Categorification of Virasoro-Magri Poisson vertex algebra

Let Σ be the direct sum of algebra of symmetric groups CΣn, n∈Z≥0. We show that the Grothendieck group K0(Σ) of the category of finite dimensional modules of Σ is isomorphic to the differential algebra of polynomials Z[∂nx|n∈Z≥0]. Moreover, we define m-th products (m∈Z≥0) on K0(Σ) which make the algebra K0(Σ) isomorphic to an integral form of the Virasoro-Magri Poisson vertex algebra. Also, we investigate relations between K0(Σ) and K0(N) where K0(N) is the direct sum of Grothendieck groups K0(Nn), n≥0, of finitely generated projective Nn-modules. Here Nn is the nil-Coxeter algebra generated by n−1 elements.