A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS

Transcription

1 A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS ALESSANDRO D ANDREA AND GIUSEPPE MARCHEI ABSTRACT. Let L be a Le pseudoalgebra, a L. We show that, f a generates a (fnte) solvable subalgebra S = a L, then one may fnd a lftng ā S of [a] S/S such that ā s nlpotent. We then apply ths result towards vertex algebras: we show that every fnte vertex algebra V admts a decomposton nto a sem-drect product V = U N, where U s a subalgebra of V whose underlyng Le conformal algebra U Le s a nlpotent self-normalzng subalgebra of V Le, and N = V [ ] s a canoncally determned deal contaned n the nlradcal Nl V. CONTENTS 1. Introducton 1 2. Prelmnares on Le pseudoalgebras Hopf algebras and Le pseudoalgebras Hopf algebra notatons The general lnear pseudoalgebra Acton of coeffcents 5 3. Representatons of solvable and nlpotent Le pseudoalgebras Weght vectors Generalzed weght submodules Nlpotent pseudoalgebras 7 4. Approxmate nlpotence of solvable subalgebras of gc M The length 2 case Proof of the general statement Structure of fnte vertex algebras Prelmnares on vertex algebras The nlradcal Root space decomposton of fnte vertex algebras A counterexample to nlpotence of fnte vertex algebras 14 References INTRODUCTION Let H be a Hopf algebra. One may make [2] the class of left H-modules nto a pseudotensor category M (H) n a non-standard way; an algebra object n M (H) s then a pseudoalgebra over H. In [1], the noton of Le pseudoalgebra over a cocommutatve Hopf algebra was ntroduced and studed. Not surprsngly, the study of fnte Le pseudoalgebras amounts to nvestgatng commutator propertes of famles of pseudolnear endomorphsms. When M s a fnte (.e., fntely generated) left H-module, the space of all pseudolnear endomorphsms of M can be gven a natural Le pseudoalgebra structure, denoted by gc M. One of the Le theoretc features of the pseudo-verson of lnear algebra s that an analogue of Le Theorem holds. However, a pseudolnear endomorphsm may fal to self-commute; as a consequence, not all pseudolnear endomorphsms can be made to stablze a flag. Ths can be The frst author was supported by AST fundngs from La Sapenza Unversty. 1

2 2 A. D ANDREA AND G. MARCHEI made precse: f f gc M, where M s a fnte H-module, then f can be put n upper trangular form f and only f f generates a solvable subalgebra f of the Le pseudoalgebra gc M; moreover, ts acton decomposes M nto a drect sum of generalzed egenspaces f and only f f s nlpotent. There are examples of f such that f s not solvable. One may also choose f so that f s solvable but not nlpotent. The frst part of ths paper s devoted to showng that whenever S = f gc M s solvable, f s not too far from generatng a nlpotent subalgebra of gc M. More precsely, one may always fnd f f mod S such that f s nlpotent. Therefore, even though f may fal to decompose M nto a drect sum of generalzed egenspaces, a (non-unque) sutable modfcaton of f certanly does. We expect ths fact to be useful towards the study of some class of subalgebras of the Le pseudoalgebra gc M, where M s a fnte H-module, e.g., subalgebras of gc M, all of whose nonzero subalgebras contan a nonzero self-commutng element. In the second half of the paper, we employ ths fact towards characterzng fnte vertex algebras by studyng the adjont representaton of the underlyng Le conformal algebra. Vertex algebras that are of nterest n physcs are very large objects, and are typcally graded vector spaces of superpolynomal growth. It s well known that fnte-dmensonal vertex algebras collapse to dfferental commutatve algebra structures; however, nfnte-dmensonal examples of low growth are less well understood. One of the authors showed n [4] that all fnte vertex algebras V possess a solvable underlyng Le conformal algebra V Le, and that all generalzed weght space (of nonzero weght) wth respect to the adjont acton of any subalgebra of V Le are nl-deals of the vertex algebra structure. A more precse descrpton can be obtaned by mmckng the root space decomposton technque n ths new settng: f V s a fnte vertex algebra, choose a generc element n V Le, and modfy t so that t generates a nlpotent subalgebra of V Le. Then, decompose V nto drect sum of generalzed weght spaces. All nonzero weghts result n abelan vertex deals, whereas the generalzed 0-weght space s a vertex subalgebra U of V wth the property that U Le s nlpotent and self-normalzng n V Le,.e., t s a Cartan subalgebra of V. Then V decomposes nto a semdrect product of U wth a canoncally determned abelan deal N; namely, N s the deal of V on whch the central seres of V Le stablzes. Fnally, we show by an explct example that N may fal to vansh. Ths shows that there exst fnte vertex algebras whose underlyng Le conformal algebra s not nlpotent. The general phlosophy s that vertex algebras naturally tend to be very large objects. Because of ths, the algebrac requrement that fntely many quantum felds close, up to C[ ]-lnear combnaton, under normally ordered product and λ-bracket, forces some form of nlpotence on the structure; we descrbe the exact form of ths nlpotence n Theorem PRELIMINARIES ON LIE PSEUDOALGEBRAS In ths paper we wll work over an algebracally closed feld k of zero characterstc. Unless otherwse specfed, all vector spaces, lnear maps and tensor product wll be consdered over k Hopf algebras and Le pseudoalgebras. Let H be a cocommutatve Hopf algebra [9] wth coproduct, (h) = h (1) h (2), count ɛ and antpode S. The tensor product H H can be made nto a rght H-module by α.h = α (h), where α H H, h H. If L s a left H-module, t makes then sense to consder (H H) H L, along wth ts natural left H H-module structure. A Le pseudoalgebra over H s a (left) H-module L together wth a pseudobracket,.e., an H H-lnear map [ ] : L L (H H) H L a b [a b]

3 satsfyng skew-commutatvty and the Jacob dentty A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS 3 [b a] = (σ H d L )[a b], (1) [[a b] c] = [a [b c]] ((σ d) H d)[b [a c]], for all choces of a, b, c L. Here, σ : H H H H denotes the permutaton of factors, σ(h k) = k h, and (1) takes place n (H H H) H L, once we extend the pseudobracket so that [((h k) H r) s] = (h k 1)( d)(f g ) H t, [r ((h k) H s)] = (1 h k)(d )(f g ) H t, f [r s] = (f g ) H t, where f, g, h, k H, r, s, t L. If L, M are Le pseudoalgebras over H, then an H-lnear map f : L M s a Le pseudoalgebra homomorphsm f [f(a) f(b)] = ((d d) H f)[a b], for all a, b L. A Le pseudoalgebra L s fnte f t s fntely generated as an H-module. Example 2.1. If H = k then H H H and = d. In ths case the noton of Le pseudoalgebra over k s equvalent to the ordnary noton of a Le algebra. Example 2.2. Let d = k be a one dmensonal Le algebra. Then H = U(d) = k[ ] has a standard cocommutatve Hopf algebra structure. In ths case the axoms of Le pseudoalgebra over H are equvalent to the axoms of Le conformal algebra [7, 8]. The equvalence between pseudobracket and λ-bracket s gven by [a b] = P ( 1, 1 ) H c [a λ b] = P ( λ, + λ)c Hopf algebra notatons. Throughout the rest of the paper d wll denote a fnte-dmensonal Le algebra, and H = U(d) ts unversal envelopng algebra. H s a Noetheran doman and possesses a standard Hopf algebra structure satsfyng ( ) = 1 + 1, S( ) =, d. If dm d = N and { } N =1 s a bass of d then (I) = N N 1!... N!, I = ( 1,..., N ) N N, s a k-bass of H, by the Poncarè-Brkoff-Wtt Theorem. The coproduct satsfes (2) ( (I) ) = (J) (K). J+K=I Recall that H has a canoncal ncreasng fltraton gven by F n H = span k { (I) I n}, n = 0, 1, 2,... where I = N f I = ( 1,..., N ). Ths fltraton satsfes F 1 H = {0}, F 0 H = k, F 1 H = k d. We wll say that elements n F H \ F 1 H have degree. Due to (2), (h) 1 h H F 1 H f h has degree. Remark 2.1. It s easy to check that h k = (hs(k (1) ) 1) (k (2) ), for all h, k H, hence every element of H H can be expressed n the form (h 1) (l ), where h, l H.

4 4 A. D ANDREA AND G. MARCHEI Smlarly, whenever M s an H-module, elements from (H H) H M can be straghtened to the form (h 1) H m. Notce that both the h and the m can be chosen to be lnearly ndependent. Lemma 2.1. The lnear map τ M : (H H) H M H M defned by τ M ((h k) H m) = hs(k (1) ) k (2) m s an somorphsm of vector spaces. Proof. It s clearly well defned, and h m (h 1) H m s ts nverse. Remark 2.2. When M = H, the above lemma shows nvertblty of the map h k hs(k (1) ) k (2). Its nverse F : h k hk (1) k (2) s a lnear endomorphsm of H H, called Fourer transform n [1]. Corollary 2.1. Let M be an H-module, α = (f g ) H m Hom k (H, H). Then the element s well defned. α γ = γ(f S(g (1)))g (2)m M (H H) H M, γ Proof. The map τ M from the prevous lemma maps α (H H) H M to f S(g(1) )) g(2) m H M. The element α γ s then obtaned by applyng µ (γ d M ), where µ : H M M s the H-module structure map. Elements of the form α γ are called coeffcents of α. Remark 2.3. It s worth notcng that f α = (h 1) H c, and the h are lnearly ndependent over k, then all elements c can be realzed as coeffcents of α; namely c = α γ, where γ (h j ) = δ j. In partcular, α les n (H H) HS, where S s an H-submodule of L, f and only f c S for all. Let L be a Le pseudoalgebra over H. For any choce of A, B L, set [A, B] to be the smallest H-submodule S L such that [a b] (H H) H S for all a A, b B. Due to Remark 2.3, [A, B] s the H-submodule generated by coeffcents of all [a b], a A, b B. A subspace S L s a subalgebra of L f [S, S] S. If X s a subset of the Le pseudoalgebra L, then X denotes the subalgebra generated by X,.e., the smallest subalgebra of L contanng X. Defne the derved seres of L as L (0) = L, L (1) = [L, L], L (n+1) = [L (n), L (n) ]. Smlarly, the central seres of L s defned by L [0] = L, L [1] = [L, L], L [n+1] = [L, L [n] ]. A Le pseudoalgebra L s solvable (resp. nlpotent) f L (n) (resp. L [n] ) equals (0) for some n; L s abelan f the derved subalgebra L = [L, L] equals (0). An deal of a Le pseudoalgebra L s a subspace I such that [L, I] I. The centre Z(L) of L s the space of all elements z L such that [z, L] = (0). Every H-submodule of Z(L) s an deal. If N s a central deal of L, then L s nlpotent f and only f L/N s nlpotent. Lemma 2.2. Let H be a cocommutatve Hopf algebra, M an H-module. Assume α H H and m M s not a torson element. Then α H m = 0 f and only f α = 0. Proof. Let Hm be the cyclc module generated by m. Assumng that m s not torson s equvalent to requrng that the map φ : H Hm, φ(h) = hm s an somorphsm of vector spaces. Let α = h k H H. By Lemma 2.1 we have τ M ((α H m)) = h S(k (1)) k (2)m H Hm H M. Applyng the nvertble map F (d H φ 1 ) : H Hm H H to ths element gves back α.

5 A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS The general lnear pseudoalgebra. Let L be a Le pseudoalgebra over H. A representaton of L, or L-module, s an H-module V endowed wth an H H-lnear acton such that, for every a, b L, v V, L V a v a v (H H) H V, [a b] v = a (b v) ((σ d) H d)(b (a v)), whch s understood as n (1). An L-module V s fnte f t s fntely generated as an H-module. Let V be a representaton of the Le pseudoalgebra L. If A L, X V, then set A X to be the smallest H-submodule N of V such that a v (H H) H N for all a A, v X. By Remark 2.3, A X s the H-submodule of V generated by all coeffcents of a v, a A, v X. An H-submodule W V s stable under the acton of a L f a W W. It s an L- submodule of V f L W W. An L-module V s rreducble f t does not contan any nontrval L-submodule. If U and V are two L-modules, then a map φ : U V s a homomorphsm of L-modules f t s H-lnear and satsfes a φ(u) = ((d d) H φ)(a u), for all a L, u U. Let V, W be two H-modules. A map f : V (H H) H W s a pseudolnear map from V to W f t s k-lnear and satsfes f(hv) = (1 h) f(v), h H, v V. The space Chom(V, W ) of all pseudolnear maps from V to W has a left H-module structure gven by (hf)(v) = (h 1) f(v). If V = W we set Cend V = Chom(V, V ). If V s a fnte H-module then there exsts a unque Le pseudoalgebra structure on Cend V makng V a representaton of Cend V va the acton f v = f(v). Ths Le pseudoalgebra s usually denoted by gc V, and makng a fnte H-module V nto a representaton of a Le pseudoalgebra L s equvalent to gvng a Le pseudoalgebra homomorphsm from L to gc V. Example 2.3. Any Le pseudoalgebra L over H s a module over tself va a b := [a b], a, b L. When L s fnte, the adjont acton defnes a Le pseudoalgebra homomorphsm ad : L gc L whose kernel equals Z(L). Notce that L s nlpotent f and only f L/Z(L) s nlpotent. Remark 2.4. If f Chom(V, W ), then f v = 0 as soon as v Tor V. The adjont acton of any gven a L nduces an element ad a Chom(L [n] /L [n+1], L [n+1] /L [n+2] ). Assume L s a fnte Le pseudoalgebra, and L [n], L [n+1] have the same rank. Then the quotent L [n] /L [n+1] s torson, whence ad a = 0 for all a L. Ths forces L [n+1] = L [n+2], and we conclude that the central seres of any fnte Le pseudoalgebra stablzes to an deal, that we denote by L [ ] Acton of coeffcents. If a, b are elements of a Le pseudoalgebra L over H, t may be useful to know the acton of coeffcents [a b] γ, as defned n Corollary 2.1, on an L-module M. Lemma 2.3. Let L be a Le pseudoalgebra over H, M an L-module. Choose a, b L, and set α = [a b]. Assume that [a b] v = (k l m ) H v, where k, l, m, H and v M. If γ Hom k (H, H), then (3) α γ v = (γ(k S(l (1)))l (2) m ) H v.

6 6 A. D ANDREA AND G. MARCHEI Proof. The assgnment k l m (γ(ks(l (1) ))l (2) m) H u extends to a well-defned lnear map φ γ : (H H H) H M (H H) H M. Moreover, f [a b] = (h 1) H c, where the h are lnearly ndependent, and c v = j (kj l j ) H v j, then (4) [a b] v =,j (h S(k j (1) ) kj (2) lj ) H v j. If we choose γ (h j ) = δ j, whch s possble by lnear ndependence of elements hj, then φ γ recovers from (4) the expresson (3) for the acton of c = c γ on v. The general statement follows by H H-lnearty of the pseudobracket. 3. REPRESENTATIONS OF SOLVABLE AND NILPOTENT LIE PSEUDOALGEBRAS In the followng two sectons we recall some results from [1] about representaton theory of solvable and nlpotent Le pseudoalgebras Weght vectors. Let L be a Le pseudoalgebra over H and M be an L-module. If φ Hom H (L, H), the weght space M φ s defned as M φ = {v M a v = (φ(a) 1) H v, for all a L}. If M φ 0, then φ s a weght for the acton of L on M. Every nonzero element of M φ s a weght vector of weght φ. Remark 3.1. The weght space M 0 s always an H-submodule of M, whereas M φ, φ 0 s just a vector subspace. However, n ths case, the H-submodule HM φ M s free over M φ. We have the followng pseudoalgebrac analogues of Le s Theorem: Theorem 3.1. Let L be a solvable Le pseudoalgebra over H. Then every fnte non-trval L-module has a weght vector. Corollary 3.1. If L s a solvable Le pseudoalgebra over H and M s a fnte L-module then M has a fnte fltraton by L-submodules (0) = M 0 M 1 M n = M such that each quotent M +1 /M s generated over H by a weght vector for the acton of L. The length of an L-module M s the mnmal length of a fltraton as above Generalzed weght submodules. Let L be a Le pseudoalgebra over H, φ Hom H (L, H) = L. Lemma 3.1. Let M be a fnte L-module. If N M φ s a vector subspace, then HN s an L-submodule of M. Proof. Let n N. Then, for any h H, a hn = (1 h)(a n) = (φ(a) h) H n (H H) H HN. We set M φ 1 = (0) and nductvely M φ +1 = span H{m M a m (φ(a) 1) H m (H H) H M φ, a L}. Then M φ 0 = HM φ and M φ +1 /M φ = H(M/M φ ) φ. The M φ form an ncreasng sequence of H- submodules of M. By Noetheranty of M ths sequence stablzes to an H-submodule M φ = M φ of M, whch s called the generalzed weght submodule relatve to the weght φ. We wll occasonally stress the dependence of M φ on the choce of the Le pseudoalgebra actng on M by wrtng M φ L, or smply M φ a when L = a s the subalgebra of gc M generated by a sngle element a. Proposton 3.1. Let N M be fnte L-modules, φ, ψ L. Then M φ s an L-submodule of M, and s a free H-module whenever φ 0.

7 A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS 7 The sum of generalzed weght submodules of M s always drect. In partcular M φ M ψ = (0) f φ ψ. N M φ f and only f N φ = N. (M/M φ ) φ = (0). If N M φ, then N = M φ f and only f (M/N) φ = (0). Proof. M φ s an L-submodule by constructon. It can be shown to be a free H-module by nducton, usng Remark 3.1. All other statements follow easly from ther Le theoretc analogues by renterpretng M as a representaton of the annhlaton Le algebra L = H H L, see [1]. Notce that the drect sum M φ may fal to equal M. Equalty, however, always holds φ L when L s nlpotent. Theorem 3.2. Let L be a nlpotent Le pseudoalgebra over H and M be a (fathful) fnte L- module. Then M decomposes as a drect sum of ts generalzed weght submodules, M = φ L M φ Nlpotent pseudoalgebras. We am to show that the converse to Theorem 3.2 also holds, at least when L s fnte. We wll frst prove the result when M concdes wth one of ts generalzed weght spaces wth respect to the acton of L. The general statement wll then follow easly. Proposton 3.2. Let M be a fnte H-module and L gc M be a Le pseudoalgebra, φ L, and assume that M concdes wth ts φ-generalzed weght space wth respect to the acton of L. Then L s a nlpotent Le pseudoalgebra. Proof. Let L [0] = L, L [] = [L, L [ 1] ], 1, be the the central seres of L and {M k } be an ncreasng famly of L-submodules of M as n Corollary 3.1. As a warm up, let us treat the case φ 0 frst. An easy nducton shows that L [] M k M k 1. Indeed, as M = M 0, we have L M k M k 1 by our choce of M k. Snce L [0] = L, ths takes care of the bass of nducton = 0. Assume now that L [] M k M k 1. Then L [+1] M k = [L, L [] ] M k = L (L [] M k ) + L [] (L M k ) L M k 1 + L [] M k 1 M k 2. In order to conclude the proof t s enough to observe that f M = M n then, for = n, we have L [n 1] M n M 0 = (0). Ths mples L [n 1] = 0,.e., L s a nlpotent Le pseudoalgebra. If φ 0, the stuaton s slghtly more delcate. For 1 r m, let m r denote the cyclc generators of the quotents M r /M r 1. Set N j k = {b L b m r (H F j H) H m r k + M r k 1, r = 1,..., N}, and N k = N j k, so that N j k N k+1 f j < 0. Notce that N 0 = L, N n+1 = (0). j N We am to prove, by nducton on p, that [L [p], N j k ] N j p 1 k. Let us start wth the bass of our nducton: [L [0], N j k ] = [L, N j k ] N j 1 k. Let a L, b N j k. We know that a m r = (φ 1) H m r mod M r 1, for r = 1,..., n. Assume that b m r = (h k ) H m r k mod M r k 1, where k F j H for all. Let us compute [a b] m r = a (b m r ) ((σ d) H d)(b (a m r )) = a ( (h k ) H m r k ) ((σ d) H d)(b ((φ 1) H m r )) = (φ h k φk(1) h k(2) ) Hm r k

8 8 A. D ANDREA AND G. MARCHEI up to terms n a M r k 1 + b M r 1 M r k 1. Now recall that, as k F j H, then (k ) 1 k H F j 1 H. We conclude that all coeffcents of [a b] le n N j 1 k. As for the nductve step, assume now that [L [p], N j k ] N j p 1 k. Then we have [L [p+1], N j k ] = [[L, L[p] ], N j k ] [L, [L[p], N j k ] + [L[p], [L, N j k ]] [L, N j p 1 k ] + [L [p], N j 1 k ] N j p 2 k. Snce L s a fnte Le pseudoalgebra, there exsts d such that N k = Nk d for all k. Then we obtan [L [d], N k ] N 1 k N k+1. As a consequence, L [(n+1)d+1] = [L [(n+1)d], N 0 ] = (0), whch proves that L s a nlpotent Le pseudoalgebra. Theorem 3.3. Let M be a fnte fathful module over a fnte Le pseudoalgebra L, and assume that M = φ L M φ. Then L s a nlpotent Le pseudoalgebra. Proof. Observe that under the assumpton M = M φ we have L gc(m φ ). By Proposton 3.2, the mage L φ of L n gc(m φ ) s nlpotent for all φ L. As a φ L φ L consequence, φ L L φ s nlpotent as t s a fnte sum of nlpotent Le pseudoalgebras. Fnally, L s a nlpotent Le pseudoalgebra as t embeds n L φ. φ L Example 3.1. The fnteness assumpton on L n the statement of Theorem 3.3 cannot be removed. Indeed, let M = Hm 1 + Hm 2 be a free H-module of rank 2, and choose L gc M to be the Le pseudoalgebra of all pseudolnear maps A gc M such that A m 1 = (φ(a) 1) H m 1, A m 2 = (φ(a) 1) H m 2 mod (H H) H m 1, for some φ(a) H. Then the central seres of L stablzes to L, whch contans all A such that φ(a) = 0. Later on, we wll deal wth fnte vertex algebras, and the followng pseudoalgebrac analogue of Engel s theorem wll turn out to be useful. Theorem 3.4. Let L be a fnte Le pseudoalgebra over H. Assume that, for every a L, the generalzed weght submodule L 0 a for the adjont acton of a equals L. Then L s a nlpotent Le pseudoalgebra. 4. APPROXIMATE NILPOTENCE OF SOLVABLE SUBALGEBRAS OF gc M In ths secton we present the followng result for 1-generated solvable subalgebras of gc M: Theorem 4.1. Let M be a fntely generated H-module. If a gc M generates a solvable subalgebra S = a, then there exsts ā S, ā a mod S, such that the subalgebra ā s nlpotent. We wll later specalze ths result to gve a characterzaton of fnte vertex algebras The length 2 case. Let M be a fnte H-module and a gc M an element generatng a solvable Le pseudoalgebra a = S. A modfcaton of a S s an element ā S such that a ā mod S. It follows by defnton that the subalgebra generated by ā s a subalgebra of S. The same ncluson holds for the correspondng derved subalgebras. As a consequence, a modfcaton of a modfcaton of a s stll a modfcaton of a. Remark 4.1. Let φ S be a weght for the acton of S = a on M. Then S = Ha + S, and the restrcton of φ to S vanshes. Ths means that φ s unquely determned by φ(a). As a consequence, φ(a) = φ(ā) whenever ā s a modfcaton of a.

9 A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS 9 Remark 4.2. Let M be a fnte H-module, S be a solvable Le pseudoalgebra generated by a gc M and N M an S-submodule. Then N s stable under the acton of any modfcaton ā of a, as ā belongs to S. Proposton 4.1. Let M be a fnte H-module and a gc M such that S = a s a solvable subalgebra. Assume that M = Hu + Hv, where u M s a φ weght vector and [v] M/Hu s a ψ weght vector for the acton of S, for some φ ψ S. Then there exsts a lftng v M of [v] such that H v s a complement of Hu n M and s stable under the acton of some modfcaton ā of a. Lemma 4.1. Under the same hypotheses as above, let b gc M be such that b u = 0, b v = (β k) H u, where β, k H, and the degree of k s K. Then some coeffcent s of [a b] satsfes s u = 0, s v = (1 k) H u mod (H F K 1 H) H u. Proof. A drect computaton gves [a b] v = (φ(a) β k ψ(a)k (1) β k (2) ) H u = (α β k) H u, up to terms n (H F K 1 H) H u. By Lemma 2.3, we have [a b] γ v = γ(αs(β (1) ))β (2) k) H u mod (H F K 1 H) H u. It now suffces to choose γ Hom k (H, H) so that γ(αs(β)) equals 1, and requrng that t vansh on all terms of lower degree. Proof of Proposton 4.1. We know that a u = (φ(a) 1) H u a v = (ψ(a) 1) H v mod (H H) H u. We may then fnd K N, and lnearly ndependent elements k H of degree K, such that a v = (h k ) H u + (ψ(a) 1) H v mod (H F K 1 H) H u. We may assume that the h are lnearly ndependent as well. By a drect computaton we obtan, modulo terms n (H H F K 1 H) H u, a (a u) = (φ(a) φ(a) 1) H u, a (a v) = (φ(a) h k ) H u + (h ψ(a)k(1) k (2) ) Hu + (ψ(a) ψ(a) 1) H v = (φ(a) h k ) H u + (h ψ(a) k ) H u + (ψ(a) ψ(a) 1) H v, so that [a a] u = 0 and [a a] v = (α h k h α k ) H u mod (H H F K 1 H) H u, where α = φ(a) ψ(a) H s a nonzero element of degree N. Let D be the maxmal degree of the h. We now proceed by nducton on K, on D and on the rank of h k H H. We dstngush three cases: (1) D > N. Then we may choose γ Hom k (H, H) such that γ(α) = 1 and obtan [a a] γ v = (γ(αs(h (1) ))h (2) k γ(h S(α (1) )α (2) k ) H u, = (h k ) H u modulo terms n (F N 1 H F K H + H F K 1 H) H u. The modfcaton a [a a] γ then leads to a coeffcent h k of lower degree n the frst tensor factor.

10 10 A. D ANDREA AND G. MARCHEI (2) N D and α / span k h. Choose γ such that γ (h ) = δ j, γ(α) = 0. Then [a a] γ v = (γ(αs(h (1) ))h (2) k γ(h S(α (1) ))α (2) k ) H u, = (α k ) H u, modulo terms n (F N H k + H F K 1 ) H u. Ths shows that some coeffcent b of [a a] acts on v so that b v = (β k ) H u mod (H F K 1 H) H u, for some nonzero β H. By Lemma 4.1 we may then fnd, for each, some element s S such that s v = (1 k ) H u mod (H F K 1 H) H u. The element a h s s then a modfcaton of a leadng to a lower value of K. (3) N D and α span k h. In ths case we can fnd c k such that α = c h. Choose j so that c j 0, and set v = v c 1 j k j u. Then a v = (φ(a) c 1 j k j ) H u + = ( φ(a) c 1 j k j + = ( c 1 j φ(a) k j + (h k ) H u + (ψ(a) 1) H v h k ) H u + (ψ(a) 1) H (c 1 j k j u) + (ψ(a) 1) H v h k ) H u + (c 1 j ψ(a)k j (1) kj (2) ) Hu + (ψ(a) 1) H v = ((h j c 1 j α) k j + j(h k )) H u + (ψ(a) 1) H v, modulo terms n (H F K 1 H) H u. The element h j c 1 j α s a lnear combnaton of the h, j, and so the rank of the H H-coeffcent multplyng u s lower than that of h k. We may now apply nducton. Remark 4.3. Notce that, n the above proof, the coeffcent h k s not unquely determned, n case u s a torson element of M. However, the proof works equally well for any gven choce of such a coeffcent Proof of the general statement. Proposton 4.2. Let M be a fnte H-module and a gc M such that S = a s a solvable subalgebra. Assume that φ ψ S are such that M/M φ = (M/M φ ) ψ. Then there exsts an H-submodule M M whch s a complement to M φ and s stable under some modfcaton of a. Proof. We start by consderng the case when the length of the S-module M/M φ s 1, and proceed by nducton on the length n of M φ. The bass of nducton n = 1 s provded by Proposton 4.1, so we assume that the length of M φ equals n > 1. Choose u M φ such that M φ /Hu has length n 1. We use nducton on the S-module M/Hu to fnd a complement N/Hu to M φ /Hu = (M/Hu) φ whch s stable under the acton of some modfcaton ã of a. Notce that N φ = N M φ = Hu and that N/Hu s somorphc to (M/Hu)/(M φ /Hu) M/M φ, hence we may apply Proposton 4.1 to N and fnd a complement M N to Hu whch s stable under some modfcaton ā of ã. Now, M φ + M = M φ + Hu + M = M φ + N = M; moreover M φ M M φ N = Hu so that M φ M M Hu = (0). We conclude that M s a complement of M φ n M that s stable under the acton of ā, whch s a modfcaton of a. We proceed now wth provng the statement when the length m of M/M φ s greater than 1. Choose N = N/M φ M/M φ of length m 1 so that M/N has length 1; as M/M φ =

11 A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS 11 (M/M φ ) ψ, then N = N ψ. Snce N φ = N M φ = M φ, we may use nducton to fnd an H-submodule N N whch s a complement to M φ and s stable under some modfcaton a of a. Consder now the quotent M = M/N. Then (M ) φ certanly contans the mage (M φ + N )/N of M φ under the canoncal projecton π : M M/N. Moreover, (M/N )/((M φ + N )/N ) s somorphc to M/(M φ + N ) and s therefore a quotent of M/M φ, whch equals ts ψ-generalzed weght space. As ψ φ, we conclude that (M ) φ = (M φ + N )/N, and that M /(M ) φ M/(M φ +N ) = M/N has length one. We may then fnd a complement M/N of (M ) φ n M whch s stable under some modfcaton ā of a. We clam that M s a complement of M φ n M. Indeed, M/N + (M ) φ = M, hence M + (M φ + N ) = M; as N M, we conclude that M = M + M φ. On the other hand, M M φ = N, hence M M φ N M φ = (0). We are now ready to prove our central result. Proposton 4.3. Let M be a fnte H-module and S be a solvable Le pseudoalgebra generated by a gc M. Then there exsts a modfcaton ā of a such that M decomposes as a drect sum of generalzed weght modules wth respect to S = ā. Proof. By nducton on the length of M. If the length equals 1, then M = M φ for some φ and there s nothng to prove. Let us assume that the length of M s n > 1. We may fnd a weght vector u M φ such that N = M/Hu has length n 1. By nductve assumpton, N decomposes as a drect sum N φ 1 N φr of (non trval) generalzed weght modules wth respect to the subalgebra S S generated by some modfcaton ã of a. Let N be the premage of N φ under the canoncal projecton π : M M/Hu, and reorder ndces so that φ φ for all r. As long as φ k φ, we may repeatedly apply Proposton 4.2 to obtan complements M to (N ) φ = Hu n N so that the sum M M k s drect and all summands are nvarant wth respect to some terated modfcaton of a. If φ r φ holds as well, we end up wth M = M 1 M r 1 M r Hu; f nstead φ r = φ, then M = M 1 M r 1 N r. In both cases, all summands are generalzed weght spaces by constructon, and are stable wth respect to some modfcaton ā of a. In the lght of Theorem 3.3, we see that Theorem 4.1 s just a restatement of Proposton 4.3. Let ā be a modfcaton of a generatng a nlpotent subalgebra of gc M. A natural queston to ask s whether the decomposton M = M φ ā depends on ā or s nstead canoncal. Ths amounts to askng f all such modfcatons of a are contaned n a sngle nlpotent subalgebra of a. We wll answer ths n the negatve at the very end of the paper. Corollary 4.1. Let L be a Le pseudoalgebra over H. If a L generates a fnte solvable subalgebra, then some modfcaton of a generates a nlpotent subalgebra. Proof. Let S = a. The adjont acton of S gves rse to a homomorphsm ad : S gc S of pseudoalgebras whose kernel equals the centre Z(S) of S. Moreover, ad S s a solvable subalgebra of gc S generated by ad a. By Theorem 4.1 we may fnd n ad S a modfcaton of ad a generatng a nlpotent subalgebra N of gc S. Such a modfcaton s of the form ad ā, where ā s a modfcaton of a. Then N s somorphc to the quotent of S = ā by a central as t s contaned n Z(S) deal. We conclude that S s nlpotent. Theorem 4.1 has some nterestng consequences. Proposton 4.4. Let M be a fnte H-module, S be a solvable Le pseudoalgebra actng on M. If φ S s nonzero and U M φ s an S-submodule, then there exsts an H-lnear secton s : M/U M. In partcular:

12 12 A. D ANDREA AND G. MARCHEI If M s torson-free, then M/U s torson-free; If M/U s free, then M s free. Proof. Choose a S such that φ(a) 0 and a s nlpotent. We may then replace S wth a, and assume M = M φ M φ 1 M φr. Then M φ /U s free as an H-module, and we can fnd a secton s : M φ /U M φ. Ths extends to a secton s : M/U M thanks to the drect sum decomposton. Corollary 4.2. Let M be a fnte H-module, S be a solvable Le pseudoalgebra actng on M. If M s not free, then some quotent of M has a 0-weght vector for the acton of S. 5. STRUCTURE OF FINITE VERTEX ALGEBRAS 5.1. Prelmnares on vertex algebras. Let V be a complex vector space. A (quantum) feld on V s a formal power seres φ(z) (EndV )[[z, z 1 ]] such that φ(z)v V ((z)) = V [[z]][z 1 ]. In other words, φ(z) = n Z φ (n) z n 1 s a quantum feld f and only f, for every choce of v V, φ (n) v = 0 for (dependng on v) suffcently hgh values of n. A vertex algebra s a complex vector space V, endowed wth a vacuum vector 1 V, an nfntesmal translaton operator T EndV, and a C-lnear state-feld correspondence Y : V (EndV )[[z, z 1 ]] mappng each element a V to some feld Y (a, z) on V, satsfyng, for all choces of a, b V, Y (1, z)a = a, Y (a, z)1 = a mod zv [[z]]; (vacuum axom) Y (T a, z) = [T, Y (a, z)] = dy (a, z)/dz; (translaton nvarance) (z w) N [Y (a, z), Y (b, w)] = 0, for some N = N(a, b). (localty) It s well known that commutators [Y (a, z), Y (b, w)] may be expanded nto a lnear combnaton of the Drac delta dstrbuton and of ts dervatves. More precsely, f δ(z w) = n Z w n z n 1, Y (a, z) = n Z a (n) z n 1, then [Y (a, z), Y (b, w)] = N(a,b) 1 j=0 Y (a (j) b, w) j! d j δ(z w). dwj It s also possble to defne the Wck, or normally ordered, product of quantum felds where : Y (a, z)y (b, z) = Y (a, z) + Y (b, z) + Y (b, z)y (a, z), Y (a, z) = n N a (n) z n 1, Y (a, z) + = Y (a, z) Y (a, z). Then one has Y (a ( n 1) b, z) = 1 n! : Y (T n a, z)y (b, z) :, for all n 0. One of the consequences of the vertex algebra axoms s the followng: Y (a, z)b = e zt Y (b, z)a (skew-commutatvty)

13 A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS 13 for all choces of a, b. Every vertex algebra has a natural C[T ]-module structure. A vertex algebra V s fnte f V s a fntely generated C[T ]-module. A C[T ]-submodule U V s a subalgebra of the vertex algebra V f 1 U and a (n) b U for all a, b U, n Z. Smlarly, a C[T ]-submodule I V s an deal f a (n) I for all a V, I, n Z. A subalgebra U V s abelan f Y (a, z)b = 0, or equvalently a (n) b = 0, for all a, b U, n Z. It s commutatve f [Y (a, z), Y (b, w)] = 0, or equvalently a (n) b = 0, for all a, b U, n N. Let U be a subalgebra, and I an deal of a vertex algebra V ; we say that V s the semdrect sum of U and I (denoted V = U I) f V = U I s a drect sum of C[T ]-submodules. Every vertex algebra becomes a Le conformal algebra, see [7], after settng = T and [a λ b] = n N λ n n! a (n)b. We have seen n Example 2.2 that the noton of Le conformal algebra s equvalent to that of Le pseudoalgebra over H = C[ ]. In ths settng, the pseudobracket s gven by [a b] = ( ) ( ) n 1 H a (n) b. n! n N If V s a vertex algebra, we wll denote by V Le the underlyng Le conformal algebra structure The nlradcal. In ths secton, we recall some propertes of nlpotent elements n a vertex algebra. Proofs can be found n [5, 6]. An element a n a vertex algebra V s nlpotent f Y (a, z 1 )Y (a, z 2 )... Y (a, z n )a = 0 for suffcently large values of n. An deal I V s a nl-deal f all of ts elements are nlpotent; clearly, every abelan deal of V s a nl-deal. Proposton 5.1. Let V be a vertex algebra. Then Every nlpotent element of V generates a nl-deal. The set Nl V of all nlpotent elements of V s an deal of V. The vertex algebra V/ Nl V contans no nonzero nlpotent elements. If V s fnte, then Nl V s a nl-deal of V. If V s a fnte vertex algebra, then V Le s always a solvable Le conformal algebra [4]. Recall that the central seres of V Le stablzes, by Remark 2.4, to a vertex deal V [ ] of V. The followng facts were proved n [4]. s a subal- Proposton 5.2. Let V be a fnte vertex algebra, S V Le a subalgebra. Then VS 0 gebra and V 0 S = V φ S s an abelan deal of V. φ S \{0} As a consequence, V 0 S Nl V. If V s fnte and Nl V = (0), then V Le s nlpotent; f moreover V s smple, then t s necessarly commutatve Root space decomposton of fnte vertex algebras. Let V be a fnte vertex algebra, a V. The subalgebra S = a V Le s always solvable. The adjont acton of S makes V nto a fnte S-module, and we can fnd submodules (0) = V 0 V 1 V n = V as n Corollary 3.1. The sngularty 1 of a s then the number of non-torson quotents V /V 1 wth a trval acton of S. Notce that the sngularty does not change under modfcatons of a. When S s nlpotent, then the sngularty of a equals the rank of Va 0 as an H-module. Theorem 5.1. Let V be a fnte vertex algebra and N = V [ ]. Then N s an abelan deal of V, and there exsts a subalgebra U V such that U Le s nlpotent and V = U N. 1 The sngularty of an element s, n other words, the multplcty of the zero egenvalue of ts adjont acton.

14 14 A. D ANDREA AND G. MARCHEI Proof. Choose an element a V of mnmal sngularty k. Up to replacng a by a sutable modfcaton, we may assume that S = a be a nlpotent subalgebra of V Le. Then, V decomposes as a drect sum of generalzed weght submodules, V = φ S V φ ā = V 0 a V 0 a. Then U = Va 0 s a vertex subalgebra of V and N = Va 0 s an abelan deal of V. We want to show that U Le s nlpotent and N = V [ ]. Say b U. As U V s a subalgebra, then U s stable under the acton of b. If the Ub 0 U, then some (generc) lnear combnaton of a and b would have lower sngularty than a, a contradcton. Thus, all elements n U Le have a nlpotent adjont acton, hence U Le s nlpotent by Theorem 3.4. It remans to prove that N = V [ ] : snce N s an deal such that V Le /N s nlpotent then V [ ] N. We prove the other ncluson by showng that N V [k] by nducton on k N, the bass of the nducton beng clear, as N V 0 = V. By constructon, [a, Va φ ] = Va φ f φ 0, hence [a, N] = N. Then N V k mples N = [a, N] [V, V [k] ] = V [k+1]. We conclude that N V [n] for all n, hence that N V [ ]. Remark 5.1. In the above statement, U s a vertex subalgebra of V wth the property that U Le s a nlpotent and self-normalzng subalgebra of V Le. As a consequence, the adjont acton of U Le on V gves a generalzed weght submodule decomposton n whch the 0-weght component s U tself. It makes sense to call every such U a Cartan subalgebra of V, and the correspondng decomposton a root space decomposton. Notce that N s the smallest nl-deal of V havng a complementary subalgebra U such that U Le s nlpotent; as N = V [ ], t s canoncally determned. If U s a Cartan subalgebra of V, then U = V/V [ ], so all Cartan subalgebras of V possess somorphc vertex algebra structures. We can be more precse. The dentfcaton of any two Cartan subalgebras U, U wth V/N gves an somorphsm φ : U U whch projects to the dentty on V/N. If we extend φ to all of V by settng t to be the dentty on N, then we obtan an automorphsm of V conjugatng U to U. Thus, all Cartan subalgebras of V are conjugated under Aut V A counterexample to nlpotence of fnte vertex algebras. The statement of Theorem 5.1 suggests how to construct a fnte vertex algebra V such that the correspondng Le conformal algebra V Le s not nlpotent. What we need s a vertex algebra U wth a nlpotent underlyng Le conformal algebra U Le, and a sutable acton on an C[ ]-module N. The smplest case s when U s a commutatve vertex algebra,.e., U Le s abelan, and N s a free C[ ]-module of rank 1. Let U = {a(t) C[[t]][t 1 ]}. U s a dfferental commutatve assocatve algebra wth 1, wth dervaton = d/dt. Hence U has a commutatve vertex algebra structure gven by (5) Y (a(t), z)b(t) = (e z a(t))b(t) = z < t a(t + z)b(t), where a(t), b(t) U and z < t (see [8]) ndcates that one should expand a(t + z) n the doman z < t,.e., usng postve powers of z/t. Let N = C[ ]n be a free C[ ]-module of rank 1. We set: (6) Y (n, z)n = 0, and defne an acton of U on N by settng (7) Y (a(t), z)n = a(z)n, where a(t) U. Theorem 5.2. There exsts a unque vertex algebra structure on the C[ ]-module V = U N such that (5), (6), (7) are satsfed. Moreover, the central seres of V Le stablzes to N.

15 A ROOT SPACE DECOMPOSITION FOR FINITE VERTEX ALGEBRAS 15 Proof. The unt element 1 U satsfes the vacuum axom, and all Y (v, z), v V are felds by defnton. Localty and translaton nvarance requre some more effort. The skew-commutatvty axom suggests that we set If further Y (n, z)a(t) = e z Y (a(t), z)n = a( z)e z n. Y (a(t), z) K n = K =1 ( ) K ( 1) a () (z) K n, then translaton nvarance s easly checked. Let us move on to provng localty. Frst of all, notce that Y (n, z)y (n, w) maps every element of V to 0, hence [Y (n, z), Y (n, w)] = 0. Takng dervatve wth respect to z and w, and usng lnearty, we obtan [Y (u, z), Y (u, w)] = 0 for all u, u N. Next, let us consder [Y (a(t), z), Y (b(t), w)]. An easy computaton gves Y (a(t), z)(y (b(t), w) n = Y (a(t), z)b(w)n = b(w)y (a(t), z)n = a(z)b(w)n, hence [Y (a(t), z), Y (b(t), w)]n = 0, for all a(t), b(t) U. As ([Y (a(t), z), Y (b(t), w)] u) = [Y (a (t), z), Y (b(t), w)] u + [Y (a(t), z), Y (b (t), w)] u + [Y (a(t), z), Y (b(t), w)] u, we conclude that [Y (a(t), z), Y (b(t), w)] vanshes on all elements from N. However, t also vanshes on U, because of ts vertex algebra structure. We are left wth showng that [Y (a(t), z), Y (n, w)] s klled by a suffcently large power of z w. Let us compute Therefore, Y (a(t), z)(y (n, w)b(t)) Y (n, w)(y (a(t), z)b(t)) = Y (a(t), z)b( w)e w n = b( w)y (a(t), z)e w n = b( w)e w (e w Y (a(t), z)e w )n = e w b( w)y (a(t), z w)n = w < z a(z w)b( w)e w n, = Y (n, w) z < t a(t + z)b(t) = z < t Y (n, w)a(t + z)b(t) = z < w a(z w)b( w)e w n. (z w) N [Y (a(t), z), Y (n, w)]b(t) = ( w < z z < w )((z w) N a(z w)b( w)e w n) s zero as soon as t N a(t) has no negatve powers of t. As [Y (a(t), z), Y (n, w)] maps every element of N to zero, localty s then proved. As for V [ ] = N, let a = a(t) = n Z a n t n 1 U such that a(t) contans some negatve power of t. Then there exsts n 0 such that a n 0, hence a(t) (n) n = a n n 0. Therefore [a, N] = N, hence N V [n] for all n. However, (V/N) Le s nlpotent, hence V [ ] N. Let us choose a fnte subalgebra of U whose conformal adjont acton on N has nonzero weghts,.e., contanng some element a(t) C[[t]]. Example 5.1. M = C[t 1 ] N U N s a fnte vertex algebra, as t s generated over C[ ] by t 1, 1 and n. However, M Le s not nlpotent. We conclude by observng that even though the nl-deal N n the decomposton stated n Theorem 5.1 s canoncally determned, the subalgebra U need not be. Indeed there may be several possble choces of U as the followng constructon shows.

16 16 A. D ANDREA AND G. MARCHEI Let M be as n Example 5.1, and choose u N. We know that u (0) s a dervaton of M, and as N s an abelan deal of M, we mmedately obtan u 2 (0) = 0. Recall that the exponental of such a nlpotent dervaton of a vertex algebra M gves an nner automorphsm of M. If we choose u = kn, k k, then exp(kn (0) )(t 1 ) = t 1 kn. Thus, f we set ψ = exp(kn (0) ), we obtan ψ(n) = N, ψ(u) = C[ ](t 1 kn) C1, and ψ(u) N = ψ(u N) = 0. We conclude that ψ(u) s another subalgebra of M whch complements N. Notce that n ths example all Cartan subalgebras can be showed to be conjugated by an nner automorphsm of V. It s not clear whether ths holds n general. One fnal comment s n order: t s easy to show that the Le conformal subalgebra of M Le generated by the element a = t 1 + n s solvable and equals C[ ]a + C[ ]n. As [a λ a] = ( + 2λ)n, we see that all elements t 1 kn, k k are modfcatons of a, and they generate nlpotent subalgebras of M Le. However, they C[ ]-lnearly span all of a, whose Le conformal algebra structure s solvable but not nlpotent. As the adjont homomorphsm ad : M Le gc M s njectve on a, we conclude that there s no sngle nlpotent subalgebra of gc M contanng all of the above modfcatons of a. REFERENCES [1] B. Bakalov, A. D Andrea, V. G. Kac, Theory of fnte pseudoalgebras, Adv. Math. 162 (2001), [2] A. Belnson, V. Drnfeld, Chral algebras, AMS Colloquum Publcatons, vol. 51, AMS, Provdence, RI, [3] R. E. Borcherds, Vertex algebras, Kac-Moody algebras and the monster, Proc. Nat. Acad. Sc. U.S.A. 83 (1986), [4] A. D Andrea, Fnte vertex algebras and nlpotence, J. of Pure and Appled Algebra 212, no. 4 (2008), [5] A. D Andrea, A remark on smplcty of vertex algebras and Le conformal algebras, J. Algebra 319, no. 5 (2008), [6] A. D Andrea, Commutatvty and assocatvty of vertex algebras, n Le Theory and ts Applcatons n Physcs VII, eds. H.-D. Doebner and V. K. Dobrev, Heron Press, Sofa. Bulg. J. Phys. 35-s1 (2008), [7] A. D Andrea, V. G. Kac, Structure theory of fnte conformal algebras, Selecta Math. (N.S.) 4 (1998), [8] V. G. Kac, Vertex Algebras for Begnners, Mathematcal Lecture Seres, AMS, Vol.10, 1996 (2nd ed. AMS, 1998). [9] M. Sweedler, Hopf algebras, Unversty Lecture Note Seres, W. A. Benjamn, Inc. New York, DIPARTIMENTO DI MATEMATICA, UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA, P.LE ALDO MORO, ROME, ITALY E-mal address: dandrea@mat.unroma1.t