Invariant theory of free bicommutative algebras

TL;DR

This paper investigates the invariant theory of free bicommutative algebras, exploring classical results like Noether's finiteness, Molien's formula, and Chevalley-Shephard-Todd theorem within this nonassociative algebra context.

Contribution

It extends classical invariant theory results to the setting of free bicommutative algebras, highlighting similarities and differences from polynomial algebra cases.

Findings

Identifies conditions under which invariants form finitely generated algebras.

Derives analogues of Molien's formula for bicommutative algebras.

Describes symmetric polynomials within free bicommutative algebras.

Abstract

The variety of bicommutative algebras consists of all nonassociative algebras satisfying the polynomial identities of right- and left-commutativity $(x_1x_2)x_3=(x_1x_3)x_2$ and $x_1(x_2x_3)=x_2(x_1x_3)$. 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 $F_d$.