Journal of Algebra 217, 496᎐527Ž. 1999 Article ID jabr.1998.7834, available online at http:rrwww.idealibrary.com on
On Free Conformal and Vertex Algebras
Michael Roitman*
CORE Department of Mathematics, Yale Uni¨ersity, New Ha¨en, Connecticut 06520 Metadata, citation and similar papers at core.ac.uk
Provided by Elsevier - Publisher Connector E-mail: [email protected]
Communicated by Efim Zelmano¨
Received April 1, 1998
Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representa- tions of the Monster. See, for example,wx 6, 10, 17, 15, 19 . In this paper we describe bases of free conformal and free vertex algebrasŽ as introduced inwx 6 ; see also w 20 x. . All linear spaces are over a field މ- of characteristic 0. Throughout this paper ޚq will stand for the set of non-negative integers. In Sections 1 and 2 we give a review of conformal and vertex algebra theory. All statements in these sections are either inwx 9, 17, 16, 15, 18, 20 or easily follow from results therein. In Section 3 we investigate free conformal and vertex algebras.
1. CONFORMAL ALGEBRAS
1.1. Definition of Conformal Algebras We first recall some basic definitions and constructions; seew 16, 15, 18, 20x . The main object of investigation is defined as follows:
DEFINITION 1.1. A conformal algebra is a linear space C endowed with a linear operator D: C ª C and a sequence of bilinear products"n : C m C ª C, n g ޚq, such that for any a, b g C one has
* Partially supported by NSF Grant DMS-9704132.
496 0021-8693r99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. CONFORMAL AND VERTEX ALGEBRAS 497
Ž.i Ž Locality . There is a non-negative integer N s Na Ž, b .such that ab"n s 0 for any n G N; Ž.ii Da Ž"""nnn b .s ŽDa . b q aDb Ž .; VWUT Ž.iii ŽDa ."n syna n y 1 b.
1.2. Spaces of Power Series Now let us discuss the main motivation for Definition 1.1. We closely followwx 14, 18 .
1.2.1. Circle Products Let A be an algebra. Consider the space of power series Azww , zy1 xx.We will write series a g Azww , zy1 xx in the form
azŽ.s Ý anz Ž.yny1 , an Ž.g A. ngޚ
ww y1 xx On Az, z there is an infinite sequence of bilinear products"n , n g ޚq, given by
s y n Ž.abz"n Ž.Res w Žawbz Ž .Ž.Ž z w ...1.1Ž.
s yny1 s Explicitly, for a pair of series azŽ.Ýng ޚ anz Ž. and bzŽ. yny1 Ýng ޚ bnzŽ. we have
ymy1 Ž.abz""nnŽ.s Ý Ž.abmzŽ. , m where
n s abmn Ž.sy Ž1 .n an Žy sbm .Žq s ..1.2 Ž . Ž." Ý ž/s ss0
There is also the linear derivation D s drdz: Azww , zy1 xxª Az ww , zy1 xx. It is easy to see that D and"n satisfy conditionsŽ. ii and Ž iii . of Definition 1.1. We can consider formulaŽ. 1.2 as a system of linear equations with unknowns akblŽ.Ž., k g ޚq, l g ޚ. This system is triangular, and its unique solution is given by
k akblŽ.Ž.s k abks Ž.Ž.q l y s .1.3 Ý ž/s Ž." ss0 498 MICHAEL ROITMAN
Remark. The term ‘‘circle products’’ appears inwx 18 , where the product ( wx ‘‘"n ’’ is denoted by ‘‘ n .’’ In 15 this product is denoted by ‘‘Ž n. .’’
1.2.2. Locality Next we define a very important property of power series, which makes them form a conformal algebra. Let again A be an algebra. y DEFINITION 1.2Ž Seewx 1, 17, 15, 18, 20 .. A series a g Azww , z 1 xx is ww y1 xx called local of order N to b g Az, z for some N g ޚq if
awbzŽ.Ž.Ž zy w .N s 0.Ž. 1.4
If a is local to b and b is local to a then we say that a and b are mutually local. Remark. Inwx 18, 20 the propertyŽ. 1.4 is called quantum commutati¨ity.
Note thatŽ. 1.4 implies that for every n G N one has ab"n s 0. We will denote the order of locality by NaŽ., b , i.e.,
s g < ᭙ G s NaŽ., b min½5n Zq k n, ab"k 0.
Note also that if A is a commutative or skew-commutative or skew-com- mutative algebra, e.g., a Lie algebra, then locality is a symmetric relation. In this case we say ‘‘a and b are local’’ instead of ‘‘mutually local.’’ s ymy1 s yny1 Let azŽ.Ýmg ޚ amz Ž . and bzŽ.Ýng ޚ bnz Ž. be some se- ries. Then the locality conditionŽ. 1.4 reads
s Ž.y1 N am Žy sbn .Žq s .s 0 for any n, m g ޚ.1.5Ž. Ý ž/s s
The locality conditionŽ. 1.4 is known to be equivalent to the formula
NaŽ., b y1 ambnŽ.Ž.s m abms Ž.Ž.q n y s .1.6 Ý ž/s Ž." ss0
The following statement is a trivial consequence of the definitions. y PROPOSITION 1.1. Let A be an algebra and let S ; Azww , z1 xx be a space of pairwise mutually local power series, which is closed under all the circle products and Ѩ. Then S is a conformal algebra. One can proveŽ see, for example,wx 15. that such families exhaust all conformal algebras. CONFORMAL AND VERTEX ALGEBRAS 499
Finally, we state here a trivial property of local series: y LEMMA 1.1. Let a, b g Azww , z1 xx be a pair of formal power series and assume a is local to b. Then each of the series a, Da, za is local to each of b, Db, zb.
1.3. Construction of the Coefficient Algebra of a Conformal Algebra Given a conformal algebra C, we can build its coefficient algebra Coeff C in the following way. For each integer n take a linear space AnˆŽ. ˆˆs [ g isomorphic to C. Let A ng ޚ AnŽ.. For an element a C we will denote the corresponding element in AnˆˆŽ.by an Ž.. Let E ; A be the subspace spanned by all elements of the form Ž.Ž.Da n q na Ž n y 1 . for any a g C, n g ޚ.1.7Ž. The underlying linear space of Coeff C is AˆrE. By an abuse of notation we will denote the image of anŽ.g Aˆ in Coeff C again by anŽ.. The following proposition defines the product on Coeff C.
PROPOSITION 1.2. Formula Ž.1.6 unambiguously defines a bilinear product on Coeff C. ClearlyŽ. 1.6 defines a product on Aˆ. To show that the product is well defined on Coeff C, it is enough to check only that ŽDa .Ž.Ž. m b n syma Ž m y 1 .Ž.bn and aŽ.Ž.Ž. m Db n syna Ž.Ž m b n y 1, . which is a straightforward calculation.
1.4. Examples of Conformal Algebras
1.4.1. Differential Algebras Take a pair Ž.A, ␦ , where A is an associative algebra and ␦: A ª A is a locally nilpotent derivation, ␦ Žab .s ␦ Ž.abq a␦ Ž.b , ␦ n Ž.a s 0 for n 4 0. Consider the ring Aw␦, ␦y1 x. Its elements are polynomials of the form ␦ i g Ýig ޚ aii, where only a finite number of a A are nonzero. Here we put a␦yn s aŽ␦y1 .n and a␦ 0 s a. The multiplication is defined by the formula
k q y a␦ klи b␦ s a␦ iŽ.b ␦ kl i. Ý ž/i iG0 500 MICHAEL ROITMAN
It is easy to check that Aw␦, ␦y1 x is a well-defined associative algebra. In fact, Aw␦, ␦y1 x is the Ore localization of the ring of differential operators Awx␦ . If in addition A has an identity element 1, then ␦Ž.1 s 0 and ␦␦y1 s ␦y1␦ s 1. g s ␦ n yny1 g w␦ ␦y1 xww y1 xx For a A denote a˜ Ýng ޚ a z A , z, z . One easily checks that for any a, b g A, a˜ and ˜b are local and & n ab˜"n ˜ s a␦ Ž.b .1.8Ž. So by Lemma 1.1 and Proposition 1.1 the series Äaa˜ < g A4 ; Aw␦, ␦y1 xww z, zy1 xx generate anŽ. associative conformal algebra; see Section 1.6. One can instead consider Aw␦, ␦y1 x to be a Lie algebra, with respect to the commutator wxp, q s pq y qp. If two series a˜ and ˜b are local in the associative sense they are local in the Lie sense, too. One computes also
& q 1 nsnn s qs ab˜"n ˜˜s a␦ Ž.b yyÝ Ž1 . Ѩ Ž.b␦ Ž.a ,1.9 Ž . sG0 s! where Ѩ s drdz. Note that inŽ. 1.9 the circle products are defined by
n s abmn ˜ Ž.sy Ž1 .n wxa␦ nysm, b␦ qs .Ž. 1.10 Ž.˜" Ý ž/s ss0
Again, it follows that Äaa˜ < g A4 ; Aw␦, ␦y1 xww z, zy1 xx generate aŽ. Lie conformal algebra; see Section 1.6. An important special case is when there is an element ¨ g A such that ␦ ¨ s ¨ s ¨␦ n yny1 g w␦ ␦y1 xww y1 xx Ž. 1. Then ˜ Ýn z A , z, z generates with re- spect to the productŽ.Ž 1.10 a centerless . Virasoro conformal algebra. It satisfies the relations
¨¨˜˜"0 s Ѩ ¨ ˜, ¨¨ ˜˜"1 s 2¨ ˜, and the rest of the products are 0.
1.4.2. Loop Algebras Let ᒄ be a Lie algebra over an algebraically closed field މ- and let : ᒄ ª ᒄ be an automorphism of finite order, p s id. Then ᒄ is decom- posed into a direct sum of eigenspaces of :
2 ikr p ᒄ s [ ᒄ < ᒄ s k , k e . kgޚrpޚ CONFORMAL AND VERTEX ALGEBRAS 501
y Define a twisted loop algebra ˜ᒄ ; ᒄ m މ- wt, t 1 x by
ᒄ s atj ag ᒄ . ˜ ½5Ý jj jmod p j The Lie product in ˜ᒄ is given by wa m t mn, b m t xws a, b xm t mqn.If y p s 1, then ˜ᒄ s ᒄ m މ- wt, t 1 x, of course. g ᒄ F - Now for any a k ,0 k p, define a˜˜s Ý atpjqk zyjy1 g ᒄ wxz, zy1 . jgޚ s g ᒄ It is easy to see that any two a˜˜, ˜˜b are local with NaŽ., b 1 and if a k g ᒄ and b l we have & wxa, b if k q l - p, ab˜"0 ˜ s & ½ zawx, b if k q l G p. As in Section 1.4.1, we conclude that Äaa˜˜< g ᒄ4 ; ᒄww z, zy1 xx generate a Ž.Lie conformal algebra. Again, see Section 1.6 for the definition of varieties of conformal algebras.
1.5. More on Coefficient Algebras Let C be a conformal algebra and let A s Coeff C. Define
Aqs SpanÄ4anŽ. ag C, n G 0,
Ays SpanÄ4anŽ. ag C, n - 0, AnŽ.s SpanÄ4an Ž. ag C . Define also for each n g ޚ linear maps Ž.n : C ª An Ž.by a ¬ an Ž., and s yny1 ª ww y1 xx s yny1 let Ýng ޚ Ž.nz : C Az, z so that a Ýng ޚ anzŽ. . Here we summarize some general properties of conformal algebras and their coefficient algebras.
PROPOSITION 1.3.Ž. a A s Ayq[ A is a direct sum of subalgebras.
Ž.b Aqy and A are filtered algebras with filtrations gi¨en by
AŽ.Ž.0 : A 1 : иии : Aqy, A s AŽ.y1 = A Ž.y2 = иии ,
DFAnŽ.s Aq, AnŽ.s 0. nG0 n-0
¡ nq1 k ~DCq D Ker DifnG 0, Ž.c Ker Žn .s¢ kG1 Ker Diyny1 fn- 0. In particular, Ž.y1 is injecti¨e. 502 MICHAEL ROITMAN
ª ww y1 xx ¨ ¬ yny1 Ž.d The map : C Az, z , gi en by a Ýng ޚ anzŽ. , is an injecti¨e homomorphism of conformal algebras; i.e., it preser¨es the circle products and agrees with the deri¨ation
Ž.ab""nns Ž.a Ž.b , ŽDa .s D Ž.a . Ž 1.11 .
Ž.e The map : C ª Azww , zy1 xx has the following uni¨ersal property: For any homomorphism : C ª Bzww , zy1 xx of C to an algebra of formal power series, there is the unique algebra homomorphism : A ª B such that the corresponding diagram commutes
Azwx, zBzy1 6 wx, zy1 6 6 C
Ž.f The formula D Ž a Ž n ..syna Ž n y 1 .defines a deri¨ation D: A ª A such that DAyy; A , DA qq; A . Proof. From formulaŽ. 1.6 for the product in A it easily follows that Aqyand A are indeed subalgebras. Also none of the linear identitiesŽ. 1.7 contains both generators with negative and non-negative index. This proves Ž.a . Similar arguments establish also Ž. b . Now we prove that Ker Ž.n is included in the right-hand side of Ž. c . Take some a g C, a / 0, and assume that anŽ.s 0. Then anŽ.g Aˆ is a linear combination of identitiesŽ.Ž 1.7 see Section 1.3. so we must have in Aˆ
kmax s q y anŽ. Ý kkŽ.ŽDa .Ž. k ka k Ž k 1. . s k kmin / F F / We can assume that k 0 for all kmink k max and that ak 0 for s s s ) k kmin and for k kmax . Assume also that k 0if k kmax or - k kmin. F F Comparing terms with index k for kmin k kmax , we get ␦ s q q kna kDa k kq1Ž.k 1 akq1 .Ž. 1.12 s y Taking inŽ. 1.12 k kmin 1, we see that there are two cases: EitherŽ. 1 s G q s / kmin 0 and n 0or2Ž.n 1 kmin 0. s y g k Case 1. Taking inŽ. 1.12 k 0,...,n 1, we get that ak DCfor F F ) F 0 k n. Now we have two subcases: kmax n and kmax n. CONFORMAL AND VERTEX ALGEBRAS 503
) s y q If kmax n we substitute inŽ. 1.12 k kmax, k max 1,...,n 1 and get y q q k ma x k 1 s s g n 1 that Dak 0. Now take k n inŽ. 1.12 and get that a DC y q Ker D k ma x n. F s s If kmax n we have nq1 0, and hence substitution k n inŽ. 1.12 gives a g D nq1C. Case 2. Here we again have two subcases: n G 0orn - 0. y q G k ma x k 1 s If n 0 then, as in the previous case, we get D ak 0 for y q F F s g k ma x n n 1 k kmax . Now taking k n inŽ. 1.12 , we get a Ker D . - s s q Finally, if n 0 then, since nn0, we have a q1Ž.n 1 anq1. s q q F Then we substitute k n 1, n 2, . . . intoŽ. 1.12 until for some k0 y q y q s k 0 k 1 s 1 we get k q10Ž.k 1 ak q1 0. It follows that Dak 0 for 00y y y q F F g k 0 n ; n 1 n 1 k k0; therefore a Ker D Ker D . This proves one inclusion inŽ. c . It also follows that Ker Ž.y1 s 0. Next we show that is a homomorphism of conformal algebras, that is, formulasŽ. 1.11 hold. For the first identity we have Ž.Da s ÝÝ Ž.Ž.Da n zyny1 sy Žnan .Žy 1 .zyny1 ngޚ ngޚ d s Ý anzŽ.yny1 . dz n The second identity reads
s abmn Ž.sy Ž1 .n an Žy sbm .Žq s ., Ž." Ý ž/s s which is precisely the formulaŽ. 1.2 . NowŽ. d is done after we notice that is injective, since Žy1is . injective. Now we can prove the other inclusion inŽ. c . If a g Ker DCk , then a is Ѩk s a solution of the differential equation z azŽ. 0. Hence a is a polynomial of degree at most k y 1, and therefore Ž.nas 0 for n G 0 and for n - yk.If a g DCk , then Ž.nas 0 for 0 F n F k y 1, by induction andŽ. 1.7 . StatementŽ. e is clear, since identities Ž 1.7 . hold for any homomorphism : C ª Bzww , zy1 xx. Finally, the formula DanŽŽ..syna Ž n y 1 . defines a derivative of Aˆ. So in order to proveŽ. f we have to show that D agrees with the identities Ž.1.7 . This is indeed the case: DDanŽ.Ž.Ž.Ž.q na n y 1 synDanŽ. Ž.Ž.Ž.Ž.y 1 q n y 1 any 2. 504 MICHAEL ROITMAN
1.6. Varieties of Conformal Algebras Consider now a variety of algebras ᑛ Žseewx 8, 13. .
DEFINITION 1.3. A conformal algebra C is a ᑛ-conformal algebra if Coeff C lies in the variety ᑛ.
The identities in ᑛ-conformal algebras are all the circle product identi- ties R such that for any integer m, RmŽ.becomes an ᑛ-algebra identity after substitution ofŽ. 1.2 for every circle product in R. Conversely, given a classical algebra identity r, we can substituteŽ. 1.6 for all products in r and get an identity of ᑛ-conformal algebras. This way we get a correspondence between classical and conformal identities. See the next section for exam- ples. Combining Propositions 1.1 and 1.3Ž. d , we get the following well-known fact:
PROPOSITION 1.4. ᑛ-conformal algebras are exhaustedŽ up to isomor- phism. by conformal algebras of formal power series S ; Azww , zy1 xx for ᑛ-algebras A.
1.7. Associati¨e and Lie Conformal Algebras The following theorem gives the explicit correspondence between con- formal and classical algebras in some important cases.
THEOREM 1.1wx 16 . Let C be a conformal algebra and let A s Coeff Cbe its coefficient algebra.
Ž.a A is associati¨e if and only if the following identity holds in C:
n s UTUT abn m csy1 n anVWVWy sbmq sc. 1.13 Ž." " Ý Ž.ž/s Ž.Ž. ss0
Ž.b The Jacoby identityww a, b x, c xs wa, wb, c xxy wb, wa, cinAis xx equi¨alent to the following conformal Jacoby identity in C:
n s abn m csy1 n Ž." " Ý Ž.ž/s ss0 UTUTUTUT =Ž.anVWVWVWVWy sbmŽ.q sc y bmq san Ž.y sc . Ž.1.14 CONFORMAL AND VERTEX ALGEBRAS 505
Ž.c The skew-commutati¨ity identitywx a, b sy wxb, a in A corresponds to the quasi-symmetry identity
1 nqsq1 s VWUT ab"n syÝ Ž.1 DbnŽ.q sa.Ž. 1.15 sG0 s!
Ž.d The commutati¨ity of A is equi¨alent to
q 1 n s s VWUT ab"n syÝ Ž.1 DbnŽ.q sa.Ž. 1.16 sG0 s!
The identitiesŽ.Ž.Ž. 1.13 , 1.14 , and 1.15 immediately imply the following:
COROLLARY 1.1. Let C be a Lie conformal or an associati¨e conformal algebra and let A s Coeff C be its coefficient algebra. Then C is an Aq- ¨ s g g ޚ module with the action gi en by aŽ. n c a"n c for a, c C, n q. More- o¨er, this action agrees with the deri¨ations on Aq and C: ŽŽ..Da n c s w D, anŽ.x c. From now on we will deal only with associative or Lie conformal algebras.
1.8. Dong’s Lemma We end this section by stating a very important property of formal power series over associative or Lie algebras. This property allows us to construct conformal algebras by taking a collection of generating series.
LEMMA 1.2. Let A be an associati¨e or a Lie algebra and let a, b, c g Azww , zy1 xx be three formal power series. Assume that they are pairwise g ޚ mutually local. Then for all n q, a"n b and c are mutually local. Moreo¨er, in the Lie algebra case,
NaŽ.Ž.""nn b, c s Nc, abF NaŽ., b q Nb Ž., c q Nc Ž., a y n y 1, Ž.1.17 and, in the associati¨e case,
NaŽ.""nn b, c F NbŽ., c , Nc Ž., abF NcŽ., a q Na Ž., b y n y 1. 506 MICHAEL ROITMAN
2. VERTEX ALGEBRAS
2.1. Fields Let now V be a vector space over މ- . Denote by glŽ. V the Lie algebra y of all މ- -linear operators on V. Consider the space FŽ.V ; gl Ž. Vww z, z 1 xx of fields on V, given by
FŽV .s Ý anz Ž.yny1 ᭙¨ g V, anŽ.¨ s 0 for n 4 0. ½5gޚ n
For azŽ.g F ŽV .denote
s yny1 s yny1 azyqŽ.ÝÝanz Ž. , azŽ.anz Ž. . n-0 nG0
| s | g | y Denote also by FŽV . FŽ.V the identity operator, such that Ž.1 s IdV ; all other coefficients are 0. Remark. Inwx 18, 20 the elements of FŽ.V are called quantum operators on V. We view glŽ. V as a Lie algebra, and Section 1.2.1 gives a collection of productsn , n g ޚ ,onFŽ.V . Now in addition to these products we " q VWUT introduce products"n for n - 0. Define first y1by VWUT azŽ.y1 bz Ž.s azbzyq Ž.Ž.q bza Ž. Ž. z.2.1 Ž .
Note that the products inŽ. 2.1 make sense, since for any ¨ g V we have anŽ.¨ s bn Ž.¨ s 0 for n 4 0. The y1st product is also known as the normally ordered product Ž.or Wick product and is usually denoted by :azbzŽ.Ž.:. Next, for any n - 0 set
1 azŽ.n bz Ž.s :Ž.Dazbzyny1 Ž. Ž.:, Ž 2.2 . " Ž.yn y 1!
s d s | where D dz . Taking b , we get UT UT a VWy1 | s a, a VWy2 | s Da.2Ž..3
It is easy to see that
| s ␦ "n a y1,n a. CONFORMAL AND VERTEX ALGEBRAS 507
We have the following explicit formula for the circle products: If Ž.Ž.Ž.Ž.s ymy1 abz""nnÝm abmz , then
sqn abmn Ž.sy Ž1 .n asbm Ž.Žq n y s . Ž." Ý ž/n y s sFn
sqn yyŽ.1 n bm Žq n y sas .Ž..2.4 Ž. Ý ž/s sG0
Note that if n ) 0 thenŽ. 2.4 becomes
nqs abmn Ž.sy Ž1 .n as Ž.Ž, bmq n y s ., Ž." Ý ž/s sG0 which is precisely formulaŽ. 1.2 for Lie algebras. It is easy to see that D is a derivation of all the circle products:
DaŽ."""nnn bs Da b q aDb.2Ž..5 Note also that the Dong’s Lemma 1.2 remains valid for negative n and the estimateŽ. 1.17 still holds.
2.2. Definition of Vertex Algebras Instead of giving a formal definition of a vertex algebra in the spirit of Definition 1.1, we present a description of these algebras similar to Proposition 1.4. For a more abstract approach see, e.g.,wx 9, 15, 18, 20 .
DEFINITION 2.1. A ¨ertex algebra is a subspace S ; FŽ.V of fields over a vector space V such that Ž.i Any two fields a, b g S are local Ž in the Lie sense . . Ž.ii S is closed under all the circle products"n , n g ޚ, given by Ž.2.4 . Ž.iii | g S. Note that fromŽ. 2.3 it follows that a vertex algebra is closed under the derivation D s drdz. Note also that a vertex algebra is a Lie conformal algebra. Let S ; FŽ.V be a vertex algebra. We introduce the left action map Y: S ª FŽ.S defined by
yny1 YaŽ.s Ý Ž.a "n и .2Ž..6 ngޚ | s | Clearly, YŽ.S FŽS.. 508 MICHAEL ROITMAN
We state here the following characterizing property of Y Žseewx 15, 20. :
PROPOSITION 2.1. The left action map Y: S ª FŽ.S is an isomorphism of ¨ertex algebras, i.e., YSŽ.; F Ž.Sisa¨ertex algebra and
s | s | YaŽ.""nn b YaŽ. Yb Ž., Y ŽS . FŽS. .2.7Ž.
FromŽ. 2.3 and Ž. 2.5 it follows that Y also agrees with D:
YDaŽ .s Ѩ Ya Ž.s D, Ya Ž..
2.3. En¨eloping Vertex Algebras of a Lie Conformal Algebra Let C be a Lie conformal algebra and let L s Coeff C be its coefficient Lie algebra.
DEFINITION 2.2wx 14, 15 .Ž. a An L-module M is called restricted if for any a g C and ¨ g M there is some integer N such that for any n G N one has anŽ.¨ s 0 Ž.bAnL-module M is called a highest weight module if it is generated over L by a single element m g M such that Lmq s 0. In this case m is called the highest weight ¨ector. Clearly any submodule and any factor module of a restricted module are restricted. Let M be a restricted L-module. Then the representation : L ª glŽ. M could be extended to the map : Lzww , zy1 xx ª FŽ.M which combined with the canonical embedding : C ª Lzww , zy1 xx Ž.Ž.see Proposition 1.3 d gives a conformal algebra homomorphism : C ª FŽ.M . Then Ž.C ; F Ž.M consists of pairwise local fields, and by Dong’s Lemma 1.2, Ž.C together | g ; with FŽ.M generates a vertex algebra SM FM Ž.. The following proposition is well known; see, e.g.,wx 11 . ¨ s PROPOSITION 2.2.Ž. a The ertex algebra S SM has the structure of a highest weight module o¨er L with the highest weight ¨ector |. The action is gi¨en by
 s  g g ޚ  g anŽ. Ž.a "n , a C, n , SM . Moreo¨er this action agrees with the deri¨ations:
Ž.DaŽ. n  s D, an Ž..
¨ Ž.b Any L-submodule of S is a ertex algebra ideal. If M12 and M are two restricted L-modules, S s S , S s S , and : S ª SisanL-mod- 1 M122 M 12 CONFORMAL AND VERTEX ALGEBRAS 509
ule homomorphism such that Ž.| s |, then is a ¨ertex algebra homo- morphism.
2.4. Uni¨ersal En¨eloping Vertex Algebras Now we build a universal highest weight module V over L, which is often referred to as a Verma module. Take the one-dimensional trivial މ- | | Lq-module VV, generated by an element . Then let
s L މ- | s m މ- | ( r Y Ind LVqqULŽ. UŽ L . V ULŽ.ULL Ž.q.
It is easy to see that V is a restricted module and hence we get an s ; ª enveloping vertex algebra S SV FŽ.V and a homomorphism : C S. Clearly, is injective, since : L ª glŽ. V is injective. ª ¨ ␣ ¬ ␣ y | THEOREM 2.1.Ž. a The map : S Vgien by Ž1 . V is an | s | L-module isomorphism, and Ž.SV. Ž.b S is the uni¨ersal en¨eloping ¨ertex algebra of C in the following sense: If : C ª U is another homomorphism of C to a ¨ertex algebra U, then there is the unique map ˆ: S ª U which makes up the following commutati¨e triangle:
ˆ 6 SU 6 6 C
s s From now on we identify V and S SV via and write V VCŽ.for the universal enveloping vertex algebra of a Lie conformal algebra C and | s | s | ª s r SV. The embedding : C V ULŽ.ULL Ž.q is then given by a ¬ aŽ.y1 |. By Proposition 1.3Ž. c , the map Ž.y1:C ª Ly, defined by a ¬ aŽ.y1 , is an isomorphism of linear spaces. Therefore, the image of C in V is equal to Ž.C s Ly| s L| ; V.
3. FREE CONFORMAL ALGEBRAS
3.1. Definition of Free Conformal and Free Vertex Algebras
Let B be a set of symbols. Consider a function N: B = B ª ޚq, which will be called a locality function. Let ᑛ be a variety of algebras. In all the applications ᑛ will be either Lie or associative algebras. Consider the category ᑝᒌᒋᒃŽ.N of ᑛ-confor- mal algebrasŽ. see Section 1.6 generated by the set B such that in any 510 MICHAEL ROITMAN conformal algebra C g ᑝ onfŽ. N one has
ab"n s 0 ᭙a, b g B ᭙n G NaŽ., b .
By an abuse of notation we will not make a distinction between B and its image in a conformal algebra C g ᑝ onfŽ. N . The morphisms of ᑝᒌᒋᒃŽ.N are, naturally, conformal algebra homo- morphisms f: C ª CЈ such that faŽ.s a for any a g B. We claim that ᑝᒌᒋᒃŽ.N has the universal object, a conformal algebra C s CNŽ., such that for any other CЈ g ᑝᒌᒋᒃŽ.N there is the unique morphism f: C ª CЈ. We call CNŽ.a free conformal algebra, correspond- ing to the locality function N. In order to build CNŽ., we first build the corresponding coefficient algebra A s Coeff C Ž.see Section 1.3 . Let A g ᑛ be the algebra presented by the set of generators
X s Ä4bnŽ. bg B, n g ޚ Ž.3.1 with relations
s Ž.y1 NbŽ., a bn Žy sam .Žq s .s 0 a, b g B, m, n g ޚ . Ý ž/s ½5s Ž.3.2
g s yny1 g wwy1 xx For any b B let ˜b Ýn bnzŽ. Az, z . FromŽ. 3.2 it follows that any two a˜ and ˜b are mutually local; therefore by Dong’s Lemma 1.2 they generate a conformal algebra C ; Azww , zy1 xx.
PROPOSITION 3.1.Ž. a A s Coeff C. Ž.b The conformal algebra C is the free conformal algebra correspond- ing to the locality function N. Proof. Ž.a Clearly, there is a surjective homomorphism A ª Coeff C, since relationsŽ. 3.2 must hold in Coeff C. Now the claim follows from the universal property of Coeff C ŽŽsee Proposition 1.3 e.. . Ž.b Take another algebra CЈ g ᑝᒌᒋᒃŽ.N and let AЈ s Coeff CЈ. Obviously, there is an algebra homomorphism f: A ª AЈ such that fbnŽŽ..s bn Ž.for any b g B and n g ޚ. It could be extended to a map f: ww y1 xx ww y1 xx < Az, z ª AЈ z, z . Now it is easy to see that the restriction f C CONFORMAL AND VERTEX ALGEBRAS 511
gives the desired conformal algebra homomorphism C ª CЈ:
f 6 Azwx, zAy1 Ј wxz, zy1 6 6
f 6 CCЈ
Indeed, due to formulaŽ. 1.2 , f preserves the circle products, and, since Ѩ is a derivation of the products, and fŽ.Ѩ a˜˜s Ѩ fa Ž., for a g C one has fŽ.Ѩ s Ѩ f Ž. for any g C. In the case when ᑛ is the variety of Lie algebras, we may consider the universal vertex enveloping algebra VCŽ.of a free Lie conformal algebra C s CNŽ.. In accordance with Theorem 2.1, we call VC Ž.a free ¨ertex algebra. Though the construction of free conformal and vertex algebras makes sense for an arbitrary locality function N: B = B ª ޚq, the results of Sections 3.4᎐3.7 are valid only for the case when N is constant.
3.2. The Positi¨e Subalgebra of Coeff CNŽ. Let again C s CNŽ.be a free conformal algebra corresponding to a locality function N: B = B ª ޚq, B being an alphabet, and let A s Coeff C. Recall that by Proposition 1.3Ž. a we have the decomposition A s Ayq[ A of the coefficient algebra into the direct sum of two subalge- s < g G ; bras. Denote Xi ÄbnŽ. b B, n i4 X.
LEMMA 3.1. The subalgebra Aqq; A is isomorphic to the algebra Aˆ
presented by the set of generators X0 and those of relations Ž.3.2 which contain only elements of X0:
s Ý Ž.y1 NbŽ., a bn Žy sam .Žq s .s 0 a, b g B, m G 0, ½ ž/s s n G NbŽ., a 5 .3.3 Ž.
Proof. Clearly, there is a surjective homomorphism : Aˆqqª A which maps X0 to itself. We prove that is in fact an isomorphism. We proceed in four steps.
Step 1. First we prove that Aq is generated by X0 in A. Indeed, we ; have X0 Aqq. On the other hand, A is spanned by elements of the form 512 MICHAEL ROITMAN
amŽ., where m G 0 and a g C is a circle product monomial in B.By s induction on the length of a it is enough to check that if a aa12"k , then amŽ.is in the subalgebra, generated by X0 , which follows fromŽ. 1.2 .
Step 2. Let ˆ : Aˆˆqqª A be the homomorphism, which acts on the ¬ q generators X0 by anŽ.an Ž1 . , so that ˆ ŽAˆˆqq .is the subalgebra of A ( generated by X1. We claim that ˆˆis injective, and therefore Ž.AˆˆqqA . މ- Indeed, ˆ acts on the free associative algebra ²:X0 . Assume that for g s g މ- some p Aˆq we have ˆŽ.p 0. Take any preimage P ²X0 :of p. s g މ- Then we have ˆŽ.P ÝiiR i, where i ²X0 :and Ri are relations Ž.3.3 , such that in all iiand R there appear only indexes greater than or X X Ј equal to 1. But then P itself must be of the form ÝiiR i, where ‘‘ ’’ stands for decreasing all indexes by 1; hence p s 0.
Step 3. Next we claim that there is an automorphism of the algebra A which acts on the generators X by the shift anŽ.¬ an Žq 1 . . Indeed, relationsŽ. 3.2 are invariant under the shift, and clearly, is invertible. For s n ( s any integer n denote An Aq. We have An Aq A0 for every n. Step 4. Now for each integer n take a copy Aˆˆn of Aq. Let ˆn: ª AˆˆnnA y1 be the isomorphism of Aˆqqonto ˆŽ.A ˆ, built in Step 1. Let Aˆ be the limit of all these Aˆnnwith respect to the maps ˆ . We identify ª generators of Aˆˆnnwith the set X . It is easy to see that : A00A ª s extends to the homomorphism : AˆˆA, such that Ž.AnnA and < X s id. Now we observe that all the defining relationsŽ. 3.2 of A hold in Aˆ; hence there is an inverse map y1 : A ª Aˆ, and therefore is an isomorphism.
3.3. The Diamond Lemma For future purposes we need a digression on the diamond lemma for associative algebras. We closely followwx 2 , but use more modern terminol- ogy. Let X be some alphabet and let K be some commutative ring. Consider the free associative algebra KX²:of non-commutative polynomials with coefficients in K. Denote by X* the set of words in X, i.e., the free semigroup with 1 generated by X. A rule on KX²:is a pair s Žw, f ., consisting of a word w g X* and a polynomial f g KX²:. The left-hand side w is called the principal part of rule . We will denote w s . Let ᑬ be a collection of rules on KX²:. For a rule s Žw, f .g R and ¨ g a pair of words u, X* consider the K-linear endomorphism ru ¨ : KX²:ª KX ²:, which fixes all words in X* except for uw¨, and sends the latter to uf¨. CONFORMAL AND VERTEX ALGEBRAS 513
A rule s Ž.w, f is said to be applicable to a word ¨ g X*if w is a subword of ¨, i.e., ¨ s ¨ Јw¨ Љ. The result of application of to ¨ is, naturally, r¨ Ј ¨ Љ Ž.¨ s ¨ Јf¨ Љ.Ifp g KX ² :is a polynomial which involves a word ¨, such that a rule is applicable to ¨, then we say that is applicable to p. A polynomial p g KX²:is called terminal if no rule from R is applicable to ¨; that is, no term of p is of the form u¨ for g R. Define a binary relation ‘‘ª ’’ on KX²:in the following way: Set ª g p q if and only if there is a finite sequence of rules 1,..., n R, and a pair of sequences of words u , ¨ g X* such that q s r ¨ иии rp ¨ Ž.. ii unnn u111
DEFINITION 3.1.Ž. a A set of rules R is a rewriting system on KX² :if there are no infinite sequences of the form
ª ª иии p12p ; i.e., any polynomial p g KX²:can be modified only finitely many times by rules from R. Ž.b A rewriting system is confluent if for any polynomial p g KX ² : there is the unique terminal polynomial t such that p ª t.
Any rule s Ž.w, f g R gives rise to an identity w y f g KX ²:. Let IŽ.R ; KX ²:be the two-sided ideal generated by all such identities. ¨ ¨ g g Let 12, X* be a pair of words. A word w X* is called a composi- ¨ ¨ s Ј Љ ¨ s Ј ¨ s Љ / tion of 12and if w w uw , 1w u, 2uw , and u 0. Finally, take a word ¨ g X*. Let us call it an ambiguity if there are two rules , g R such that either ¨ is a composition of and or if ¨ s and is a subword of . Now we can state the lemma.
LEMMA 3.2Ž.Ž. Diamond Lemma . a A rewriting system R is confluent if and only if all terminal monomials form a basis of K²:Ž. X rI R . Ž.b A rewriting system is confluent if and only if it is confluent on all the ambiguities; that is, for any ambiguity ¨ g X* there is the unique terminal t g K²: X such that ¨ ª t.
Remark. StatementŽ. a appears inwx 21 . A variant of Lemma 3.2 appears inwx 3, 4 . It was also known to ShirshovŽ seewx 25. . The name ‘‘diamond’’ is due to the following graphical description of the confluency property; see wx21 . Let R be a rewriting system in the sense of Definition 3.1Ž. a , and let ª g ‘‘ ’’ be defined as above. Assume p, q12, q KX²:are such that ª ª g ª p q12and p q . Then there is some t KX²:such that q1t and 514 MICHAEL ROITMAN
ª q2 t:
6q1 6
pt6 6
q2
Bergman inwx 2 uses the existence of a semigroup order with descending chain condition on the set of words X*. Though in our case there is an order on the setŽ. 3.1 , this order does not satisfy the descending chain condition, so we slightly modify the argument inwx 2 .
Proof of Lemma 3.2.Ž. a Assume that the rewriting system R is confluent. Define a map r: KX²:ª KX ²:by taking rp Ž.to be the unique terminal monomial such that p ª rpŽ.. The crucial observation is s y ¨ that r is a K-linear endomorphism of KX²:.Soif p Ýiiiuw Ž if i . i g g ¨ g g s y ¨ IŽ.R , iiiiiK, u , X*, Žw , f .R, then rp Ž.Ýiiiiiiru ŽŽ w f .. s 0; therefore the terminal monomials are linearly independent modulo IŽ.R . Form the other side, if R is not confluent, then there are a polynomial g g ª ª p KX²:and terminals q12, q KX ²:such that p q1, p q 2, and / y g q12q , and then q12q IŽ.R . Ž.b Take a polynomial p g KX² :. We prove that there is the unique terminal t such that p ª t by induction on the number npŽ.s ࠻Äqp< ª q4. ConditionŽ. a of Definition 3.1 assures that npŽ.is always finite. If npŽ.s 0 then p is a terminal itself and there is nothing to prove. By induction, without loss of generality we can assume that there are at least two different rules , g R which are applicable to p. This means that ¨ g / / there are some words u, , x, y X* such that rpu ¨Ž.p, rpx y Ž. p, / and rpu ¨Ž.rpx yu Ž.. By induction, both rp ¨Ž.and rpx y Ž.are uniquely ª ª reduced to terminals, say, rpup¨Ž. t1 and rpx yŽ. t2 . We need to s show that t12t . Consider two cases: when and have common symbols in p, and thus u¨ s x y is a word in p; and when and are disjoint. In the first case, let w g X* be the union of and in p. Then w is an ambiguity. By assumption, there is the unique terminal s g KX²:such that w ª s. Let q g KX²:be obtained from p by substituting w by s. CONFORMAL AND VERTEX ALGEBRAS 515
Then we have
6rpu ¨Ž. 6
pq6 Ž.3.4 6
rpx yŽ.
By induction, q is uniquely reduced to a terminal t, and therefore one has ª ª rpu ¨Ž.t and rpx y Ž. t. s In the second case, note that rrx yu ¨Ž. p rru ¨ x y Ž. p. Denote this polynomial by q. Then relationsŽ. 3.4 still hold, and we finish by the same argument as in the first case.
3.4. Basis of a Free Vertex Algebra Return to the setup of Section 3.1. From now on we take the locality function NaŽ., b to be constant: Na Ž., b ' N. Let C s CN Ž.be the free Lie conformal algebra and let L s Coeff C be its Lie algebra of coeffi- cients; see Proposition 3.1. In this section we build a basis of the universal enveloping algebra ULŽ.of L and a basis of the free vertex algebra V s VCŽ.. We start by endowing B with an arbitrary linear order. Then we define a linear order on the set X of generators of L, given byŽ. 3.1 , in the following way: amŽ.- bn Ž.m m - n or Žm s n and a - b.Ž..3.5 On the set X* of words in X introduce the standard lexicographical order: For u, ¨ g X*if <
Ž.amŽ.Ž.q N , am s ž amŽ.Ž.Ž.Ž.q Nam, amamq N Ž.Ny1 r2 1 s yyŽ.1 N an Žy s .Ž, amq s .. Ž 3.7b . 2 Ý ž/s ss1 / Denote the set of all such rules by R:
R s Ä4 Ž.bnŽ., am Ž . Ž3.6 . holds .Ž. 3.8
LEMMA 3.3. The set of rules R is a confluent rewriting system on މ- ²:X . We prove this lemma in Section 3.5. Here we derive from it and from Lemma 3.2 the following theorem.
THEOREM 3.1.Ž. a Let C s C Ž N . be the free Lie conformal algebra generated by a linearly ordered set B corresponding to a constant locality function N. Let L s Coeff C be the Lie algebra of coefficients and let U s UŽ. L be its uni¨ersal en¨eloping algebra. Then a basis of U is gi¨en by all monomials иии g g ޚ anan112Ž.Ž. 2 ankk Ž., a iB, n i ,3.9Ž.
such that for any 1 F i - k one has
y ) s N 1 if aiiaoraq1 Ž.iiaq1 and N is odd , n y n q F ii1 ½ N otherwise. Ž.3.10
Ž.b A basis of the algebra UŽ. Lq is gi¨en by all monomials Ž.3.9 G satisfying the condition Ž.3.10 and such that all ni 0. Ž.c Let V s VŽ. C be the corresponding free ¨ertex algebra. Then a basis of V consists of elements иии | g g ޚ anan112Ž.Ž. 2 ankkk Ž ., a iB, n i ,Ž. 3.11 - such that the condition Ž.3.10 holds and, in addition, nk 0. Proof. StatementŽ. a is a direct corollary of Lemmas 3.3 and 3.2, becauseŽ. 3.9 is precisely the set of all terminal monomials with respect to R. CONFORMAL AND VERTEX ALGEBRAS 517
StatementŽ. b follows immediately from Lemma 3.1, since any subset of rules R is also a confluent rewriting system. Note also that for a rule given byŽ. 3.7 if the principal term contains only elements from X0 then so does the whole rule . For the proof ofŽ. c recall that V ( UrULq as linear spaces Ž and even as L-modules. , where ULqqis the left ideal generated by L ; see Section 2.4. By Lemma 3.1, this ideal is the linear span of all monomials иии G anan112Ž.Ž. 2 ankk Ž.such that n k 0. But under the action of the rewriting system R the index of the rightmost symbol in a word can only increase; hence the linear span of these monomials in މ- ²:X is stable under R. It follows that the terminal monomials with a non-negative rightmost index form a basis of ULq. This provesŽ. b .
3.5. Proof of Lemma 3.3 First we prove that the set of rules R, given byŽ. 3.8 , is a rewriting މ- s иии g g މ- system on ²:X . Take a word u am11Ž.Ž.amkk X*. Let p ²:X be such that u ª p. Then any word ¨ that appears in p lies in the finite set
s иии g G s Wu bn11Ž.bnkk Ž. X* n imin Ä4m jand ÝÝn im i. ½51FjFk Ž.3.12
Therefore conditionŽ. a of Definition 3.1 holds. Thus we are left to prove that R is confluent. According to Lemma 3.2, it is enough to check that it is confluent on a composition w s ckbjaiŽ.Ž.Ž. of principal parts of a pair of rules Žbj Ž ., ai Ž .., Žck Ž ., bj Ž .. g R. Thus it is sufficient to prove the following claim.
LEMMA 3.4. Let u s ckbjaiŽ.Ž.Ž.g X* be a word of length 3. Then R is confluent on u; i.e., there is a unique terminal rŽ. w g މ- ²X : such that u ª rwŽ..
Proof. Assume for simplicity that the three rules ŽŽ.bn, am Ž .., ŽŽcp ., bn Ž.., and ŽŽcp ., am Ž ..are of the form Ž 3.7a . . The general case is essentially the same, but requires some additional calculations.
Consider the set Wu, given byŽ. 3.12 . We prove that the lemma holds for g all w Wu by induction on w.If w is sufficiently small then it is a terminal itself. By induction, it is enough to consider w s cpbnamŽ.Ž.Ž. g Wu such that R is applicable to both bnamŽ.Ž .and c Ž.Ž. pbn. Apply 518 MICHAEL ROITMAN
Žbn Ž ., am Ž ..and Žcp Ž ., bn Ž .. to w and take the difference of the results:
N s ¨ s bncŽ.Ž.Ž. pamyy Ž1 .N cp Žy s ., bn Žq sam .Ž. Ý ž/s ss1 N s y cpambnŽ.Ž.Ž.qy Ž1 .N cp Ž. bn Žy s ., am Žq s .. Ý ž/s ss1
By induction, ¨ is reduced uniquely to a terminal t and we only have to show that t s 0. First we apply the rules Žbn Ž ., am Ž .., Žcp Ž ., bn Ž .., and ŽŽcp ., am Ž ..to ¨ several times and get
N s ¨ ª yyŽ1 .N bn Ž. c Ž py s ., am Žq s .q bnamc Ž.Ž .Ž. p Ý ž/s ss1 N s qyŽ1 .N cp Žy s ., am Žq sbn . Ž.y amc Ž .Ž.Ž. pbn Ý ž/s ss1 N s yyŽ.1 N cp Žy s .Ž, bnq sam .Ž. Ý ž/s ss1 N s qyŽ.1 N cp Ž.Ž bny s .Ž, amq s . Ý ž/s ss1 N s ª Ž.y1 N am Ž.Ž, cpy s .Ž, bnq s . Ý ž/s ss1 N s qyŽ.1 N cp Žy s .Ž, amq s .Ž., bn Ý ž/s ss1 N s qyŽ.1 N cp Ž.Ž, bny s .Ž, amq s .. Ž 3.13 . Ý ž/s ss1
Next we introduce two rules acting on the linear combinations of Ž.formal commutators: For any am Ž.Ž.Ž., bn, cpg X let
s Ž amŽ., bn Ž.Ž., cp , am Ž., bn Ž., cp Ž. q bnŽ., am Ž ., cp Ž. ., N s s bnŽ., am Ž ., yy Ž1 .N bn Žy s ., am Žq s . . Ý ž/s ž/ss1 CONFORMAL AND VERTEX ALGEBRAS 519
The rule is the locality relation, and is nothing else but the Jacoby identity. The lemma will be proved after we show two things: Ž.1 There always exists a finite sequence of applications of the rules and that reduces Ž.3.13 to 0. Ž.2 All words which appear in the process of reduction in Ž.1 are smaller than the initial word u s cŽ.Ž.Ž. p b n a m with respect to the order Ž.3.5 .
Indeed, assumeŽ. 1 and Ž. 2 hold. Denote the polynomial in Ž 3.13 . by p0. Let ª ª иии ª p01p 0 be the reduction, guaranteed byŽ. 1 . By Ž. 2 and by the induction hypothe- ª sis, any two neighboring polynomials piip q1 from this sequence are uniquely R-reduced to a terminal, and this terminal must be the same, ªR ªR since either piip q1 or piq1 pi. Denote the three last terms inŽ. 3.13 by a , b , and c . In Fig. 1 we present a scheme of how and should be applied in order to reduce Ž.3.13 to 0. Each box in Fig. 1 stands for a sum of commutators:
j sy r
N sqt N syŽ.1 N cpŽy s y t .Ž, amq t .Ž, bnq s ., Ý ž/s ž/t s, ts1
FIG. 1. Application of rules and . 520 MICHAEL ROITMAN
N sqt N k syŽ.1 N cpŽy s ., bn Žq s y t .Ž, amq t ., Ý ž/s ž/t s, ts1 N sqt N l sy t syŽ.1 N Ý ž/s ž/t s, ts1 = cpŽ.Ž.Žy s , bny t , amq s q t ., N sqt N m syŽ.1 N bnŽq t .Ž, cpy s y t .Ž, amq s ., Ý ž/s ž/t s, ts1 N sqt N n sy q syŽ.1 N Ý ž/s ž/t s, ts1 = bnŽ.Ž.Ž.y s q t , cpy t , amq s , N sqt N o syŽ.1 N bnŽy s .Ž, amq s q t .Ž, cpy t ., Ý ž/s ž/t s, ts1 N sqtqr N v sy y syŽ.1 NN Ý ž/srž/t ž/ s, t, rs1 bnŽ.Ž.Ž.q s y t , amq t q r , cpy s y r , N sqtqr N w syŽ.1 NN Ý ž/srž/t ž/ s, t, rs1 = amŽ.Ž.Ž.q s q r , bnq t y r , cpy s y t , N sqtqr N x sy z syŽ.1 NN Ý ž/srž/t ž/ s, t, rs1 = amŽ.Ž.Ž.q s q t , cpy t y r , bny s q r . One can see that all terminal boxes in the above scheme cancel, so that a q b q c ª 0. ClaimŽ. 2 also holds, since every symbol in every box in Fig. 1 is less than cpŽ..
3.6. Digression on Hall Bases
Let again B be some linearly ordered alphabet, N g ޚq, C s CNŽ. the free Lie conformal algebra generated by B with respect to the constant locality N, and L s Coeff CNŽ.. A basis of the Lie algebra L could be obtained by modifying the construction of a Hall basis of a free Lie algebra; seewx 12, 23, 24 . Here we review the latter construction. We closely followwx 22 , except that all the order relations are reversed. CONFORMAL AND VERTEX ALGEBRAS 521
As in Section 3.3, take an alphabet X and a commutative ring K. Let TXŽ.be the set of all binary trees$ with leaves from X. For typographical reasons we will write the tree xy as ²:x, y . Assume that TXŽ.is endowed with a linear order such that ²:x, y ) minÄ4x, y for any x, y g TXŽ..
DEFINITION 3.2. A Hall set H ; TXŽ.is a subset of all trees h g TXŽ. satisfying the followingŽ. recursive properties: 1. If h s ²:x, y then y, x g H and x ) y. 2Ifh s ²²x, y :, z :then z G y, so that ²x, y : ) z G y. In particular, X ; H. Introduce two maps ␣: TXŽ.ª X* and : TX Ž.ª KX ²:in the following recursive way: For a g X set ␣Ža .s Ža .s a and ␣ Ž²x, y :. s ␣Ž.x ␣ Ž.y , Ž²x, y :.s wŽ.x , Ž.y x. It is a well-known factŽ see, e.g.,wx 22. that
Ž.a ŽH .is a basis of the free Lie algebra generated by X and < Ž.b ␣ H is injecti¨e. A word w g ␣Ž.H is called a Hall word. On the set X* of words in X introduce aŽ. lexicographic order as follows: If u is a prefix of ¨ then u ) ¨; otherwise u ) ¨ whenever for ) ¨ s ¨ - some index i one has uiiand u jjfor all j i.
DEFINITION 3.3wx 25, 7 . A word ¨ g X* is called Lyndon᎐Shirsho¨ if it is bigger than all its proper suffices. ␣ PROPOSITION 3.2.Ž. a There is a Hall set HLS such that Ž.HLS is the set of all Lyndon᎐Shirsho¨ words and ␣: TXŽ.ª X* preser¨es the order. g ␣ Ž.b For any tree h HLS the biggest term in Ž.his Ž.h .
3.7. Basis of the Algebra of Coefficients of a Free Lie Conformal Algebra Here we apply general results from Section 3.6 to the situation of Section 3.1. Recall that starting from a set of symbols B and a number N ) 0, we build the free conformal algebra C s CNŽ.generated by B such that ab"n s 0 for any two a, b g B and n G N. Let L s Coeff C be the corresponding Lie algebra of coefficients. It is generated by the set X s ÄanŽ.< ag B, n g ޚ4 subject to relationsŽ. 3.2 . The set of generators X is equipped with the linear order defined by Ž.3.5 . W define the order on X* as in Section 3.6. Consider the set of all s ; Lyndon words in X* and let H HLS TXŽ.be the corresponding Hall 522 MICHAEL ROITMAN
set. Recall that there is a rewriting system R on މ- ²:X , given by Ž 3.8 . . Define
s g ␣ Hterm Ä4h H Ž.h is terminal .
¨ F иии F ¨ LEMMA 3.5.Ž. a Let 1 n be a non-decreasing sequence of ᎐ ¨ s ¨ иии ¨ g terminal Lyndon Shirsho words. Then their concatenation w 1 n X* is a terminal word. Ž.b Each terminal word w g X* can be uniquely represented as a s ¨ иии ¨ ¨ F иии F ¨ concatenation w 1 n, where 1 n is a non-decreasing se- quence of terminal Lyndon᎐Shirsho¨ words. ᎐ ¨ F ¨ Proof. Ž.a Take two terminal Lyndon Shirshov words 12. Let g ¨ g ¨ x X be the last symbol of 12and let y X be the first symbol of . ¨ ᎐ Then, since a word is less than its prefix and since 1 is a Lyndon Shirshov word, we get - ¨ F ¨ - x 12y.
¨¨ Therefore, xy is a terminal, and hence 12is a terminal, too. Ž.b Take a terminal word w g X*. Assume it is not Lyndon᎐Shirshov. Let ¨ be the maximal among all proper suffices of w. Then ¨ is Lyndon᎐Shirshov, ¨ ) w, and w s u¨ for some word u. By induction, s ¨ иии ¨ ᎐ u 1 ny1 for a non-decreasing sequence of Lyndon Shirshov words ¨ F иии F ¨ ¨ G ¨ 1 ny1. We are left to show that ny1. ¨ - ¨ ¨ ) ¨¨¨ Assume on the contrary that ny1. Then, since ny1 , ny1 ¨ ¨ s ¨¨Ј ¨ Ј ) ¨ must be a prefix of so that ny1 . But then which contra- dicts the Lyndon᎐Shirshov property of ¨. The uniqueness is obvious. Let : މ- ²:X ª UL Ž.be the canonical projection with the kernel IŽ.R . THEOREM 3.2. The set ŽŽHterm ..is a basis of L. s ; Proof. Let s Ä4h1,...,hn Hterm be a non-decreasing sequence of s иии g މ- ␣ s terminal Hall trees. Let Ž.s Žh1 . Žhn . ²X :and Ž.s ␣ иии ␣ g Ž.h1 Ž.hn X*. By the Poincare´᎐Birkhoff᎐Witt theorem it is sufficient to prove that the set ÄŽŽ..s 4, when s ranges over all non-decreasing sequences s of terminal Hall trees, is a basis of ULŽ.. By Proposition 3.2Ž. b , Ž.s s ␣ Ž.s q O Ž␣ Ž..s , where O Ž.¨ stands for a sum of terms which are less than ¨. Now let tsŽ.g މ- ²X :be a terminal such that Ž.s ª ts Ž.. One can view ts Ž.as the decomposition of ŽŽ..s in basisŽ 3.9 . . By Lemma 3.5, ␣ Ž.s is a terminal monomial; hence ts Ž.has CONFORMAL AND VERTEX ALGEBRAS 523
form tsŽ.s ␣ Ž.s q fs Ž.where fs Ž.is a sum of terms ¨ g X* satisfying the following properties: 1. ¨ is terminal and ¨ - ␣Ž.s . 2. If ¨ contains a symbol anŽ.g X then a appears in ␣ Ž.s and F F nmin n nmax , where nmin and nmax are, respectively, minimum and maximum of all indices that appear in ␣Ž.s . Indeed, due to Proposition 3.2Ž. b properties 1 and 2 are satisfied by all the terms in Ž.s y ␣ Ž.s , and they cannot be broken by an application of the rules R. Property 1 implies that all tsŽ.and, therefore, ŽŽ..s are linearly independent. So we are left to show that they span ULŽ.. For that purpose we show that any terminal word w g X* can be represented as a linear combination of tsŽ.. By Lemma 3.5Ž. b any terminal word w could be written as w s ␣ Ž.s for some non-decreasing sequence s of terminal Hall trees. So we can write w s tsŽ.y fs Ž.. Now do the same with any term ¨ that appears in fs Ž., and so on. This process should terminate, because every term ¨ that appears during this process must satisfy properties 1 and 2 and there are only finitely many such terms. Remark. Alternatively we could use the theorem of Bokut’ and Mal- colmsonwx 5 . As in Theorem 3.1Ž. b , we deduce that all the elements of ŽŽHterm .. containing only symbols from X0 form a basis of Lq. Note that we have an algorithm for building a basis of the free Lie conformal algebra C s CNŽ.. Let L s Coeff C, V s VC Ž., and U s UL Ž.. Recall that the image if C in V under the canonical embedding : C ª V is Ž.C s Ly| s L| ; V. So, the algorithm goes as follows: Take the basis of L provided by Theorem 3.2. Decompose its element in basisŽ. 3.9 of the universal enveloping algebra ULŽ., and then cancel all terms of the иии G form an11Ž.ankk Ž.where n k 0. What remains, being interpreted as elements of the vertex algebra V, form a basis of Ž.C ; V.
3.8. Basis of the Algebra of Coefficients of a Free Associati¨e Conformal Algebra
Let again B be some alphabet, and let N: B = B ª ޚq be a locality function, not necessarily constant and not necessarily symmetric. By Proposition 3.1, the coefficient algebra A s Coeff CNŽ.of the free asso- ciative conformal algebra CNŽ.corresponding to the locality function N is presented in terms of generators and relations by the set of generators X s ÄbnŽ.< bg B, n g ޚ4 and relationsŽ. 3.2 . 524 MICHAEL ROITMAN
THEOREM 3.3.Ž. a A basis of the algebra A is gi¨en by all monomials of the form
иии an11Ž.ananly1 Žly1 .Ž.ll,Ž. 3.14
g where ai B and
y y Nii1 N 1 yFn F , 22i s s y NiiiNaŽ., aq1 for i 1,...,l 1.
Ž.b A basis of the algebra Aq is gi¨en by all monomials of the form
иии an11Ž.ananly1 Žly1 .Ž.ll,Ž. 3.15
g where ai B and
F F y s s y 0 niiN 1, N iNaŽ. ii, aq1 for i 1,...,l 1.
COROLLARY 3.1. Assume that the locality function N is constant. Con- sider the homogeneous component Ak, l of A, spanned by all the words of s ly1 length l and of the sum of indexes k. Then dim Ak, l N . Proof of Theorem 3.3.Ž. a Introduce a linear order on B and define an order on the set of generators X by the rule
amŽ.) bn Ž.m < In particular, for some a g B we have aŽ.0 - a Žy1 .- a Ž.1 - a Žy2 .- a Ž.2 - иии . For any relation r fromŽ. 3.2 take the biggest term r and consider the rule Ž.r, r y r . This way we get a collection of rules NbŽ., a y 1 R s Ž.bnŽ., am Ž . a, b g B, n ) ½51 2 NbŽ., a y 1 j Ž.bnŽ., am Ž . n- y , ½52 2 CONFORMAL AND VERTEX ALGEBRAS 525 where 1Ž.bnŽ., am Ž . NbŽ., a q s bnamŽ.Ž ., Žy1 .s 1 NbŽ., a bnŽy sam .Žq s ., Ý s ž/ss1 ž/ 2 Ž.bnŽ., am Ž . NbŽ., a q s bnamŽ.Ž ., Žy1 .s 1 NbŽ., a bnŽq sam .Žy s .. Ý s ž/ss1 ž/ By Lemma 3.2, we have to prove that these rules form a confluent rewriting system on މ- ²:X . Clearly R is a rewriting system, since it decreases the order, and each subset of މ- ²:X , containing only finitely many different letters from B, has the minimal element, in contrast to the situation of Section 3.5. As before, it is enough to check that R is confluent on any composition w s cŽ.Ž.Ž. pbnam, of the principal parts of rules from R. Consider the set W s ÄckbjaiŽ.Ž.Ž.< k, j, i g ޚ4 ; X*. We prove by induction on w g W that R is confluent on w.If w is sufficiently small, then it is terminal. Assume that w s ckbjaiŽ.Ž.Ž. is an ambiguity, for example, that 12Žcp Ž ., bn Ž ..and Žbn Ž ., am Ž .. are both applicable to w. Other cases are done in the same way. Let s w11Ž.cpŽ., bn Ž.Ž. w NcŽ., b s syŽ.1 NcŽ., b cp Žy sbn .Žq sam .Ž., Ý ž/s ss1 s w22Ž.bnŽ., am Ž .Ž. w NbŽ., a t NbŽ., a syÝ Ž.1 cpbn Ž.Žq tam .Žy t .. s ž/t t 1 q s Applying 21ŽŽbn s ., am Ž ..for s 1,...,NbŽ., a to w gives the same q s result as we get from applying 1ŽŽcp ., bn Žt ..for t 1,...,NcŽ., b to w2 , namely, sqt NbŽ., a Ý Ž.y1 NcŽ., b cpŽy sbn .Žq s q tam .Žy t .. G ž/s ž/t s, t 1 Ž.3.16 526 MICHAEL ROITMAN y By the induction assumption, w12w is uniquely reduced to a terminal, and since all monomials inŽ. 3.16 are smaller than w, we conclude that this terminal must be 0. Ž.b Follows at once from Lemma 3.1. 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