Invariant theory of relatively free right-symmetric and Novikov 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.