Invariant theory of free bicommutative algebras

2022/10/15 by Vesselin Drensky, Drensky, Vesselin
Mathematics · #13A50 #15A72 #17A30 #17A36 #17A50 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2210.08317

openalex publication_date 2022/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The variety of bicommutative algebras consists of all nonassociative algebras satisfying the polynomial identities of right- and left-commutativity (x₁x₂)x₃=(x₁x₃)x₂ and x₁(x₂x₃)=x₂(x₁x₃). Let F_d be the free d-generated bicommutative algebra over a field K of characteristic 0. We study the algebra F_d^G of invariants of a subgroup G of the general linear group GL_d(K). When G is finite we search for analogies of classical results of invariant theory of finite groups acting on polynomial algebras: the Endlichkeitssatz of Emmy Noether, the Molien formula and the Chevalley-Shephard-Todd theorem and show the similarities and the differences in the case of bicommutative algebras. We also describe the symmetric polynomials in F_d.

Related

• Classical invariant theory for free metabelian Lie algebras
• Noncommutative invariants of dihedral groups
• Polynomial identities of bicommutative algebras, Lie and Jordan elements
• Symmetric polynomials in the free metabelian Lie algebras
• Invariant theory of relatively free right-symmetric and Novikov algebras