On functors between module categories for associative algebras and for N-graded vertex algebras

On functors between module categores for assocatve algebras and for N-graded vertex algebras Y-Zh Huang and Jnwe Yang Abstract We prove that the weak assocatvty for modules for vertex algebras are equvalent to a resdue formula for terates of vertex operators, obtaned usng the weak assocatvty and the lower truncaton property of vertex operators, together wth a known formula expressng products of components of vertex operators as lnear combnatons of terates of components of vertex operators. By requrng that these two formulas nstead of the commutator formula hold, we construct a functor S from the category of modules for Zhu s algebra of a vertex operator algebra V to the category of N-gradable weak V -modules. We prove that S has a unversal property and the functor T of takng top levels of N-gradable weak V -modules s a left nverse of S. In partcular, S s equal to a functor mplctly gven by Zhu and explctly constructed by Dong, L and Mason and we obtan a new constructon wthout usng relatons correspondng to the commutator formula. The hard part of ths new constructon s a techncal theorem statng roughly that n a module for Zhu s algebra, the relaton correspondng to the resdue formula mentoned above can n fact be obtaned from the relatons correspondng to the acton of Zhu s algebra. 1 Introducton In the study of vertex operator algebras and ther modules, commutator formulas for vertex operators often play an mportant role. Because of the commutator formula, the components of the vertex operators of a vertex operator algebra form a Le algebra. The vertex operator algebra tself and ts modules become modules for ths Le algebra. Ths fact allows one to apply many technques n the representaton theory of Le algebras to the study of vertex operator algebras and ther modules. For example, the proof by Zhu [Z] of the modular nvarance of the space spanned by the q-traces of products of vertex operators depends heavly on the commutator formula, especally n the case of q-traces of products of at least two vertex operators. 1

On the other hand, the man nterestng mathematcal objects n the representaton theory of vertex operator algebras are ntertwnng operators, not just vertex operators for vertex operator algebras and for modules. For ntertwnng operators, n general there s no commutator formula and therefore the Le-algebra-theoretc technques cannot be appled. Thus t s necessary to develop dfferent, non-le-algebra-theoretc method to study ntertwnng operators. One example s the modular nvarance of the space spanned by the q-traces of products of at least two ntertwnng operators. Ths modular nvarance was conjectured frst by physcsts and was used explctly by Moore and Seberg n [MS] to derve ther polynomal equatons. But ts proof was gven by the frst author n the preprnt of [H2] 13 years after Zhu s proof of hs modular nvarance result n [Z]. The dffculty les manly n the development of new method wthout usng the commutator formula on whch Zhu s proof depends heavly. Instead of any commutator formula (whch n fact does not exst n general, the frst author used the assocatvty for ntertwnng operators proved n [H1]. Because of the lmtaton of the commutator formula dscussed above, t s desrable to prove results n the representaton theory of vertex operator algebras, even only for modules, wthout usng the commutator formula so that t wll be relatvely easy to see whether such results can be generalzed to ntertwnng operators. Note that for modules for vertex operator algebras, the weak assocatvty can be taken to be the man axom. It s natural to expect that all the results for modules can be proved usng the weak assocatvty wthout the commutator formula. However, n many problems n the representaton theory of vertex operator algebras and ts applcatons, one has to work wth components of vertex operators nstead of vertex operators drectly. Thus we need formulas expressed usng resdues of formal seres obtaned from products and terates of vertex operators that are equvalent to the weak assocatvty. It s known from [DLM] and [L] (see also [LL] that for a vertex algebra, the weak assocatvty for vertex operators gves a formula that expresses products of components of vertex operators as lnear combnatons of terates of components of vertex operators. Unfortunately, ths formula s not equvalent to the weak assocatvty. In the present paper, we gve a formula for the resdues of certan formal seres nvolvng terates of vertex operators obtaned usng the weak assocatvty and the lower truncaton property of vertex operators. Then we prove that the weak assocatvty s equvalent to ths formula together wth the formula gven n [DLM] and [L]. Thus, n prncple, all the results for modules for vertex operator algebras can be proved usng these two formulas wthout the help of the commutator formula. We apply ths result to gve a new constructon of N-gradable weak modules for an N- graded vertex algebra V from modules for Zhu s algebra A(V. We construct a functor S from the category of A(V -modules to the category of N-gradable weak V -modules. We prove that S has a unversal property and the functor T of takng top levels of N-gradable weak V -modules s a left nverse of S. In partcular, S s equal to the functor constructed by Dong, L and Mason n [DLM] n the case n = 0 and we acheve our goal of fndng a new constructon wthout usng relatons correspondng to the commutator formula. The hard part of ths new constructon s a techncal theorem statng roughly that n a module for 2

Zhu s algebra, the relaton correspondng to the resdue formula gven n the present paper (see above can n fact be obtaned from the relatons correspondng to the acton of Zhu s algebra. The mportance of our constructon s that t allows us to use the method n the present paper to gve constructons and prove results n the cases that there s no commutator formula. The man example that motvated the present paper s a constructon of generalzed twsted modules assocated to an nfnte order automorphsm of a vertex operator algebra n the sense of the frst author n [H3]. For these generalzed twsted modules, the twsted vertex operators nvolve the logarthm of the varable and thus do not have a commutator formula. We shall dscuss ths constructon n a future publcaton. Ths paper s organzed as follows: We recall and derve some resdue formulas for vertex operators from the weak assocatvty n Secton 2. We prove that these resdue formulas are equvalent to the weak assocatvty n Secton 3. In Secton 4, we construct our functor S. In Secton 5, we prove a unversty property of S and prove that the functor T mentoned above s a left nverse of S. Acknowledgments The results n ths paper was presented by the second author n the conference Representaton theory XIII, Dubrovnk, Croata, June 21-27, 2013. The second author would lke to thank Drazen Adamovc and Antun Mlas for nvtng hm to gve the talk n the conference. Ths research s supported n part by NSF grant PHY-0901237. 2 Resdue formulas from the weak assocatvty In ths secton, we recall and derve some consequences of the weak assocatvty. Let (V, Y, 1 be a vertex algebra and (W, Y W a V -module. (Note that snce V as a vertex algebra mght not have a gradng, a V -module also mght not have a gradng. The weak assocatvty for W states that for u, v V, w W and l a nonnegatve nteger such that u n w = 0 for n l, we have (x 0 + x 2 l Y W (u, x 0 + x 2 Y W (v, x 2 w = (x 0 + x 2 l Y W (Y (u, x 0 v, x 2 w. We recall the followng result from [DLM] and [L] (Proposton 4.5.7 n [LL]: Proposton 2.1 Let W be a V -module and let u, v V, p, q Z and w W. Let l be a nonnegatve nteger such that u n w = 0 for n l and let k be a nonnegatve nteger such that k q s postve and v n w = 0 for n k. Then u p v q w = k q 1 l ( p l j=0 ( l j (u p l +j v q+l+ j w. (2.1 3

Remark 2.2 Let p(x 0, x 2 = Then the formula (2.1, the formula and the formula k q 1 ( p l Res x1 Res x2 x p 1x q 2Y (u, x 1 Y (v, x 2 w x p l 0 x 2. = Res x0 Res x2 p(x 0, x 2 x q 2(x 0 + x 2 l Y (Y (u, x 0 v, x 2 w (2.2 Res x0 Res x2 (x 0 + x 2 p x q 2Y (u, x 0 + x 2 Y (v, x 2 w = Res x0 Res x2 p(x 0, x 2 x q 2(x 0 + x 2 l Y (Y (u, x 0 v, x 2 w (2.3 are equvalent to each other. See the proof of Proposton 4.5.7 n [LL]. From the lower truncaton property for vertex operators, we have the followng: Proposton 2.3 For W, u, v, w, l, k as n Proposton 2.1, we have for q k. Res x2 x q 2(x 0 + x 2 l Y W (u, x 0 + x 2 Y W (v, x 2 w = 0, (2.4 Res x2 x q 2(x 0 + x 2 l Y W (Y (u, x 0 v, x 2 w = 0 (2.5 Proof. Note that x q 2(x 0 +x 2 l Y W (u, x 0 +x 2 contans only terms wth the powers of x 2 larger than or equal to q k. Hence the left hand sde of (2.4 are lnear combnatons of u m v n w for m Z and n k wth coeffcents n C((x 0. Snce v n w = 0 for n k, (2.4 holds. By the weak assocatvty, the left-hand sde of (2.5 s equal to the left-hand sde of (2.4. Thus (2.5 also holds. Remark 2.4 Note that the proof of (2.4 uses only the property of Y W that Y W (v, xw W ((x for v V and w W. Thus for a vector space W and a lnear map Y W from V W to W ((x, (2.4 stll holds. On the other hand, (2.5 does not hold n general for such a vector space W and such a map Y W. From (2.5, we obtan mmedately: Corollary 2.5 For W, u, v, p, q, w, l, k as n Proposton 2.1 except that k q does not have to be postve, we have for k q. Res x0 Res x2 x p l 0 x q+ 2 (x 0 + x 2 l Y W (Y (u, x 0 v, x 2 w = 0 (2.6 4

Remark 2.6 Assume that V s a Z-graded vertex algebra and W s an N-gradable weak V -module. Here by a Z-graded vertex algebra we mean a vertex algebra V = n Z V (n such that for u V, [d, Y (u, x] = x d Y (u, x + Y (du, x, dx where d : V V s defned by du = nu for u V (n. For u V (n, we use wt u to denote n. By an N-gradable weak V -module, we mean a module W = n N W (n for V when V s vewed as a vertex algebra such that for u V (m u n = Res x x n Y W (u, x maps W (k to W (k+m n 1. For w W (n, we use deg w to denote n. When u, v and w are homogeneous, we take k = wt v + deg w and l = wt u + deg w and set K = p + q + 2 wt u wt v and m = p l. Then (2.6 becomes Res x0 Res x2 x m K+wt v m deg w 2 0 x2 (x 0 + x 2 wt u+deg w Y W (Y (u, x 0 v, x 2 w = 0 (2.7 for m M 2 deg w 2. The formula (2.7 can be further wrtten wthout usng the weghts of u and v as Res x0 Res x2 x m K m deg w 2 0 x2 (x 0 + x 2 deg w Y W (Y ((x 0 + x 2 L(0 u, x 0 x L(0 2 v, x 2 w = 0, (2.8 whch s n fact holds for general u and v, not necessarly homogeneous. The component form of (2.7 s wt u+deg w j=0 for m K 2 deg w 2. ( wt u + deg w j (u j+m v K+wt u+wt v j m 2 w = 0 (2.9 3 The equvalence between the resdue formulas and the weak assocatvty In ths secton, we prove that (2.3 and (2.6 mply, and therefore are equvalent to, the weak assocatvty. Theorem 3.1 Let V be a vertex operator algebra, W a vector space and Y W a lnear map from V W to W ((x. For v V and w W, as for a V -module, we denote the mage of u w under Y W by Y W (u, xw and Res x x n Y W (u, xw by u n. Let u, v V and w W and let k, l Z such that v n w = 0 for n k (3.1 and u n w = 0 for n l. (3.2 Assume that ( the formulas (2.3 (or equvalently, (2.2 or (2.1 holds for p, q Z such that q < k and ( (2.6 holds for for p, q Z such that k q. Then (x 0 + x 2 l Y W (u, x 0 + x 2 Y W (v, x 2 w = (x 0 + x 2 l Y W (Y (u, x 0 v, x 2 w. (3.3 5

Proof. The Laurent polynomal p(x 0, x 2 n Remark 2.2 s n fact the frst k q 1 terms of the formal seres (x 0 + x 2 p l. But from (2.6, we obtan Res x0 Res x2 ((x 0 + x 2 p l p(x 0, x 2 x q 2((x 0 + x 2 l Y W (Y (u, x 0 v, x 2 w = 0. (3.4 Combnng (2.3 and (3.4, we obtan or equvalently, Res x0 Res x2 (x 0 + x 2 p l x q 2((x 0 + x 2 l Y W (u, x 0 + x 2 Y W (v, x 2 w = Res x0 Res x2 (x 0 + x 2 p l x q 2((x 0 + x 2 l Y W (Y (u, x 0 v, x 2 w, Res x0 Res x2 (x 0 +x 2 p l x q 2((x 0 +x 2 l (Y (u, x 0 +x 2 Y (v, x 2 Y W (Y (u, x 0 v, x 2 w = 0, (3.5 On the other hand, by Remark 2.4, (2.4 holds for m k. In partcular, we have Res x0 Res x2 x p l 0 x q+ 2 (x 0 + x 2 l Y W (u, x 0 + x 2 Y W (v, x 2 w = 0 (3.6 for k q. From (2.6 and (3.6, we obtan Res x0 Res x2 x p l 0 x q+ 2 (x 0 + x 2 l (Y W (u, x 0 + x 2 Y W (v, x 2 Y W (Y (u, x 0 v, x 2 w = 0 (3.7 for k q. Combnng (3.5 and (3.7, we obtan k q 1 ( p l Res x0 Res x2 x p l 0 x q+ 2 = 0. We now use nducton on k q 1 to prove ((x 0 + x 2 l (Y (u, x 0 + x 2 Y (v, x 2 Y W (Y (u, x 0 v, x 2 w (3.8 Res x0 Res x2 x p l 0 x q 2((x 0 + x 2 l (Y (u, x 0 + x 2 Y (v, x 2 Y W (Y (u, x 0 v, x 2 w = 0 (3.9 for p Z and q < k. When k q 1 = 0, (3.8 becomes Res x0 Res x2 x p l 0 x q 2((x 0 + x 2 l (Y (u, x 0 + x 2 Y (v, x 2 Y W (Y (u, x 0 v, x 2 w = 0. Assumng that (3.9 holds when 0 k q 1 < n. When k q 1 = n, 0 k q 1 < n for = 1,..., n = k q 1. Snce p s arbtrary, we can replace p by p for any Z n (3.9. Thus by the nducton assumpton, Res x0 Res x2 x p l 0 x q+ 2 ((x 0 + x 2 l (Y (u, x 0 + x 2 Y (v, x 2 Y W (Y (u, x 0 v, x 2 w = 0 (3.10 for = 1,..., n = k q 1. From (3.10 for = 1,..., n = k q 1 and (3.8, we obtan Res x0 Res x2 x p l 0 x q 2((x 0 + x 2 l (Y (u, x 0 + x 2 Y (v, x 2 Y W (Y (u, x 0 v, x 2 w k q 1 ( p l = Res x0 Res x2 x p l 0 x q+ 2 = 0, ((x 0 + x 2 l (Y (u, x 0 + x 2 Y (v, x 2 Y W (Y (u, x 0 v, x 2 w 6

provng (3.9 n ths case. Takng = 0 n (2.6, we see that (3.9 also holds for p Z and q k. Thus (2.6 holds for p, q Z. But ths means that that s, (3.3 holds. ((x 0 + x 2 l (Y (u, x 0 + x 2 Y (v, x 2 Y W (Y (u, x 0 v, x 2 w = 0, The followng result follows mmedately from Theorem 3.1 above and Theorems 3.6.3 and 3.11.8 n [LL]. Theorem 3.2 Let V be a Z-graded vertex algebra, W = n N W (n an N-graded vector space and Y W a lnear map from V W to W ((x. For v V and w W, we denote the mage of u w under Y W by Y W (u, xw and Res x x n Y W (u, xw by u n. Assume that u n maps W (k to W (k+m n 1 for for u V (m and n Z and Y W (1, x = 1 W. Also assume that for u, v V, w W, there exst k, l Z such that (3.1 and (3.2 hold, and for p, q Z, u, v V, w W, the formulas (2.2 (or equvalently, (2.1 or (2.3 and (2.6 (or equvalently, (2.7 hold. Then (W, Y W s an N-gradable weak V -module. 4 A functor S In ths and the next secton, (V, Y, 1 s a Z-graded vertex algebra such that V (n = 0 for n < 0. We shall call such a Z-graded vertex algebra an N-graded vertex algebra. In ths secton, by dvdng relatons such that (2.3 and (2.6 hold, we construct a functor S from the category of modules for Zhu s algebra A(V to the category of N-gradable weak V -modules. We frst recall the defnton of Zhu s algebra n [Z]. Let O(V be the subspace of V spanned by elements of the form Res x x n Y ((x + 1 L(0 u, xv for u, v V and n 2. Zhu s algebra assocated to V s the quotent space A(V = V/O(V equpped wth the multplcaton u v = Res x x 1 Y ((x + 1 L(0 u, xv for u, v V. It was proved n [Z] that A(V equpped wth the product s an assocatve algebra. Let (W, Y W be an N-gradable V -module. We shall use the same notatons as n the precedng secton. Let T (W be the subspace of elements w of W such that u n w = 0 when wt u n 1 < 0. The subspace T (W s called the top level of W. For u V, let o(u = Res x x 1 Y W (x L(0 u, x. Then for w T (W, o(uw T (W. In fact, t was proved n [Z] that u + O(V o(u gves a A(V -module structure on T (W. Clearly, takng the top level of an N-gradable V - module gves a functor T from the category of A(V -modules to the category of N-gradable V -modules. 7

Consder the affnzaton V [t, t 1 ] = V C[t, t 1 ] of V and the tensor algebra T (V [t, t 1 ] generated by V [t, t 1 ]. (Note that although we use the same notaton, T (V [t, t 1 ] has nothng to do wth the top level of an N-gradable weak V -module. For smplcty, we shall denote u t m for u V and m Z by u(m and we shall omt the tensor product sgn when we wrte an element of T (V [t, t 1 ]. Thus T (V [t, t 1 ] s spanned by elements of the form u 1 (m 1 u k (m k for u V and m Z, = 1,..., k. Let M be an A(V -module and ρ : A(V End M the correspondng representaton of A(V. Consder T (V [t, t 1 ] M. Agan for smplcty we shall omt the tensor product sgn. So T (V [t, t 1 ] M s spanned by elements of the form u 1 (m 1 u k (m k w for u V, m Z, = 1,..., k and w M and for any u V, m Z, u(m acts from the left on T (V [t, t 1 ] M. For homogeneous u V, m Z, = 1,..., k and w M, we defne the degree of u 1 (m 1 u k (m k w to be (wt u 1 m 1 1 + (wt u k m k 1. In partcular, the degrees of elements of M are 0. For u V, let be defned by Y t (u, x : T (V [t, t 1 ] M T (V [t, t 1 ] M[[x, x 1 ]] For a homogeneous element u V, let Y t (u, x = m Z u(mx m 1. o t (u = u(wt u 1. Usng lnearty, we extend o t (u to non-homogeneous u. Let I be the Z-graded T (V [t, t 1 ]-submodule of T (V [t, t 1 ] W generated by elements of the forms u(mw for homogeneous u V, w T (V [t, t 1 ] M, wt u m 1 + deg w < 0, for u V, w M and where or equvalently, u(pv(qw o t (uw ρ(u + O(V w Res x1 Res x2 x p 1x q 2Y t (u, x 1 Y t (v, x 2 w Res x0 Res x2 p(x 0, x 2 x q 2(x 0 + x 2 wt u+deg w Y t (Y (u, x 0 v, x 2 w, (4.1 p(x 0, x 2 = wt v+deg w q 1 wt v+deg w q 1 wt u+deg w j=0 ( p wt u deg w p wt u deg w x0 x 2, ( p wt u deg w ( wt u + deg w j (u p wt u deg w +j v(q + wt u + deg w + jw (4.2 8

for homogeneous u, v V, w T (V [t, t 1 ] M, p, q Z such that wt v + deg w > q and wt u p 1 + wt v q 1 + deg w 0. Let S 1 (M = (T (V [t, t 1 ] M/I. Then S 1 (M s also a Z-graded T (V [t, t 1 ]-module. In fact, by defnton of I, we see that S 1 (M s spanned by elements of the form u(mw + I for homogeneous u V, m Z such that m < wt u 1 and w M. In partcular, we see that S 1 (M has an N-gradng. Note that snce I M = {0}, M can be embedded nto S 1 (M and the homogeneous subspace (S 1 (M (0 of degree 0 s the mage of M under ths embeddng. We shall dentfy M wth (S 1 (M (0. In partcular, M s now vewed as a subspace of S 1 (M. The vertex operator map Y t nduces a vertex operator map Y S1 (M : V S 1 (M S 1 (M[[x, x 1 ]] by Y S1 (M(u, xw = Y t (u, xw. Let J be the N-graded T (V [t, t 1 ]-submodule of S 1 (M generated by p wt u deg w Res x0 Res x2 x0 x q+ 2 (x 0 + x 2 wt u+deg w Y S1 (M(Y (u, x 0 v, x 2 w. (4.3 for homogeneous u, v V, w S 1 (M, p, q Z and wt v + deg w q. By Remark 2.6, J s generated by Res x0 Res x2 x m K m deg w 2 0 x2 (x 0 + x 2 deg w Y S1 (M(Y ((x 0 + x 2 L(0 u, x 0 x L(0 2 v, x 2 w for homogeneous u, v V, w S 1 (M, K, m Z satsfyng m K 2 deg w 2, or equvalently, by wt u+deg w j=0 ( wt u + deg w j (u j+m v(k + wt u + wt v j m 2w, (4.4 for homogeneous u, v V and the same w, K and m. Let S(M = S 1 (M/J. Then S(M s also an N-graded T (V [t, t 1 ]-module. We can stll use elements of T (V [t, t 1 ] M to represent elements of S(M. But note that these elements now satsfy more relatons. We equp S(M wth the vertex operator map Y S(M : V S(M S(M[[x, x 1 ]] gven by u w Y S(M (u, xw = Y S1 (M(u, xw = Y t (u, xw. Theorem 4.1 The par (S(M, Y S(M s an N-gradable weak V -module. Proof. As n S 1 (M, for u V and w S(M, we have u(mw = 0 when m > wt u + wt w 1. Clearly, Y (1, x = 1 S(M, where 1 S(M s the dentty operator on S(M. By Theorems 3.1 and 3.2, S(M s an N-gradable weak V -module. Let M 1 and M 2 be N-gradable weak V -modules and f : M 1 M 2 a module map. Then F nduces a lnear map from T (V [t, t 1 ] M 1 to T (V [t, t 1 ] M 2. By defnton, ths 9

nduced lnear map n turn nduces a lnear map S(f from S(M 1 to S(M 2. Snce Y S(M1 and Y S(M2 are nduced by Y t on T (V [t, t 1 ] M 1 and T (V [t, t 1 ] M 2, respectvely, we have S(f(Y S(M1 (u, xw 1 = Y S(M2 (u, xs(f(w 1 for u V and w 1 S(M 1. Thus S(f s a module map. The followng result s now clear: Corollary 4.2 Let V be a N-graded vertex algebra. Then the correspondence sendng an A(V -module M to an N-gradable weak V -module (S(M, Y S(M and an A(V -module map M 1 M 2 to a V -module map S(f : S(M 1 S(M 2 s a functor from the category of A(V -modules to the category of N-gradable weak V -modules. 5 A unversal property and T as a left nverse of S In ths secton, we prove that S satsfes a natural unversal property and thus s the same as the functor constructed n [DLM]. In partcular, we acheve our goal of constructng N- gradable weak V -modules from A(V -modules wthout dvdng relatons correspondng to the commutator formula for weak modules. We also prove that the functor T of sendng an N-gradable weak V -module W to ts top level T (W s a left nverse of S. As n the precedng secton, we let V be a N-graded vertex algebra. We shall need the followng lemma: Lemma 5.1 In the settng of Theorem 3.1, f we assume only (2.3 (or equvalently, (2.1, then for u, v V, homogeneous w W and p, q Z, q < deg w, we have Res x1 Res x2 x p 1 x q 2 Y W (x L(0 1 u, x 1 Y W (x L(0 2 v, x 2 w = Res x0 Res x2 q(x 0, x 2 x q 2 (x 0 + x 2 deg w Y W (Y ((x 0 + x 2 L(0 u, x 0 x L(0 2 v, x 2 w, (5.5 where q(x 0, x 2 C[x 0, x 1 0, x 2 ] such that when w s homogeneous, q(x 0, x 2 = deg w q 1 ( p deg w x p deg w 0 x 2. Proof. In the case that u, v are also homogeneous, the lemma follows from Remark 2.2 by takng p = p + wt u, q = q + wt v, k = wt v + deg w and l = wt u + deg w. For an A(V -module M, let I, S 1 (M and J be the same as n the precedng secton. The followng theorem s the man techncal result n ths secton and ts proof s the hardest n the present paper: Theorem 5.2 Let M be a an A(V -module. Then n S 1 (M, J M = 0. 10

Proof. Note that elements of the form (4.2 are n I and that M s dentfed wth (S 1 (M (0. Usng the defntons of I and J and notcng that the degree of elements of M s 0, we see that elements of J are lnear combnatons of elements of the form a(wt a K + N 1Res x0 Res x2 x m 0 x2 K m N 2 (x 0 + x 2 N Y S1 (M(Y ((x 0 + x 2 L(0 u, x 0 x L(0 2 v, x 2 b(wt b N 1w + I wt u+n ( wt u + N = a(wt a K + N 1 j j=0 (u j+m v(k + wt u + wt v j m 2b(wt b N 1w + I (5.6 for homogeneous a, b, u, v V, w M, K, N, m Z satsfyng m K 2N 2. To prove ths theorem, we need only prove that (5.6 s 0 n S 1 (M. When N < 0, deg b(wt b N 1w = N < 0. So b(wt b N 1w I and hence (5.6 s 0 n S 1 (M. When N 0 but K > N, deg(u j+m v(k + wt u + wt v j m 2b(wt b N 1w = N K < 0. So (u j+m v(k + wt u + wt v j m 2b(wt b N 1w I and hence (5.6 s 0 n S 1 (M. We now prove that (5.6 s 0 n S 1 (M n the case N 0 and N K. We rewrte (5.6 as Res y1 Res y2 Res x0 Res x2 y (K N 1 1 y N 1 2 x m 0 x K m N 2 2 (x 0 + x 2 N 1 a, y 1 Y S1 (M(Y ((x 0 + x 2 L(0 u, x 0 x L(0 2 v, x 2 Y S1 (M(y L(0 2 b, y 2 w (5.7 where a, b, u, v V, w M, K, N, m Z satsfyng m K 2N 2. Note that n S 1 (M, Y S1 (M satsfes (5.5 wth Y W replaced by Y S1 (M. We shall use (5.5 and other formulas that stll hold to show that (5.7 s n fact a lnear combnaton of an element of O(V actng on w M and hence equals 0 n S 1 (M. Let w = Res y2 y2 N 1 Y S1 (W (y L(0 2 b, y 2 w. Usng the propertes of vertex operators, changng varables and usng (5.5, we see that 11

(5.7 s equal to Res y1 Res x0 Res x2 y (K N 1 1 x m 0 x K m N 2 2 (x 0 + x 2 N 1 a, y 1 Y S1 (M(Y ((x 0 + x 2 L(0 u, x 0 x L(0 2 v, x 2 w = Res y1 Res x0 Res x2 y (K N 1 1 x m 0 x K m 2 2 (x 0 x 1 2 + 1 N 1 a, y 1 Y S1 (M(x L(0 2 Y ((x 0 x 1 2 + 1 L(0 u, x 0 x 1 2 v, x 2 w = Res x0 x m 0 (x 0 + 1 N Res y1 Res x2 y (K N 1 1 x K 1 2 1 a, y 1 Y S1 (M(x L(0 2 Y ((x 0 + 1 L(0 u, x 0 v, x 2 w = Res x0 x m 0 (x 0 + 1 N Res y0 Res x2 q 1 (y 0, x 2 x K 1 2 (y 0 + x 2 N Y S1 (M(Y ((y 0 + x 2 L(0 a, y 0 x L(0 2 Y ((x 0 + 1 L(0 u, x 0 v, x 2 w, (5.8 where q 1 (y 0, x 2 = N K N K = x K 1 2 ( K 1 ( K 1 = x K 1 2 q 1 (y 0 x 1 2, 1. y K 1 0 x 2 (y 0 x 1 2 K 1 The rght-hand sde of (5.8 can now be wrtten as Res x0 x m 0 (x 0 + 1 N Res y0 Res x2 q 1 (y 0 x 1 2, 1x N 2 2 (y 0 x 1 2 + 1 N Y S1 (M(x L(0 2 Y ((y 0 x 1 2 + 1 L(0 a, y 0 x 1 2 Y ((x 0 + 1 L(0 u, x 0 v, x 2 w = Res x0 x m 0 (x 0 + 1 N Res y0 Res x2 q 1 (y 0, 1x N 1 2 (y 0 + 1 N Y S1 (M(x L(0 2 Y ((y 0 + 1 L(0 a, y 0 Y ((x 0 + 1 L(0 u, x 0 v, x 2 w. (5.9 Usng the commutator formula for V, the rght-hand sde of (5.9 s equal to Res y0 q 1 (y 0, 1(y 0 + 1 N Res x2 Res x0 x N 1 2 x m 0 (x 0 + 1 N Y S1 (M(x L(0 2 Y ((x 0 + 1 L(0 u, x 0 Y ((y 0 + 1 L(0 a, y 0 v, x 2 w +Res y0 q 1 (y 0, 1(y 0 + 1 N Res x2 Res x0 Res x x N 1 2 x m 0 (x 0 + 1 N x 1 0 ( y0 x x 0 Y S1 (M(x L(0 2 Y (Y ((y 0 + 1 L(0 a, x(x 0 + 1 L(0 u, x 0 v, x 2 w. The frst term of (5.10 s a lnear combnaton of the elements of the form (5.10 Res x2 Res x0 x N 1 2 x n 0(x 0 + 1 N Y S1 (M(x L(0 2 Y ((x 0 + 1 L(0 ũ, x 0 ṽ, x 2 w (5.11 12

for n = m N 2, ũ = u, ṽ V. The second term of (5.10 s equal to ( Res x2 Res x0 Res x Res y0 q 1 (x 0 + x, 1(x 0 + x + 1 N x N 1 2 x m 0 (x 0 + 1 N x 1 y0 x 0 Y S1 (M(x L(0 2 Y ((x 0 + 1 L(0 Y ((1 + x(x 0 + 1 1 L(0 a, x(x 0 + 1 1 u, x 0 v, x 2 w = Res x2 Res x0 Res x q 1 (x 0 + x, 1(x 0 + x + 1 N x N 1 2 x m 0 (x 0 + 1 N Y S1 (M(x L(0 2 Y ((x 0 + 1 L(0 Y ((1 + x(x 0 + 1 1 L(0 a, x(x 0 + 1 1 u, x 0 v, x 2 w = Res x2 Res x0 Res y q 1 (x 0 + y(x 0 + 1, 1(x 0 + 1 N (1 + y N x N 1 2 x m 0 (x 0 + 1 N+1 Y S1 (M(x L(0 2 Y ((x 0 + 1 L(0 Y ((1 + y L(0 a, yu, x 0 v, x 2 w. (5.12 By the defnton of q 1 (y 0, x 2 above, we see that q 1 (x 0 + y(x 0 + 1, 1 s a Laurent seres n x 0 and y such that the largest power of x 0 n q 1 (x 0 + y(x 0 + 1, 1 s K 1. Then the powers of x 0 n q 1 (x 0 + y(x 0 + 1, 1x m 0 (x 0 + 1 N+1 are less than or equal to N 2. Thus the rght-hand sde of (5.12 s a lnear combnaton of the elements of the form (5.11 for n N 2 and ũ, ṽ = v V. It remans to prove that (5.11 for n N 2 and ũ, ṽ V s 0 n S 1 (M. We prove that elements of the form (5.11 for n N 2 and ũ, ṽ V are lnear combnatons of elements obtaned from the acton of O(V on M. Then by the defnton of S 1 (M, (5.11 for n N 2 and ũ, ṽ V s 0. Wrtng the element w explctly and usng (5.5, we see that (5.11 s equal to Res x0 x n 0(x 0 + 1 N Res x2 Res y2 x N 1 2 y N 1 2 Y S1 (M(x L(0 2 Y ((x 0 + 1 L(0 ũ, x 0 ṽ, x 2 Y S1 (M(y L(0 2 b, y 2 w = Res x0 x n 0(x 0 + 1 N Res y0 Res y2 q 2 (y 0, y 2 y N 1 2 Y S1 (M(Y ((y 0 + y 2 L(0 Y ((x 0 + 1 L(0 ũ, x 0 ṽ, y 0 y L(0 2 b, y 2 w = Res x0 x n 0(x 0 + 1 N Res y0 Res y2 q 2 (y 0, y 2 y N 1 2 2 Y ((y 0 y 1 2 + 1 L(0 Y ((x 0 + 1 L(0 ũ, x 0 ṽ, y 0 y 1 2 b, y 2 w = Res x0 x n 0(x 0 + 1 N Res y Res y2 q 2 (yy 2, y 2 y N 2 2 Y ((y + 1 L(0 Y ((x 0 + 1 L(0 ũ, x 0 ṽ, yb, y 2 w = Res x0 x n 0(x 0 + 1 N Res y Res y2 q 2 (yy 2, y 2 y N 2 2 Y (Y ((x 0 + 1 L(0 (y + 1 L(0 ũ, x 0 (y + 1(y + 1 L(0 ṽ, yb, y 2 w = Res y2 Res y Res x x n (x + y + 1 N q 2 (yy 2, y 2 y N 2 (y + 1 N 1 n 2 Y (Y ((x + y + 1 L(0 ũ, x(y + 1 L(0 ṽ, yb, y 2 w = Res y2 Res y Res x1 (x 1 y n (x 1 + 1 N q 2 (yy 2, y 2 y N 2 (y + 1 N 1 n 2 Y ((x 1 + 1 L(0 ũ, x 1 Y ((y + 1 L(0 ṽ, yb, y 2 w Res y2 Res y Res x1 ( y + x 1 n (x 1 + 1 N q 2 (yy 2, y 2 y N 2 (y + 1 N 1 n 2 Y ((y + 1 L(0 ṽ, yy ((x 1 + 1 L(0 ũ, x 1 b, y 2 w 13 x 0 (5.13

Where q 2 (y 0, y 2 = N ( N 1 y0 N 1 y 2. From the expresson of q 2 (y 0, y 2, we have q 2 (yy 2, y 2 = y N 1 2 q 2 (y, 1. Thus the frst term n the rght-hand sde of (5.13 s a lnear combnaton of terms of the form Res y2 y2 1 Y S1 (M(y L(0 2 Res x1 xñ1y ((x 1 + 1 L(0 ũ, x 1 ṽ, y 2 w for ñ 2 and ṽ V. By defnton, and, by defnton, ũ = Res x1 xñ1y ((x 1 + 1 L(0 ũ, x 1 ṽ O(V Res y2 y 1 Snce o(ũw = 0 n S 1 (M, Res y2 y 1 term n the rght-hand sde of (5.13 s 0. The largest power of y n 2 Y S1 (M(y L(0 2 2 Y S1 (M(y L(0 2 ũ, y 2 = o(ũ. ( y + x 1 n q 2 (y, 1(y + 1 N 1 n ũ, y 2 w s also 0. Ths proves that the frst s 2. Thus the second term n the rght-hand sde of (5.13 s a lnear combnaton of terms of the form Res y2 y2 1 Y S1 (M(y L(0 2 Res y yñy ((y + 1 L(0 ṽ, yũ, y 2 w for ñ 2 and ũ V. Then the same argument as above shows that the second term n the rght-hand sde of (5.13 s 0. Remark 5.3 Note that though the proof of the theorem above does not need the commutator formula for Y S1 (M, the commutator formula for V s ndeed used. It wll be nterestng to see whether there s a proof wthout usng the commutator formula for V. Corollary 5.4 Let M be an A(V -module. Then the embeddng from M to S 1 (M nduces an somorphsm e M of A(V -modules from M to T (S(M. Proof. Gven an N-gradable weak V -module M, we have an embeddng from M to S 1 (M. By Theorem 5.2, the nduced map from M to S(M = S 1 /J s stll an embeddng. By defnton, T (S(M s exactly the mage of M n S(M. Clearly, when we vew the nduced map from M to S(M = S 1 /J as a map from M to T (S(M, t s an A(V -module map and thus s a an somorphsm of A(V -modules. 14

Theorem 5.5 The functor S has the followng unversal property: Let M be an A(V - module. For any N-gradable weak V -module W and any A(V -module map f : M T (W, there exsts a unque V -module map f : S(M W such that f T (S(M = f e 1 M. Proof. Elements of S(M are lnear combnatons of elements of the form u(mw for homogeneous u V, w M and m Z satsfyng wt m 1 + deg w 0. We defne f(u(mw = u m f(w. By defnton, relatons among elements of the form u(mw for homogeneous u V, w M and m Z satsfyng wt m 1 + deg w 0 are also satsfed by elements of W of the form u m f(w. Thus the map f s well defned. Usng (2.2 (or equvalently, (2.1 or (2.3, we see that f s ndeed a V -module map. By defnton, f M = f e 1 M. If g : S(M W s another such V -module map, then g(u(mw = u m g(w = u m f(w = f(u(mw, provng the unqueness of f. We also have the followng: Corollary 5.6 The functor T s a left nverse of S, that s, T S s naturally somorphc to s the dentty functor on the category of A(V -modules. Proof. For an A(V -module M, by Corollary 5.4, we have an somorphsm e 1 M from (T S(M to M. For an A(V -module map f : M 1 M 2, by defnton, the A(V -module map (T S(f : (T S(M 1 (T S(M 2 s e M2 f e 1 M 1. Thus we obtan a natural transformaton e 1 from T S to the dentty functor on the category of A(V -modules. Ths natural transformaton e 1 s clearly a natural somorphsm snce t has an nverse e whch s gven by the somorphsm e M from an A(V -module M to T (S(M. In [Z], n hs proof that the set of equvalence classes of rreducble N-gradable weak V -modules s n bjecton wth the set of equvalence classes of rreducble modules for A(V, Zhu n fact obtaned mplctly a functor from the category of A(V -modules to the category of N-gradable weak V -modules. In [DLM], ths functor, denoted by M 0, was constructed explctly and was proved to satsfy the same unversal property above. The followng result acheves our goal of constructng ths functor wthout dvdng relatons correspondng to the commutator formula for weak modules: Corollary 5.7 The functor S s equal to the functor M 0 constructed n [DLM]. Proof. Ths result follows mmedately from the unversal property. REFERENCES [DLM] C. Dong, H. L and G. Mason, Vertex operator algebras and assocatve algebras, J. Algebra 206 (1998, 67-96. 15

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