Categorification of Virasoro-Magri Poisson vertex algebra

Let S be the direct sum of algebra of symmetric groups C S_n for a non-negative integer n. We show that the Grothendieck group K_0(S) of the category of finite dimensional modules of S is isomorphic to the differential algebra of polynomials Z[D^n x]. Moreover, for a non-negative integer m, we define m-th products on K_0(S) which make the algebra K_0(S) isomorphic to an integral form of the Virasoro-Magri Poisson vertex algebra. Also, we investigate relations between K_0(S) and K_0(N) where K_0(N) is the direct sum of Grothendieck groups K_0(N_n) of finitely generated projective N_n-modules. Here N_n is the nil-Coxeter algebra generated by n-1 elements.