Categorification of Virasoro-Magri Poisson vertex algebra

Abstract

Let Sigma be the direct sum of algebra of symmetric groups C Sigma(n), n is an element of Z(>= 0). We show that the Grothendieck group K-0(Sigma) of the category of finite dimensional modules of Sigma is isomorphic to the differential algebra of polynomials Z[partial derivative(n)x vertical bar n is an element of Z(>= 0)]. Moreover, we define m-th products (m is an element of Z(>= 0)) on K-0(Sigma) which make the algebra K-0(Sigma) isomorphic to an integral form of the Virasoro-Magri Poisson vertex algebra. Also, we investigate relations between K-0(Sigma) and K-0(N) where K-0(N) is the direct sum of Grothendieck groups K-0(N-n), n >= 0, of finitely generated projective N-n-modules. Here N-n is the nil-Coxeter algebra generated by n - 1 elements. (C) 2016 Elsevier Inc. All rights reserved.