Automorphisms of Veronese subalgebras of polynomial algebras and free Poisson algebras

TL;DR

This paper studies the automorphisms and derivations of Veronese subalgebras of polynomial and free Poisson algebras, showing they are induced by automorphisms and derivations of the original algebras.

Contribution

It proves that all automorphisms and locally nilpotent derivations of these Veronese subalgebras originate from those of the parent polynomial and Poisson algebras.

Findings

Automorphisms of Veronese subalgebras are induced by automorphisms of the original algebras.

Locally nilpotent derivations of Veronese subalgebras come from those of the original algebras.

Results hold over fields closed under taking all d-th roots.

Abstract

The Veronese subalgebra $A_0$ of degree $d\geq 2$ of the polynomial algebra $A=K[x_1,x_2,\ldots,x_n]$ over a field $K$ in the variables $x_1,x_2,\ldots,x_n$ is the subalgebra of $A$ generated by all monomials of degree $d$ and the Veronese subalgebra $P_0$ of degree $d\geq 2$ of the free Poisson algebra $P=P\langle x_1,x_2,\ldots,x_n\rangle$ is the subalgebra spanned by all homogeneous elements of degree $kd$, where $k\geq 0$. If $n\geq 2$ then every derivation and every locally nilpotent derivation of $A_0$ and $P_0$ over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $A$ and $P$, respectively. Moreover, we prove that every automorphism of $A_0$ and $P_0$ over a field $K$ closed with respect to taking all $d$-roots of elements is induced by an automorphism of $A$ and $P$, respectively.