Identities of Conformal Algebras and Pseudoalgebras

For a given conformal algebra C, we write down the correspondence between identities of the coefficient algebra Coeff C and identities of C itself as of pseudoalgebra. In particular, we write down the defining relations of Jordan, alternative and Mal’cev conformal algebras, and show that the analogue of Artin’s Theorem does not hold for alternative conformal algebras. 1. Conformal algebras In this note, we present a proof of a technical statement which concerns the relation between identities of a conformal algebra C and its coefficient algebra Coeff C. This relation was mentioned in [8], where some particular cases (associativity, commutativity, Jacobi identity) were considered. Although the approach of [8] is quite general, it is still technically difficult to write down the conformal identities corresponding to a given variety of ordinary algebras. We propose another approach which uses the language of pseudoproduct [1], in order to obtain the correspondence between identities of C and Coeff C in a very explicit form. This approach was mentioned in [1], where the most important cases (associativity, commutativity, Jacobi identity) were considered. We prove the general statement for any homogeneous multilinear identity. As an application, we write down the identities of Jordan, alternative and Mal’cev conformal algebras and derive their elementary properties. Definition 1.1 ([3]). Let k be a field of zero characteristic, and let k[D] be the polynomial algebra in one variable. A conformal algebra C is a unital left k[D]module endowed with a family of k-bilinear operations (· ◦n ·) (n ranges the set of non-negative integers) satisfying the following properties: a ◦n b = 0 for sufficiently large n, a, b ∈ C; (1.1) Da ◦n b = −na ◦n−1 b, a ◦n Db = D(a ◦n b) + na ◦n−1 b, n ≥ 0. (1.2) The conditions (1.1) and (1.2) are called locality and sesqui-linearity, respectively. This definition is a formalization of the following structure (appeared in mathematical physics) [4, 5]. Let A be an algebra over k (in general, A is non-associative), and let A[[z, z]] be the space of formal distributions over A: A[[z, z]] = A ⊗ k[[z, z]]. An ordered pair of distributions 〈a(z), b(z)〉 is said to be local, if a(w)b(z)(w − z) = 0 (1.3) for some N ≥ 0. If 〈a(z), b(z)〉 is a local pair, then the product a(w)b(z) ∈ A[[z, z, w, w]] could be presented as a finite sum [3] a(w)b(z) = ∑ n≥0 cn(z) 1 n! ∂ z δ(w − z), (1.4) Partially supported by RFBR and SSc-2269.203. 1