 3.4, Gr ( W ) is an irreducible graded A N ( V )-module. Thus we obtain a map f : [ W ] irr → [ G ] irr given by f ([ W ]) = [ Gr ( W )], where [ W ] ∈ [ W ] irr is the equivalence class containing theirreducible lower-bounded generalized V -module W and [ Gr ( W )] ∈ [ G ] irr is the equiva-lence class containing the irreducible graded A N ( V )-module Gr ( W ). By Proposition 3.3,[ Gr ( W )] = [ W ] in [ G ] irr , where W is viewed as a graded A ∞ ( V )-module.Given an irreducible graded A ∞ ( V )-module G , we have a lower-bounded generalized V -module S ( G ). By Proposition 3.5, Gr ( S ( G )) is equivalent to G . Since G is irreduible, Gr ( S ( G )) is also irreducible. Then by Theorem 3.4, S ( G ) is an irreducible lower-boundedgeneralized V -module. Thus we obtain a map g : [ G ] irr → [ W ] irr given by g ([ G ]) = [ S ( G )].We still need to show that f and g are inverse to each other. By Proposition 3.5, Gr ( S ( G ))is equivalent to G for an irreducible graded A ∞ ( V )-module G . We obtain [ Gr ( S ( G ))] = [ G ].This means f ( g ([ G ])) = [ G ]. So we have f ◦ g = 1 [ G ] irr .Let W be an irreducible lower-bounded generalized V -module. By Theorem 3.4, Gr ( W )is an irreducible graded A ∞ ( V )-module. We then have a lower-bounded generalized V -module S ( Gr ( W )). By Proposition 3.5, Gr ( S ( Gr ( W ))) is equivalent to Gr ( W ) as a graded A ∞ ( V )-module. Since Gr ( W ) is irreducible, Gr ( S ( Gr ( W ))) is also irreducible. By Theo-rem 3.4, S ( Gr ( W )) is an irreducible lower-bounded generalized V -module. Since both W and S ( Gr ( W )) are irreducible, by Proposition 3.3, W and S ( Gr ( W )) are graded A ∞ ( V )-modules and are equivalent to Gr ( W ) and Gr ( S ( Gr ( W ))), respectively. But we alreadyknow that Gr ( S ( Gr ( W ))) is equivalent to Gr ( W ) as a graded A ∞ ( V )-module. So both W and S ( Gr ( W )) are equivalent to Gr ( W ) as graded A ∞ ( V )-modules. Since vertex opera-tors on W and S ( Gr ( W )) can be expressed using the actions of elements of A ∞ ( V ), we seethat W and S ( Gr ( W )) are also equivalent as lower-bounded generalized V -modules. Thus[ S ( Gr ( W ))] = [ W ], or g ( f ([ W ])) = [ W ]. So g ◦ f = 1 [ W ] irr . A ∞ ( V ) We give some very special subalgebras of A ∞ ( V ) and prove that they are isomorphic to theZhu algebra A ( V ) [Z] and its generalizations A N ( V ) for N ∈ N by Dong-Li-Mason [DLM]in Subsection 4.1. Then we introduce the main interesting and new subalgebras A N ( V ) for N ∈ N of A ∞ ( V ) in Subsection 4.2. Note that we use the superscript N instead of subscript N to distinguish this algebra from A N ( V ) in [DLM].24 .1 Zhu algebra and the generalizations by Dong-Li-Mason Let U ( V ) = { [ v ] | v ∈ V } ⊂ U ∞ ( V ) . Then U ( V ) can be canonically identified with V through the map i : U ( V ) → V givenby i ([ v ] ) = v for v ∈ V . Since by (2.3),[ u ] ⋄ [ v ] = Res x x − (cid:2) Y V ((1 + x ) L (0) u, x ) v (cid:3) ,U ( V ) is closed under the product ⋄ . Let A ( V ) = { [ v ] + Q ∞ ( V ) | v ∈ V } . Theorem 4.1
The subspace A ( V ) of A ∞ ( V ) is closed under ⋄ and is thus a subalgebra of A ∞ ( V ) with [ ] + Q ∞ ( V ) as its identity. The associative algebra A ( V ) is isomorphic tothe Zhu algebra A ( V ) in [Z] and, in particular, [ ω ] + Q ∞ ( V ) is in the center of A ( V ) if V is a vertex operator algebra with the conformal vector ω . Since this result is a special case of the result on the generalizations A N ( V ) in [DLM],we will not give a proof. The proof is the special case N = 0 of the proof of Theorem 4.2below for A N ( V ).Fix N ∈ N . Let U NN = { [ v ] NN | v ∈ V } ⊂ U ∞ ( V ) . By (2.3),[ u ] NN ⋄ [ v ] NN = Res x T N +1 (( x + 1) − N − )(1 + x ) N (cid:2) Y V ((1 + x ) L (0) u, x ) v (cid:3) NN for u, v ∈ V . So U NN ( V ) is closed under the product ⋄ . Let A NN ( V ) = { [ v ] NN + Q ∞ ( V ) | v ∈ V } ⊂ A ∞ ( V ) . Theorem 4.2
The subspace A NN ( V ) of A ∞ ( V ) is closed under ⋄ and is thus a subalgebra of A ∞ ( V ) with [ ] NN + Q ∞ ( V ) as the identity. The associative algebra A NN ( V ) is isomorphic tothe associative algebra A N ( V ) of Dong, Li and Mason in [DLM] and, in particular, [ ω ] NN + Q ∞ ( V ) is in the center of A NN ( V ) if V is a vertex operator algebra with the conformal vector ω .Proof. By (2.3), we have([ u ] NN + Q ∞ ( V )) ⋄ ([ v ] NN + Q ∞ ( V ))= Res x T N +1 (( x + 1) − N − )(1 + x ) N (cid:2) Y V ((1 + x ) L (0) u, x ) v (cid:3) NN + Q ∞ ( V ) ∈ A NN ( V )for u, v ∈ V . Thus A NN ( V ) is closed under ⋄ and is a subalgebra of A ∞ ( V ). Let f NN : U NN ( V ) → A N ( V ) be defined by f NN ([ v ] NN ) = v + O N ( V ) for v ∈ V .25e now view A N ( V ) as an A N ( V )-module. We construct a lower-bounded generalized V -module S ( A N ( V )) from A N ( V ) using the construction in Section 5 of [H3] as follows: Takethe generating fields for the grading-restricted vertex algebra V to be Y V ( v, x ) for v ∈ V .Take M in Section 5 of [H3] to be A N ( V ). We define the operator L M (0) on M to be themultiplication by the scalar N . So M itself is an eigenspace of L M (0) with eigenvalue N .Take the automrophism g of V in Section 5 of [H3] to be 1 V since we are interested only inuntwisted modules. Take B in in Section 5 of [H3] to be 0. Then we obtain a lower-boundedgeneralized V -module c M [1 V ]0 , which shall be denoted by \ A N ( V ) here. By Theorem 3.3 in[H4] and the construction in Section 5 of [H3], \ A N ( V ) is spanned by elements of the form( Y \ A N ( V ) ) n ( u ) L \ A N ( V ) ( − p ( v + O N ( V ))for homogeneous u, v ∈ V , p ∈ N and n ∈ wt u + N + p − − N . Let J be the generalized V -submodule of \ A N ( V ) generated by elements of the form( Y \ A N ( V ) ) wt u − ( u )( v + O N ( V )) − u ∗ N v + O N ( V )for u, v ∈ V . Let S ( A N ( V )) = \ A N ( V ) /J . Then S ( A N ( V )) = a n ∈ N ( S ( A N ( V ))) [ n ] is a lower-bounded generalized V -module such that ( S ( A N ( V ))) [ N ] = A N ( V ). From theconstruction in Section 5 of [H3] and the definition of S ( A N ( V )) above, elements of the form( Y S ( A N ( V )) ) wt u − N ( u )( v + O N ( V ))for homogeneous nonzero u ∈ V and nonzero v ∈ V are not 0. Thus for nonzero v ∈ V , v + O N ( V ) ∈ A N ( V ) is not in Ω N − ( S ( A N ( V ))). In other words, if v + O N ( V ) ∈ Ω N − ( S ( A N ( V ))), then v = 0. On the other hand, we know that A N ( V ) = ( S ( A N ( V ))) [ N ] ⊂ Ω N ( S ( A N ( V ))).Let W be a lower-bounded generalized V -module. Then ker ϑ Gr ( W ) is a two-sided idealof U ∞ ( V ). So ker ϑ Gr ( W ) ∩ U NN ( V ) is a two-sided ideal of U NN ( V ). From [DLM], the map o W : V → End Ω N ( W ) defined by o W ( v ) = ( Y W ) wt v − ( v ) = Res x x − Y W ( x L V (0) v, x ) givesΩ N ( W ) an A N ( V )-module structure. In particular, o W ( O N ( V )) = 0. So O N ( V ) ⊂ ker o W .We take W = S ( A N ( V )). By the definition of o S N ( A N ( V )) , we have o S N ( A N ( V )) ( u )( v + O N ( V )) = u ∗ N v + O N ( V )for u, v ∈ V . For u ∈ ker o S N ( A N ( V )) , we have o S N ( A N ( V )) ( u )( v + O N ( V )) = 0for v ∈ V . So we have u ∗ N v + O N ( V ) = 0 or u ∗ N v ∈ O N ( V ). In particular, for v = , wehave u ∗ N ∈ O ( V ). But modulo O N ( V ), u ∗ N is equal to u . So u ∈ O N ( V ). This meansker o S N ( A N ( V )) ⊂ O N ( V ) and thus ker o S N ( A N ( V )) = O N ( V ).26or v ∈ V , we have shown that v + O N ( V ) ∈ Ω N ( S ( A N ( V ))) and that v + O N ( V ) ∈ Ω N − ( S ( A N ( V ))) implies v = 0. So using our notation above, we see that[ v + O N ( V )] N = ( v + O N ( V )) + Ω N − ( S ( A N ( V )))is an element of Gr N ( S N ( A N ( V ))) and if it is equal to 0 ∈ Gr N ( S N ( A N ( V ))), then v = 0.By defintion, for u, v ∈ V , ϑ Gr N ( S N ( A N ( V ))) ([ u ] NN )[( v + O N ( V ))] N = [Res x x − Y S N ( A N ( V )) ( x L V (0) u, x ) v + O N ( V )] N = [ o S ( A N ( V )) ( u ) v + O N ( V )] N . Then ϑ Gr N ( S N ( A N ( V ))) ([ u ] NN )[( v + O N ( V ))] N = 0 if and only if o S N ( A N ( V )) ( u ) v = 0. If [ u ] NN ∈ Q ∞ ( V ), then ϑ S N ( A N ( V )) ([ u ] NN )[( v + O N ( V ))] N = 0 for all v ∈ V . So o S N ( A N ( V )) ( u ) v = 0 forall v ∈ V . Thus o S N ( A N ( V )) ( u ) = 0 and u ∈ ker o S N ( A N ( V )) = O N ( V ). Then f NN ([ u ] NN ) = u + O N ( V ) = 0 + O N ( V ) or in other words, [ u ] NN ∈ ker f NN . On the other hand, if[ u ] NN ∈ ker f NN , that is, f NN ([ u ] NN ) = 0 + O N ( V ), then u ∈ O N ( V ). From the definitionsof O N ( V ) and O ∞ ( V ), we have [ u ] NN ∈ O ∞ ( V ) ⊂ Q ∞ ( V ). This shows that ker f NN = Q ∞ ( V ) ∩ U NN ( V ). It is clear that f NN is surjective. In particular, f NN induces a linearisomorphism, still denoted by f NN , from A NN ( V ) to A ( V ).For u, v ∈ V , f NN ([ u ] NN ⋄ ([ v ] NN )= Res x T N +1 (( x + 1) − N − )(1 + x ) N f NN ([ Y V ((1 + x ) L (0) u, x ) v ] NN )= Res x T N +1 (( x + 1) − N − )(1 + x ) N Y V ((1 + x ) L (0) u, x ) v + O N ( V )= Res x N X m =0 (cid:18) − N − m (cid:19) x − N − m − (1 + x ) N Y V ((1 + x ) L (0) u, x ) v + O N ( V )= u ∗ N v + O N ( V )= ( u + O N ( V )) ∗ N ( v + O N ( V )) . Therefore f NN is an isomorphism of associative algebra. Since + O N ( V ) is the identity of A N ( V ), [ ] NN + O ∞ ( V ) is the identity of A NN ( V ). If V is a vertex operator algebra withthe conformal vector ω , since ω + O N ( V ) is in the center of A N ( V ), [ ω ] NN + O ∞ ( V ) is in thecenter of A NN ( V ). We now introduce new subalgebras of A ∞ ( V ). For N ∈ N , let U N ( V ) be the space of all( N + 1) × ( N + 1) matrices with entries in V . It is clear that U N ( V ) can be canonicallyembedded into U ∞ ( V ) as a subspace. We shall view U N ( V ) as a subspace of U ∞ ( V ) inthis paper. As a subspace of U ∞ ( V ), U N ( V ) consists of infinite matrices in U ∞ ( V ) whose( k, l )-th entries for k > N or l > N are all 0 and is spanned by elements of the form [ v ] kl for v ∈ V , k, l = 0 , . . . , N . 27et N = N X k =0 [ ] kk , that is, N is the element of U N ( V ) with the only nonzero entries to be equal to at thediagonal ( k, k )-th entries for k = 0 , . . . , N . By (2.2), we have N ⋄ [ v ] kl = Res x T k + l +1 (( x + 1) − l − )(1 + x ) l [ Y V ((1 + x ) L (0) , x ) v ] kl = [ v ] kl for v ∈ V and k, l = 0 , . . . , N . So N is a left identity of U N ( V ) with respect to the product ⋄ . Note that for v ∈ V and k, l = 0 , . . . , N ,[ v ] kl ⋄ N = Res x T k + l +1 (( x + 1) − k − )(1 + x ) l [ Y V ((1 + x ) L (0) v, x ) ] kl = [ v ] kl ⋄ ∞ . This formula together with (2.21) immediately gives[ v ] kl ⋄ N = k X m =0 (cid:18) − k − m (cid:19) (cid:20)(cid:18) L V ( −
1) + L V (0) + lk + m (cid:19) v (cid:21) kl (4.1)for v ∈ V and k, l = 0 , . . . , N .By (2.3), for u, v ∈ V and k, n, l = 0 , . . . , N ,[ u ] kn ⋄ [ v ] nl = Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l (cid:2) Y V ((1 + x ) L (0) u, x ) v (cid:3) kl ∈ U N ( V ) . (4.2)So U N ( V ) is closed under the product ⋄ . Let A N ( V ) = { v + Q ∞ ( V ) | v ∈ U N ( V ) } = π A ∞ ( V ) ( U N ( V )) , where π A ∞ ( V ) is the projection from U ∞ ( V ) to A ∞ ( V ). Then A N ( V ) is spanned by elementsof the form [ v ] kl + Q ∞ ( V ) for v ∈ V and k, l = 0 , . . . , N . Proposition 4.3
The subspace A N ( V ) is closed under ⋄ and is thus a subalgebra of A ∞ ( V ) with the identity N + Q ∞ ( V ) .Proof. By (4.2), we have([ u ] kn + Q ∞ ( V )) ⋄ ([ v ] nl + Q ∞ ( V ))= Res x T k + l +1 (( x + 1) − k + n − l − )(1 + x ) l (cid:2) Y V ((1 + x ) L (0) u, x ) v (cid:3) kl + Q ∞ ( V ) ∈ A N ( V )for u, v ∈ V and k, n, l = 0 , . . . , N . Thus A N ( V ) is closed under ⋄ and is thus a subalgebraof A ∞ ( V ).Since N is a left identity of U N ( V ) with respect to the product ⋄ , N + Q ∞ ( V ) is a leftidentity of A N ( V ). Since[ v ] kl ⋄ N = [ v ] kl ⋄ ∞ ≡ [ v ] kl mod Q ∞ ( V ) , N + Q ∞ ( V ) is also a right identity of A N ( V ). In particular, it is the identity of A N ( V ).28 emark 4.4 We have derived A N ( V ) as a subalgebra of A ∞ ( V ). One can certainly obtain A N ( V ) directly starting with the space U N ( V ) of ( N + 1) × ( N + 1) matrices with entriesin V . Remark 4.5
It is clear from the definition that A nn ( V ) for n = 0 , . . . , N are subalgebrasof A N ( V ). In particular, the Zhu algebra A ( V ) in [Z] and its generalizations A n ( V ) for n = 0 , . . . , N by Dong, Li and Mason in [DLM] can be viewed as subalgebras of A N ( V ). Inthe case N = 0, A is equal to A ( V ) and is thus isomorphic to the Zhu algebra A ( V ) byTheorem 4.1.We say that V is of positive energy if V = ` n ∈ N V ( n ) and V (0) = C . (In some papers, V is of positive energy is said to be of CFT type.) We recall that for n ∈ N , V is C n -cofiniteif dim V /C n ( V ) < ∞ , where C n ( V ) is the subspace of V spanned by elements of the form( Y V ) − n ( u ) v for u, v ∈ V . Theorem 4.6
Assume that V is of positive energy and C -cofinite. Then A N ( V ) is finitedimensional.Proof. By Theorem 11 in [GN] (see Proposition 5.5 in [AN]), V is also C n -cofinite for n ≥ V is C k + l +2 -cofinite for k, l = 0 , . . . , N . By definition, C k + l +2 ( V ) are spannedby elements of the form ( Y V ) − k − l − ( u ) v for u, v ∈ V . Since V is C k + l +2 -cofinite, there existsa finite dimensional subspace X k + l of V such that X + C k + l +2 ( V ) = V . Let U N ( X ) be thesubspace of U N ( V ) consisting matrices in U N ( V ) whose entries are in X . Since X is finitedimensional, U N ( X ) is also finite dimensional. We now prove U N ( X ) + ( O ∞ ( V ) ∩ U N ( V )) = U N ( V ). To prove this, we need only prove that every element of U N ( V ) of the form [ v ] kl for v ∈ V and 0 ≤ k, l ≤ N , can be written as [ v ] kl = [ v ] kl + [ v ] kl , where v ∈ X k + l and v ∈ V such that [ v ] kl ∈ O ∞ ( V ). We shall denote the subspace of V consisting of elements v suchthat [ v ] kl ∈ O ∞ ( V ) by O ∞ kl ( V ). Then what we need to prove is V = X k + l + O ∞ kl ( V ).We can always take X k + l to be a subspace of V containing . We use induction on theweight of v . When wt v = 0, v is proportional to and can indeed be writte as v = v + 0,where v ∈ X and 0 ∈ O ∞ kl ( V ).Assume that when wt v = p < q , v = v + v , where v ∈ X k + l and v ∈ O ∞ kl ( V ). Thensince V is C k + l +2 -cofinite, for v ∈ V ( q ) , there exists homogeneous u ∈ X k + l and homogeneous u i , v i ∈ V for i = 1 , . . . , m such that v = u + P mi =1 u i − k − l − v i . Moreover, we can always findsuch u and u i , v i ∈ V for i = 1 , . . . , m such that wt u = wt u i − k − l − v i = wt v = q . Sincewt u in − k − l − v i < wt u i − k − l − v i = wt v = q for i = 1 , . . . , m and n ∈ Z + , by induction assumption, u in − k − l − v i ∈ X k + l + O ∞ kl ( V ) for29 = 1 , . . . , m and k ∈ Z + . Thus v = u + m X i =1 u i − k − l − v i = u + m X i =1 Res x x − k − l − (1 + x ) l Y ((1 + x ) L (0) u i , x ) v i − m X i =1 X n ∈ Z + (cid:18) wt u i + ln (cid:19) u in − k − l − v i . By definition, [Res x x − k − l − (1 + x ) l Y ((1 + x ) L (0) u i , x ) v i ] kl ∈ O ∞ ( V ) . Thus Res x x − k − l − (1 + x ) l Y ((1 + x ) L (0) u i , x ) v i ∈ O ∞ kl ( V ) . Thus we have v = v + v , where v ∈ X k + l and v ∈ O ∞ kl ( V ). By induction principle, wehave V = X k + l + O ∞ kl ( V ).We now have proved U N ( X ) + ( O ∞ ( V ) ∩ U N ( V )) = U N ( V ). Since O ∞ ( V ) ∩ U N ( V ) ⊂ Q ∞ ( V ) ∩ U N ( V ), we also have U N ( X ) + ( Q ∞ ( V ) ∩ U N ( V )) = U N ( V ). Since U N ( X ) is finitedimensional, A N ( V ) is finite dimensional. V -modules and graded A N ( V ) -modules By Theorem 2.8, the associated graded space Gr ( W ) of a filtration of a lower-boundedgeneralized V -module W is a graded A ∞ ( V )-module. In this section, for N ∈ N , we give an A N ( V )-module structure to a subspace of Gr ( W ) and use it to study W .Let N ∈ N . Let W be a lower-bounded generalized V -module. Since A N ( V ) is a subal-gebra of A ∞ ( V ), Gr ( W ) as an A ∞ ( V )-module is also an A N ( V )-module. Let Gr N ( W ) = N a n =0 Gr n ( W ) ⊂ Gr ( W ) . By the definition of ϑ Gr ( W ) , we see that for v ∈ A N ( V ) and [ w ] n ∈ Gr N ( W ), ϑ Gr ( W ) ( v )[ w ] n ∈ Gr N ( W ). Thus Gr N ( W ) is an A N ( W )-submodule of Gr ( W ). But Gr N ( W ) has some addi-tional structures and properties and we are only interested in those A N ( W )-modules havingthese additional structures and properties. Similar to Definition 3.1, we have the followingnotion: Definition 5.1
Let M be an A N ( V )-module M with the A N ( V )-module structure on M given by ϑ M : A N ( V ) → End M . We say that M is a graded A N ( V ) -module if the followingconditions are satisfied: 30. M = ` Nn =0 G n ( M ) such that for v ∈ V and k, l = 0 , . . . , N , ϑ M ([ v ] kl + Q ∞ ( V )) maps G n ( M ) for 0 ≤ n ≤ N to 0 when n = l and to G k ( M ) when n = l .2. For w ∈ G l ( M ), if ϑ M ([ v ] l + Q ∞ ( V )) w = 0 for all v ∈ V , then w = 0.3. M is a direct sum of generalized eigenspaces of of an operator L M (0) on M and thereal parts of the eigenvalues of L M (0) has a lower bound.4. There is a linear map L M ( −
1) : ` N − n =0 G n − ( M ) → ` Nn =1 G n ( M ) mapping G n ( M ) to G n +1 ( M ) for n = 0 , . . . , N − L M (0) , L M ( − L M ( − , [ L M (0) , ϑ M ([ v ] kl + Q ∞ ( V ))] = ( k − l ) ϑ M ([ v ] kl + Q ∞ ( V )) , [ L M ( − , ϑ M ([ v ] pl + Q ∞ ( V ))] = ϑ M ([ L V ( − v ] ( p +1) l + Q ∞ ( V ))hold for v ∈ V , k, l = 0 , . . . , N and p = 0 , . . . , N − M and M be graded A N ( V )-modules. An graded A N ( V ) -module map from M to M is an A N ( V )-module map f : M → M such that f ( G n ( M )) ⊂ G n ( M ) for n =0 . . . , N , f ◦ L M (0) = L M (0) ◦ f and f ◦ L M ( −
1) = L M ( − ◦ f . A graded A N ( V ) -submodule of a graded A N ( V )-module M is an A N ( V )-submodule M of M such that withthe A N ( V )-module structure, the N -grading induced from M and the operators L M (0) (cid:12)(cid:12) M and L M ( − (cid:12)(cid:12) M , M is a graded A N ( V )-module. A graded A ∞ ( V )-module M is said to be generated by a subset S if M is equal to the smallest graded A N ( V )-submodule containing S , or equivalently, M is spanned by homogeneous elements obtained by applying elements of A N ( V ), L M (0) and L M ( −
1) to homogeneous summands of elements of S . A graded A N ( V )-module is said to be irreducible if it has no nonzero proper graded A N ( V )-modules. A graded A N ( V )-module is said to be completely reducible if it is a direct sum of irreducible graded A N ( V )-modules.From the discussion above and the property of Gr N ( W ), we obtain immediately: Proposition 5.2
For a lower-bounded generalized V -module W , Gr N ( W ) is a graded A N ( V ) -module. Let W and W be lower-bounded generalized V -modules and f : W → W a V -module map. Then f induces a graded A N ( V ) -module map Gr N ( f ) : Gr N ( W ) → Gr N ( W ) . We have the following results on irreducible and completely reducible lower-boundedgeneralized V -modules without additional conditions: Proposition 5.3
Let W be a lower-bounded generalized V -module. If W is irreducible orcompletely reducible, then Gr N ( W ) is equivalent to T N ( W ) as an A N ( V ) -module and is alsoirreducible or completely reducible, respectively. roof. Let W be irreducible. By Proposition 3.3, Ω n ( W ) = T n ( W ) for n = 0 , . . . , N . Then T N ( W ) is a graded A N ( V )-module equivalent to Gr N ( W ). We need to prove that the graded A N ( V )-module T N ( W ) is irreducible.Let M be a nonzero graded A N ( V )-submodule of T N ( W ). We use the construction inSection 5 of [H3] to construct a universal lower-bounded generalized V -module c M from M .We take the generating fields for the grading-restricted vertex algebra V to be Y V ( v, x ) for v ∈ V . By definition, M is a direct sum of generalized eigenspaces of L M (0) and the realparts of the eigenvalues of L M (0) have a lower bound B ∈ R . We take M and B in Section5 of [H3] to be the given graded A N ( V )-module M and the lower bound B above. Using theconstruction in Section 5 of [H3], we obtain a universal lower-bounded generalized V -module c M [1 V ] B . For simplicity, we shall denote it simply by c M . By the universal property of c M (Theorem 5.2 in [H3]), for the embedding map e M : M → T N ( W ), there is a unique V -module map c e M : c M → W such that c e M (cid:12)(cid:12) M = e . Then c e M ( c M ) is a generalized V -submoduleof W generated by M . It is nonzero since M ⊂ c e M ( c M ). Since W is irreducible, it mustbe W . Then W is generated by M . In particular, T N ( W ) is obtained by applying thecomponents of the vertex operators on W , L W (0) and L W ( −
1) to elements of M . Since thecomponents of the vertex operators on W and the operators L W (0) and L W ( −
1) preserving T N ( W ) are by definition the actions of elements of A N ( V ), L W (0) and L M ( −
1) preserving T N ( W ), we see that as a graded A N ( V )-module, T N ( W ) is generated by M . But M itselfis an A N ( V )-submodule of T N ( W ). So we have M = T N ( W ). Thus T N ( W ) as a graded A N -module is irreducible.If W is completely reducible, by Proposition 3.3 again, Ω n ( W ) = T n ( W ) for n = 0 , . . . , N .Then T N ( W ) is a graded A N ( V )-module equivalent to Gr N ( W ). Since W is completelyreducible, W = ` µ ∈M W µ , where W µ for µ ∈ M are irreducible lower-bounded generalized V -modules. By the definition of T N ( W ), we have T N ( W ) = ` µ ∈M T N ( W µ ). From what wehave proved above, for µ ∈ M , T N ( W µ ) is an irreducible graded A N ( V )-module. Thus wesee that T N ( W ) is completely reducible.Let M be a graded A N ( V )-module given by the map ϑ M : A N ( V ) → End M andoperators L M (0) and L M ( − V -module S N ( M ) from M . We use the construction in Section 5 of [H3]. We take the generating fieldsfor the grading-restricted vertex algebra V to be Y V ( v, x ) for v ∈ V . By definition, M isa direct sum of generalized eigenspaces of L M (0) and the real parts of the eigenvalues of L M (0) has a lower bound B ∈ R . We take M and B in Section 5 of [H3] to be the givengraded A N ( V )-module M and the lower bound B above. Using the construction in Section5 of [H3], we obtain a universal lower-bounded generalized V -module c M [1 V ] B . For simplicity,we shall denote it simply by c M .By Theorem 3.4 in [H4] and the construction in Section 5 of [H3] and by identifyingelements of the form ( ψ a c M ) − , with basis elements w a ∈ M for a ∈ A for a basis { w a } a ∈ A of M , we see that c M is generated by M (in the sense of Definition 3.1 in [H4]). Moreover,Theorems 3.3 and 3.4 in [H4] state that elements of the form L c M ( − p w a for p ∈ N and32 ∈ A are linearly independent and c M is spanned by elements obtained by applying thecomponents of the vertex operators to these elements. In particular, we identify M as asubspace of c M . Let J M be the generalized V -submodule of c M generated by elements of theforms Res x x l − k − Y c M ( x L V (0) v, x ) w (5.1)for l = 0 , . . . , N , k ∈ − Z + and w ∈ G l ( M ),Res x x l − k − Y c M ( x L V (0) v, x ) w − ϑ M ([ v ] kl ) w (5.2)for v ∈ V , k, l = 0 , . . . , N and w ∈ G l ( M ) and L c M ( − w − L M ( − w (5.3)for w ∈ ` N − n =0 G n ( M ).Let S N ( M ) = c M /J M . Then S N ( M ) is a lower-bounded generalized V -module. Let π S N ( M ) be the projection from c M to S N ( M ). Since c M is generated by M (in the sense ofDefinition 3.1 in [H4]), S N ( M ) is generated π S N ( M ) ( M ) (in the same sense). In particular, S N ( M ) is spanned by elements of the formRes x x ( l + p ) − n − Y S N ( M ) ( x L V (0) v, x ) L S N ( M ) ( − p π S N ( M ) ( w ) (5.4)for v ∈ V , l = 0 , . . . , N , n, p ∈ N and w ∈ G l ( M ). For n ∈ N , let G n ( S N ( M )) be thesubspace of S N ( M ) spanned by elements of the form (5.4) for v ∈ V , l = 0 , . . . , N , p ∈ N and w ∈ G l ( M ). Proposition 5.4
Let M be a graded A N ( V ) -module.1. For ≤ n ≤ N , G n ( S N ( M )) = π S N ( M ) ( G n ( M )) and for ≤ n , n ≤ N , n = n , G n ( S N ( M )) ∩ G n ( S N ( M )) = 0 . Moreover, S N ( M ) = ` n ∈ N G n ( S N ( M )) and π S N ( M ) ( M ) = ` Nn =0 G n ( S N ( M )) .2. For n = 0 , . . . , N , π S N ( M ) n a j =0 G j ( M ) ! = n a j =0 G j ( S N ( M )) ⊂ Ω n ( S N ( M )) (5.5) and π S N ( M ) N a j = n G j ( M ) ! ∩ Ω n ( S N ( M )) = N a j = n G j ( S N ( M )) ! ∩ Ω n ( S N ( M )) = 0 . (5.6) M is equivalent to a graded A N ( V ) -submodule of Gr N ( S N ( M )) . roof. By definition, G n ( S N ( M )) for 0 ≤ n ≤ N is spanned by elements of the form (5.4)for v ∈ V , l = 0 , . . . , N , p ∈ N and w ∈ G l ( M ). Using the L ( − Y S N ( M ) , we see that it is also spanned by elements of the formRes x x l − k − L S N ( M ) ( − p Y S N ( M ) ( x L V (0) v, x ) π S N ( M ) ( w ) (5.7)for v ∈ V , l, k = 0 , . . . , N , p = 0 , . . . , n − k and w ∈ G l ( M ). Since elements of the forms(5.2) and (5.3) are in J M , we see that (5.7) is in fact equal to π S N ( M ) ( L M ( − p ϑ M ([ v ] kl + Q ∞ ( V )) w ) ∈ π S N ( M ) ( G n ( M )) . (5.8)Since L M ( − p ϑ M ([ v ] k ( l + p ) + Q ∞ ( V )) w for v ∈ V , l, k = 0 , . . . , N , p = 0 , . . . , n − k and w ∈ G l ( M ) certainly span G n ( M ) (in fact, we need only v = , k = l = n , p = 0 and w ∈ G n ( W )) and elements of the form (5.7) for v ∈ V , l, k = 0 , . . . , N , p = 0 , . . . , n − k and w ∈ G l ( M ) span G n ( S ( G )) for 0 ≤ n ≤ N , we see that elements of the form (5.8)for v ∈ V , l, k = 0 , . . . , N , p = 0 , . . . , n − k and w ∈ G l ( M ) also span G n ( S N ( M )). Thuswe obtain G n ( S N ( M )) = π S N ( M ) ( G n ( M )) for n = 0 , . . . , N . When n = n , we know G n ( M ) ∩ G n ( M ) = 0. Then G n ( S N ( M )) ∩ G n ( S N ( M )) = π S ( G ) ( G n ( M ) ∩ G n ( M )) = 0.Since S N ( M ) is spanned by elements of the form (5.4) for v ∈ V , l = 0 , . . . , N , n, p ∈ N and w ∈ G l ( M ), by the definition of G n ( S N ( M )), we have S N ( M ) = ` n ∈ N G n ( S N ( M )). Since G n ( S N ( M )) = π S N ( M ) ( G n ( M )) for n = 0 , . . . , N , we have π S N ( M ) ( M ) = N a n =0 π S N ( M ) ( G n ( M )) = N a n =0 G n ( S N ( M )) . By definition, for 0 ≤ j ≤ n ≤ N , G j ( S N ( M )) ⊂ Ω n ( S N ( M )). Then for j = 0 , . . . , n , π S N ( M ) ( G j ( M )) = G j ( S N ( M )) ⊂ Ω j ( S N ( M )) ⊂ Ω n ( S N ( M )) . So we obtain (5.5). By Condition 2 in Definition 5.1 of graded A N ( V )-module, nonzeroelements of G j ( M ) for N ≥ j > n are not in Ω n ( c M ). From the construction of c M , nonzeroelements of the form (5.1), (5.2) or (5.3) are not in M ⊂ c M . In particular, we see that theintersection of J ( M ) with M is 0. So π S N ( M ) (cid:12)(cid:12) M is injective. Since π S N ( M ) (cid:12)(cid:12) M is injective,we see that nonzero elements of G j ( S N ( M )) = π S N ( M ) ( G j ( M )) for N ≥ j > n are not inΩ n ( S N ( M )). Thus we obtain (5.6).For 0 ≤ n ≤ N and w ∈ G n ( M ), we define f M ( w ) = π S N ( M ) ( w ) + Ω n − ( S N ( M )).Since π S N ( M ) ( w ) ∈ Ω n ( S N ( M )), f M ( w ) ∈ Gr n ( S N ( M )). Therefore we obtain linear map f M : M → Gr N ( S N ( M )). It is clear from the definition that f M is in fact a graded A N ( V )-module map. If for some 0 ≤ n ≤ N and w ∈ G n ( M ), f M ( w ) = 0, then π S N ( M ) ( w ) ∈ Ω n − ( S N ( M )). But we have proved above that nonzero elements of π S N ( M ) ( G n ( M )) are notin Ω n − ( S N ( M )). So π S N ( M ) ( w ) = 0. Since π S N ( M ) (cid:12)(cid:12) M is injective, we obtain w = 0. So f M isinjective. Thus M is equivalent to the graded A N ( V )-submodule f M ( M ) of Gr N ( S N ( M )).34 emark 5.5 As in the case of S ( G ) in Section 3, our construction of the lower-boundedgeneralized V -module c M depends on the lower bound B of the real parts of the eigenvaluesof L M (0). But by Proposition 5.4, S N ( M ) depends only on M , not on B . Theorem 5.6
For N ∈ N , the set of the equivalence classes of irreducible lower-boundedgeneralized V -modules is in bijection with the set of the equivalence classes of irreduciblegraded A N ( V ) -modules.Proof. Recall the set [ W ] irr of the equivalence classes of irreducible lower-bounded general-ized V -modules in the proof of Theorem 3.7. Let [ M N ] irr be the set of the equivalence classesof irreducible graded A N ( V )-modules. Given an irreducible lower-bounded generalized V -module W , by Theorem 5.3, Gr N ( W ) = T N ( W ) is an irreducible graded A N ( V )-module.Thus we obtain a map f : [ W ] irr → [ M N ] irr given by f ([ W ]) = [ T N ( W )], where [ W ] ∈ [ W ] irr is the equivalence class containing the irreducible lower-bounded generalized V -module W and [ T N ( W )] ∈ [ M N ] irr is the equivalence class containing the irreducible graded A N ( V )-module T N ( W ).Given an irreducible graded A N ( V )-module M , we have the lower-bounded generalized V -module S N ( M ) generated by π S N ( M ) ( M ). The main difference of the proof here andthe the proof of Theorem 3.7 is that we do not know whether S N ( M ) is irreducible. Sowe need to take a quotient of S N ( M ). Since M is an irreducible graded A N ( V )-module,it is generated by any nonzero element. Since S N ( M ) is generated by π S N ( M ) ( M ), it isalso generated by any element w ∈ π S N ( M ) ( M ). Then by Theorem 4.7 in [H4], there is amaximal generalized V -submodule J π SN ( M ) ( M ) ,w of S N ( M ) such that J π SN ( M ) ( M ) ,w does notcontain w and S N ( M ) /J π SN ( M ) ( M ) ,w is irreducible. The maximal generalized V -submodule J π SN ( M ) ( M ) ,w is in fact independent of w ∈ π S N ( M ) ( M ). We prove this fact by proving thatno nonzero element of π S N ( M ) ( M ) is in J π SN ( M ) ( M ) ,w . In fact, if a nonzero w ∈ π S N ( M ) ( M )is also in J π SN ( M ) ( M ) ,w , since the actions of components of vertex operators on w are equalto the actions of elements of A N ( V ) and M is generated also by w , we see that w must alsobe in J π SN ( M ) ( M ) ,w . Contradiction. Thus J π SN ( M ) ( M ) ,w is in fact the maximal generalized V -submodule of S N ( M ) such that it does not contain nonzero elements of M . We denoteit by e J M , which depends only on π S N ( M ) ( M ), or equivalently, M . Thus we obtain a map g : [ M N ] irr → [ W ] irr given by g ([ M ]) = [ S N ( M ) / e J M ].We still need to show that the two maps above are inverses of each other. Let M be an irreducible graded A N ( V )-module. Since S N ( M ) / e J M is irreducible, by Proposition3.3, Gr N ( S N ( M ) / e J M ) is an irreducible graded A N ( V )-module. By Proposition 5.4, M isequivalent to a graded A N ( V )-submodule of Gr N ( S N ( M )). As in the proof of Proposition5.4, we denote this equivalence by f M . Let π e J M : S N ( M ) → S N ( M ) / e J M be the projectionmap. Since e J M ∩ π S N ( M ) ( M ) = 0, π e J M (cid:12)(cid:12) π SN ( M ) ( M ) is injective and in particular, is not 0.The V -module map π e J M induces a graded A N ( V )-module map Gr N ( π e J M ) : Gr N ( S N ( M )) → Gr N ( S N ( M ) / e J M ). Since π e J M (cid:12)(cid:12) π SN ( M ) ( M ) is not 0, the restriction Gr N ( π e J M ) (cid:12)(cid:12) f M ( M ) of Gr N ( π e J M )to the image of M under f M is also not 0. Consider the A N ( V )-module map Gr N ( π e J M ) ◦ f M :35 → Gr N ( S N ( M ) / e J M ). Since f M is injective and Gr N ( π e J M ) (cid:12)(cid:12) f M ( M ) = 0, Gr N ( π e J M ) ◦ f M is not 0. But both M and Gr N ( S N ( M ) / e J M ) are irreducible. So Gr N ( π e J M ) ◦ f M must bean equivalence of graded A N ( V )-modules. Moreover, by Proposition 5.3, Gr N ( S N ( M ) / e J M )is equivalent to T N ( S N ( M ) / e J M ). So M is equivalent to T N ( S N ( M ) / e J M ). Thus [ M ] =[ T N ( c M / e J M )]. This means f ( g ([ M ])) = [ M ]. So we obtain f ◦ g = 1 [ M N ] irr .Let W be an irreducible lower-bounded generalized V -module. By Theorem 5.3, T N ( W )is an irreducible A N ( V )-module. We then have a lower-bounded generalized V -module S N ( T N ( W )). By the universal property of \ T N ( W ), there is a unique V -module map \ T N ( W ) : \ T N ( W ) → W such that \ T N ( W ) ) (cid:12)(cid:12) T N ( W ) = 1 T N ( W ) , where 1 T N ( W ) is the identity operatoron T N ( W ). Since W is irreducible, the image of \ T N ( W ) under \ T N ( W ) is either 0 or W .Since \ T N ( W ) ) (cid:12)(cid:12) T N ( W ) = 1 T N ( W ) , the image of \ T N ( W ) under \ T N ( W ) cannot be 0 and thusmust be W . In particular, \ T N ( W ) is surjective. Moreover, since J T N ( W ) is generated by(5.1), (5.2) and (5.3) with M = T N ( W ), the image of J T N ( W ) under \ T N ( W ) is 0, that is, J T N ( W ) ∈ ker \ T N ( W ) . In particular, \ T N ( W ) induces a surjective V -module map f T N ( W ) : S N ( T N ( W )) = \ T N ( W ) /J T N ( W ) → W . Since J T N ( W ) ∩ T N ( W ) = 0, f T N ( W ) ( T N ( W )) = T N ( W ).We have a maximal generalized V -submodule e J T N ( W ) of S N ( T N ( W )) as in the construc-tion above such that T N ( W ) ∩ e J T N ( W ) = 0 and S N ( T N ( W )) / e J T N ( W ) is irreducible. Since f T N ( W ) ( T N ( W )) = T N ( W ), ker f T N ( W ) is a generalized V -submodule of S N ( T N ( W )) thatdoes not contain nonzero elements of M . Hence ker f T N ( W ) ⊂ e J T N ( W ) . Thus we obtain asurjective V -module map from S N ( T N ( W )) / e J T N ( W ) to W . Since both S N ( T N ( W )) / e J T N ( W ) and W are irreducible, this surjective V -module map must be an equivalence. So we obtain[ S N ( T N ( W )) / e J T N ( W ) ] = [ W ], that is, g ( f ([ W ])) = [ W ]. So we obtain g ◦ f = 1 [ W ] irr . Thisfinishes the proof that [ W ] irr is in bijection with [ M N ] irr . Corollary 5.7
For N , N ∈ N or equal to ∞ , the set of the equivalence classes of irreduciblegraded A N ( V ) -modules is in bijection with the set of the equivalence classes of irreduciblegraded A N ( V ) -modules. We now assume that V is a M¨obius vertex algebra, that is, a grading-restricted vertexalgebra equipped with an operator L V (1) such that L V (1), L V (0) and L V ( −
1) satisfyingthe usually commutator relations for the standard basis of sl and the usual commutatorformula between L V (1) and vertex operators for a vertex operator algebra. See, for example,Definition 7.1 in [H4] for the precise definition. In this case, a lower-bounded generalized V -module should also have an operator L W (1) satisfying the same relations as L V (1). Weassume that V is a grading-restricted M¨obius vertex algebra in the remaining part of thepaper because in this case, a lowest weight of a lower-bounded generalized V -module is welldefined. See Remark 7.3 in [H4]. 36 roposition 5.8 Let V be a M¨obius vertex algebra. Assume that A N ( V ) for all N ∈ N arefinite dimensional (for example, when V is C -cofinite and of positive energy by Theorem4.6). Then every irreducible lower-bounded generalized V -module is an ordinary V -moduleand every lower-bounded generalized V -module of finite length is grading restricted.Proof. Since for N ∈ N , A N ( V ) is finite dimensional, there are only finitely many irreducible A N ( V )-modules. By Theorem 5.6, there are also finitely many irreducible lower-boundedgeneralized V -modules. For an irreducible lower-bounded generalized V -module W withlowest weight h W and N ∈ N , T N ( W ) is an irreducible graded A N ( V )-module by Proposition5.3. Since A N ( V ) is finite diemensional, T N ( W ) is also finite dimensional. Thus G N ( W ) = W [ h W + N ] ⊂ T N ( W ) is also finite dimensional. Since this is true for N ∈ N , we see that W is grading restricted. Since W is irreducible, L W (0) must act semisimply on W . So W is anirreducible ordinary V -module.Since as a graded vector space, a lower-bounded generalized V module W of finite lengthis a finite sum of irreducible lower-bounded generalized V -modules, which are all ordinary V -modules from what we have proved above. Then W must be grading restricted.Since V is a M¨obius verex algebra, the associative algebras A ∞ ( V ) and A N ( V ) for N ∈ N have an additional operator L V (1) induced from the operator L V (1) acting on V . For alower-bounded generalized V -module W , there is also an operator L Gr ( W ) (1) on the A ∞ ( V )-module Gr ( W ) induced from L W (1) on W such that L Gr ( W ) (1) maps Gr n ( W ) to Gr n − ( W ).Restricting L Gr ( W ) (1) to Gr N ( W ), we obtain an operator L Gr N ( W ) (1) on Gr N ( W ). Definition 5.9
Let V be a M¨obius vertex algebra. A graded A N ( V ) -module is a graded A N ( V )-module M when V is viewed as a grading-restricted vertex algebra together with anoperator L M (1) satisfying the following conditions:1. L M (1) maps G n ( M ) to G n − ( M ) for n = 0 , . . . , N , where G − ( M ) = 0.2. The operators L M (1) satisfies the commutator relations[ L M (0) , L M (1)] = − L M (1) , [ L M (1) , L M ( − L M (0) , [ L M (1) , ϑ M ([ v ] kl + Q ∞ ( V ))] = ϑ M ([( L V (1) + 2 L V (0) + L V ( − v ] ( k − l + Q ∞ ( V )) . Let M and M be graded A N ( V )-modules. An graded A N ( V ) -module map from M to M is an A N ( V )-module map f : M → M such that f ( G n ( M ) ⊂ G n ( M ) for n = 0 . . . , N , f ◦ L M (1) = L M (1) ◦ f , f ◦ L M (0) = L M (0) ◦ f and f ◦ L M ( −
1) = L M ( − ◦ f . A graded A N ( V ) -submodule of a graded A N ( V )-module M is an A N ( V )-submodule M of M such that with the A N ( V )-module structure and the N -grading induced from M and theoperators L M (1) (cid:12)(cid:12) M , L M (0) (cid:12)(cid:12) M and L M ( − (cid:12)(cid:12) M , M is a graded A N ( V )-module. A graded A ∞ ( V )-module M is said to be generated by a subset S if M is equal to the smallest graded A N ( V )-submodule containing S , or equivalently, M is spanned by homogeneous elementsobtained by applying elements of A N ( V ), L M (1) and L M ( −
1) to homogeneous summands37f elements of S . Irreducible and completely reducible graded A N ( V )-module are defined inthe same way as in the case that V is a grading-restricted vertex algebra.From Proposition 5.2 and the property of L W (1), we immediately obtain the following: Proposition 5.10
Let V be a M¨obius vertex algebra. For a lower-bounded generalized V -module W , Gr N ( W ) is a graded A N ( V ) -module. Let W and W be lower-bounded generalized V -modules and f : W → W a V -module map. Then f induces a graded A N ( V ) -modulemap Gr N ( f ) : Gr N ( W ) → Gr N ( W ) . As is mentioned above, in the remaining part of this paper, we assume that V is a M¨obiusvertex algebra. We shall not repeat this assumption except in the statements of propositions,theorems, corollaries and so on. Lower-bounded generalized V -modules and graded A N ( V )-modules always mean those for V as a M¨obius vertex algebra, not as a grading-restrictedvertex algebra. All the results that we have obtained above certainly still hold.We recall the notion of lower-bounded generalized V -module of finite length. A lower-bounded generalized V -module W is said to be of fnite length if there is a composition series W = W ⊃ · · · ⊃ W l +1 = 0 of lower-bounded generalized V -modules such that W i /W i +1 for i = 0 , . . . , l are irreducible lower-bounded generalized V -modules. Proposition 5.11
Let V be a M¨obius vertex algebra. Assume that the differences betweenthe real parts of the lowest weights of the irreducible lower-bounded generalized V -modulesare all less than or equal to N ∈ N . Then a lower-bounded generalized V -module W of finitelength is generated by a ℜ ( h W ) ≤ℜ ( n ) ≤ℜ ( h W )+ N W [ n ] ⊂ Ω N ( W ) , where h W is a lowest weight of W .Proof. Let W = W ⊃ W ⊃ · · · ⊃ W l +1 = 0 be a finite composition series such that W i /W i +1 for i = 0 , . . . , l are irreducible lower-bounded generalized V -modules. As a gradedvector space, W is isomorphic to ` li =0 W i /W i +1 . In particular, the lowest weight of one ofthe irreducible lower-bounded generalized V -modules W i /W i +1 for i = 0 , . . . , l is a lowestweight h W of W .Let w i ∈ W i be homogeneous for i = 0 , . . . , l such that w i + W i +1 is a lowest weightvector of W i /W i +1 . Then by assumption, the differences between the real parts of the lowestweights of W i /W i +1 for i = 0 , . . . , l are less than or equal to N . Since one of these lowestweights is a lowest weight h W of W , we see that the the differences between the real partsof the lowest weights of W i /W i +1 for i = 0 , . . . , l and ℜ ( h W ) are less than or equal to N .In particular w i ∈ ` ℜ ( h W ) ≤ℜ ( n ) ≤ℜ ( h W )+ N W [ n ] . Since for each i , W i /W i +1 is generated by w i + W i +1 , W i is generated by w i and W i +1 . Thus W is generated by w i for i = 0 , . . . , l .Since w i ∈ ` ℜ ( h W ) ≤ℜ ( n ) ≤ℜ ( h W )+ N W [ n ] , W is generated by ` ℜ ( h W ) ≤ℜ ( n ) ≤ℜ ( h W )+ N W [ n ] . It isclear that ` ℜ ( h W ) ≤ℜ ( n ) ≤ℜ ( h W )+ N W [ n ] is a subspace of Ω N ( W ).38 orollary 5.12 Let V be a M¨obius vertex algebra. Assume that A N ′ ( V ) for all N ′ ∈ N arefinite dimensional (for example, when V is C -cofinite and of positive energy by Theorem4.6). Let N ∈ N such that the differences between the real parts of the lowest weights of thefinitely many (inequivalent) irreducible ordinary V -modules are less than or equal to N . Thena lower-bounded generalized V -module W of finite length or a grading-restricted generalized V -module W is generated by a ℜ ( h W ) ≤ℜ ( n ) ≤ℜ ( h W )+ N W [ n ] ⊂ Ω N ( W ) . Proof.
Since by Proposition 5.8, the finitely many (inequivalent) irreducible lower-boundedgeneralized V -modules are all ordinary V -modules, the condition in Proposition 5.11 is sat-isfied. Also, by Corollary 3.16 in [H1], every grading-restricted generalized V -module is offinite length. Thus W is generated by ` ℜ ( h W ) ≤ℜ ( n ) ≤ℜ ( h W )+ N W [ n ] . Theorem 5.13
Let V be a M¨obius vertex algebra. Assume that the differences between thereal parts of the lowest weights of the irreducible lower-bounded generalized V -modules are allless than or equal to N ∈ N . Then a lower-bounded generalized V -module W of finite lengthis irreducible or completely reducbible if and only if the graded A N ( V ) -module Gr N ( W ) isirreducible or completely reducible, respectively.Proof. By Proposition 5.3, we already know that if W is irreducible, Gr N ( W ) = T N ( W )is irreducibile. Conversely, assume that the graded A N ( V )-module Gr N ( W ) is irreducible.Let W be a nonzero generalized V -submodule of W . Let e W : W → W be the embeddingmap. Then we have a graded A N ( V )-moudle map Gr ( e W ) : Gr N ( W ) → Gr N ( W ) given by( Gr ( e W ))( w + Ω n − ( W )) = w + Ω n − ( W ) for n = 0 , . . . , N and w ∈ Ω n ( W ). Since e W isinjective, Gr ( e W ) is also injective . So ( Gr ( e W ))( Gr N ( W )) is a graded A N ( V )-submoduleof Gr N ( W ). Since W is nonzero, Gr N ( W ) is nonzero. Since Gr N ( W ) is irreducible and Gr ( e W ) is injective, ( Gr ( e W ))( Gr N ( W )) is equal to Gr N ( W ). We now prove W = W .In fact, for n = 0 , . . . , N , ( Gr ( e W ))( Gr n ( W )) = { w + Ω n ( W ) | w ∈ Ω n ( W ) } . So Gr n ( W ) = { w + Ω n − ( W ) | w ∈ Ω n ( W ) } . For n = 0, we obtain Ω ( W ) = Gr ( W ) = Gr ( W ) = Ω ( W ). Assume that Ω n − ( W ) = Ω n − ( W ) for n < N . Given w ∈ Ω n ( W ), w + Ω n − ( W ) ∈ Gr n ( W ). By Gr n ( W ) = { w + Ω n − ( W ) | w ∈ Ω n ( W ) } , there exists w ∈ Ω n ( W ) such that w + Ω n − ( W ) = w + Ω n − ( W ), or equivalently, w − w ∈ Ω n − ( W ) =Ω n − ( W ). Thus w ∈ Ω n ( W ). This shows Ω n ( W ) = Ω n ( W ) for n = 0 , . . . , N . In particular,Ω N ( W ) = Ω N ( W ). But by Proposition 5.11, W and W are generated by Ω N ( W ) andΩ N ( W ), respectively. Since Ω N ( W ) = Ω N ( W ), we must have W = W . So W is irreducible.If W is completely reducible, then by Proposition 5.3, Gr N ( W ) = T N ( W ) is completelyreducible. Conversely, assume that the graded A N ( V )-module Gr N ( W ) is completely re-ducible. Then Gr N ( W ) = ` µ ∈M M µ , where M µ for µ ∈ M are irreducible graded A N ( V )-submodules of Gr N ( W ). For µ ∈ M , since M µ is a graded A N ( V )-submodule of Gr N ( W ),we have M µn ⊂ Gr n ( W ) = Ω n ( W ) / Ω n − ( W ) for n = 0 , . . . , N . Let W µ be the generalized V -submodule of W generated by the set of elements of the form w µ ∈ Ω n ( W ) such that39 µ + Ω n − ( W ) ∈ M µn for for n = 0 , . . . , N . Since W µ is a generalized V -submodule of W ,for v ∈ V , k, l ∈ N and w µ ∈ Ω l ( W ) such that w µ + Ω l − ( W ) ∈ M µl ,Res x x l − k − Y W ( x L V (0) v, x ) w µ + Ω k − ( W ) ∈ Gr k ( W µ ) . By the definition of W µ , we see that Res x x l − k − Y W ( x L V (0) v, x ) w µ ∈ W µ . Since w µ ∈ Ω l ( W ),Res x x l − k − Y W ( x L V (0) v, x ) w µ = 0 for k ∈ − Z + . Therefore Res x x l − k − Y W ( x L V (0) v, x ) w µ ∈ W µ for k ∈ N are all the nonzero coefficients of Y W ( v, x ) w µ . So W µ is closed under the actionof the vertex operators on W . Since M µ is invariant under the actions of L Gr ( W ) (0) and L Gr ( W ) ( −
1) and is a direct sum of generalized eigenspaces of L Gr ( W ) (0), W µ is invariantunder the actions of L W (0) and L W ( −
1) and is a direct sum of generalized eigenspaces of L W (0). Thus W µ is a generalized V -submodule of W .Let w µ + Ω n − ( W µ ) ∈ Gr n ( W µ ), where 0 ≤ n ≤ N and w µ ∈ Ω n ( W µ ) ⊂ Ω n ( W ). By thedefinition of W µ , we see that since w µ is an element of W µ , w µ + Ω n − ( W ) ∈ G n ( M µ ). So weobtain a linear map from Gr n ( W µ ) to G n ( M µ ) given by w µ + Ω n − ( W µ ) w µ + Ω n − ( W )for w µ + Ω n − ( W µ ) ∈ Gr n ( W µ ). These maps for n = 0 , . . . , N give a map from Gr N ( W µ )to M µ . It is clear that this map is a graded A N ( V )-module map. If for 0 ≤ n ≤ N ,the image w µ + Ω n − ( W ) of w µ + Ω n − ( W µ ) ∈ Gr n ( W µ ) under this map is 0 in M µ , then w µ ∈ Ω n − ( W ). But w µ ∈ Ω n ( W µ ) ⊂ W µ . So w µ ∈ Ω n − ( W µ ) and w µ + Ω n − ( W µ ) is 0in Gr N ( W µ ). This means that this graded A N ( V )-module map is injective. In particular,the image of Gr N ( W µ ) under this map is a nonzero graded A N ( V )-submodule of M µ . But M µ is irreducible. So Gr N ( W µ ) must be equivalent to M µ and is therefore also irreducible.From what we have proved above, since Gr N ( W µ ) is irreducible, W µ is irreducible. Thus W is complete reducible.From Corollary 5.12 and Theorem 5.13, we obtain the following result: Corollary 5.14
Let V be a M¨obius vertex algebra. Assume that A N ′ ( V ) for all N ′ ∈ N arefinite dimensional (for example, when V is C -cofinite and of positive energy by Theorem4.6). Let N ∈ N such that the differences between the real parts of the lowest weights ofthe finitely many (inequivalent) irreducible ordinary V -modules are less than or equal to N . Then a lower-bounded generalized V -module W of finite length or a grading-restrictedgeneralized V -module W is a direct sum of irreducible ordinary V -modules if and only if thegraded A N ( V ) -module Gr N ( W ) is completely reducible.Proof. Since A N ′ ( V ) is finite dimensional, there are only finitely many (inequivalent) ir-recucible graded A N ′ ( V )-modules. By Theorem 5.6, there are finitely many irreduciblelower-bounded generalized V -modules. By Corollary 5.12, these finitely many irreduciblelower-bounded generalized V -modules are all irreducible ordinary V -modules. There eixts N ∈ N such that the differences between the real parts of the lowest weights of the finitelymany irreducible ordinary V -modules are less than or equal to N . For such N , the conditionin Theorem 5.13 holds. So by Theorem 5.13, a lower-bounded generalized V -module W offinite length is a direct sum of irreducible ordinary V -modules if and only if Gr N ( W ) iscompletely reducible as a graded A N ( V )-module.40y Corollary 3.16 in [H1], every grading-restricted generalized V -module is of finitelength. Thus the conclusion holds also for a grading-restricted generalized V -module W . References [AN] T. Abe and K. Nagatomo, Finiteness of conformal blocks over the projective line,in:
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E-mail address : [email protected]@math.rutgers.edu