Symmetric polynomials in the variety generated by Grassmann algebras

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Akdoğan N., Flndlk Ş.

Journal of Algebra and its Applications, vol.22, no.1, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 22 Issue: 1
  • Publication Date: 2023
  • Doi Number: 10.1142/s0219498823500196
  • Journal Name: Journal of Algebra and its Applications
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Keywords: Grassmann algebras, symmetric polynomials
  • Istanbul Technical University Affiliated: Yes

Abstract

© 2023 World Scientific Publishing Company.Let 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 [zi,zj] = zizj-zjzi. Consider the free algebra F3 in generated by X3 = {x1,x2,x3}. We call a polynomial p F3 symmetric if it is preserved under the action of the symmetric group S3 on generators, i.e. p(x1,x2,x3) = p(xζ1,xζ2,xζ3) for each permutation ζ S3. The set of symmetric polynomials forms the subalgebra F3S3 of invariants of the group S3 in F3. The commutator ideal F3′ of the algebra F3 has a natural left K[X3]-module structure, and (F3′)S3 is a left K[X3]S3-module. We give a finite free generating set for the K[X3]S3-module (F3′)S3.