Automorphisms of Veronese subalgebras of polynomial algebras and free Poisson algebras
Bakhyt Aitzhanova, Leonid Makar-Limanov, and Ualbai Umirbaev

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 of degree of the polynomial algebra over a field in the variables is the subalgebra of generated by all monomials of degree and the Veronese subalgebra of degree of the free Poisson algebra is the subalgebra spanned by all homogeneous elements of degree , where . If then every derivation and every locally nilpotent derivation of and over a field of characteristic zero is induced by a derivation and a locally nilpotent derivation of and , respectively. Moreover, we prove that every automorphism of and over a field closed with respect to taking all -roots of elements is induced by an automorphism of and , respectively.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Advanced Topics in Algebra · Carbohydrate Chemistry and Synthesis
