Invariant theory of relatively free right-symmetric and Novikov algebras

TL;DR

This paper investigates the invariants of free right-symmetric and Novikov algebras under linear group actions, revealing that for many groups, these invariants are not finitely generated, which impacts understanding of their algebraic structure.

Contribution

It demonstrates that the invariant algebras of free right-symmetric and Novikov algebras are generally not finitely generated for a broad class of groups, extending to certain subvarieties.

Findings

Invariants are not finitely generated for many groups G.

Results apply to subvarieties containing left-nilpotent algebras.

The study advances understanding of invariant theory in non-associative algebras.

Abstract

Algebras with the polynomial identity (x,y,z)=(x,z,y), where (x,y,z)=x(yz)-(xy)z is the associator, are called right-symmetric. Novikov algebras are right-symmetric algebras satisfying additionally the polynomial identity x(yz)=y(xz). We consider the d-generated free right-symmetric algebra F(R) and the free Novikov algebra F(N) over a filed K of characteristic 0. The general linear group GL(K,d) with its canonical action on the d-dimensional vector space with basis the generators of F(R) and F(N) acts on F(R) and F(N) as a group of linear automorphisms. For a subgroup G of GL(K,d) we study the algebras of G-invariants in F(R) and F(N). For a large class of groups G we show that these algebras of invariants are never finitely generated. The same result holds for any subvariety of the variety R of right-symmetric algebras which contains the subvariety L of left-nilpotent of class 3 algebras in R.