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arXiv · 2210.08317

Invariant theory of free bicommutative algebras

Vesselin Drensky
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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 $F_d$.

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Vesselin Drensky. 2022-10-15. Invariant theory of free bicommutative algebras. https://arxiv.org/abs/2210.08317

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