ON THE GRASSMANN T-SPACE

Abstract

We study the Grassmann T-space, S3, generated by the commutator [x1,x2,x3] in the free unital associative algebra K 〈x1,x2,… 〉 over a field of characteristic zero. We prove that S3 = S2 ∩ T3, where S2 is the commutator T-space generated by [x1,x2] and T3 is the Grassmann T-ideal generated by S3. We also construct an explicit basis for each vector space S3 ∩ Pn, where Pn represents the space of all multilinear polynomials of degree n in x1,…,xn, and deduce the recursive vector space decomposition T3 ∩ Pn = (S3 ∩ Pn) ⊕ (T3 ∩ Pn-1)xn.