Automorphisms of Veronese subalgebras of polynomial algebras and free Poisson algebras

Bakhyt Aitzhanova, Leonid Grigor'evich Makar-Limanov, Ualbai Umirbaev

2023

Data confidence 85 high

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Abstract

The Veronese subalgebra [Formula: see text] of degree [Formula: see text] of the polynomial algebra [Formula: see text] over a field [Formula: see text] in the variables [Formula: see text] is the subalgebra of [Formula: see text] generated by all monomials of degree [Formula: see text] and the Veronese subalgebra [Formula: see text] of degree [Formula: see text] of the free Poisson algebra [Formula: see text] is the subalgebra spanned by all homogeneous elements of degree [Formula: see text], where [Formula: see text]. If [Formula: see text] then every derivation and every locally nilpotent derivation of [Formula: see text] and [Formula: see text] over a field [Formula: see text] of characteristic zero is induced by a derivation and a locally nilpotent derivation of [Formula: see text] and [Formula: see text], respectively. Moreover, we prove that every automorphism of [Formula: see text] and [Formula: see text] over a field [Formula: see text] closed with respect to taking all [Formula: see text]-roots of elements is induced by an automorphism of [Formula: see text] and [Formula: see text], respectively.