On the kernel of the projection map $T(V)\to S(V)$

Abstract

Let V be a vector space over some field F and let ρ T,S : T (V ) → S(V ) be the projection map, given byIn this paper we give a descrption of ker ρ S,T in terms of generators and relations. Namely, we will define a Z ≥2 -gradedIn a related result, we define the algebra S ′ (V ) as T (V ) factorised by the bilateral ideal generated by x ⊗ y ⊗ z − y ⊗ z ⊗ x, with x, y, z ∈ V , and we prove that there is a short exact sequence,When considering the homogeneous components of degree 2, we have M 2 (V ) = Λ 2 (V ) and S ′2 (V ) = T 2 (V ) so in both cases we get the well known exact sequence 0 → Λ 2 (V ) → T 2 (V ) → S 2 (V ) → 0.