Symmetric polynomials in the variety generated by Grassmann algebras

2021/04/20 by Akdogan, Nazan, Findik, Sehmus
#13A50 #15A75 #16R30 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2104.09781

Abstract

Let G denote the variety generated by infinite dimensional Grassmann algebras; i.e., the collection of all unitary associative algebras satisfying the identity [[z₁,z₂],z₃]=0, where [z_i,z_j]=z_iz_j-z_jz_i. Consider the free algebra F₃ in G generated by X₃=\x₁,x₂,x₃\. The commutator ideal F₃' of the algebra F₃ has a natural K[X₃]-module structure. We call an element p∈ F₃ symmetric if p(x₁,x₂,x₃)=p(x_ξ1,x_ξ2,x_ξ3) for each permutation ξ∈ S₃. Symmetric elements form the subalgebra F₃^S₃ of invariants of the symmetric group S₃ in F₃. We give a free generating set for the K[X₃]^S₃-module (F₃')^S₃.

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