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On Free Conformal and Vertex Algebras 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 AnÃŽ. ÃÃs [ g isomorphic to C. Let A ng ޚ AnŽ.. For an element a C we will denote the corresponding element in AnÃÃŽ.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 AÃrE. By an abuse of notation we will denote the image of anŽ.g AÃ 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 AÃ. 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 aÄ Ýng ޚ a z A , z, z . One easily checks that for any a, b g A, aÄ and Äb are local and & n abÄ"n Ä s a␦ Ž.b .1.8Ž. So by Lemma 1.1 and Proposition 1.1 the series ÄaaÄ < 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 aÄ and Äb are local in the associative sense they are local in the Lie sense, too. One computes also & q 1 nsnn s qs abÄ"n ÄÄs 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 Ä Ž.sy Ž1 .n wxa␦ nysm, b␦ qs .Ž. 1.10 Ž.Ä" Ý ž/s ss0 Again, it follows that ÄaaÄ < 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.
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