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Access Type
WSU Access
Date of Award
January 2023
Degree Type
Dissertation
Degree Name
Ph.D.
Department
Mathematics
First Advisor
Ualbai Umirbaev
Abstract
We study the derivations, locally nilpotent derivations and automorphisms of the affine Veronese surface $V_d$ for all $d\geq 2$ by purely algebraic methods \cite{AU}. We show that over a field $K$ of characteristic zero every derivation and every locally nilpotent derivation of the algebra $K[x^d, x^{d-1}y,\ldots,xy^{d-1}, y^d]$ is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Using the proof of the Rentschler's Theorem \cite{RR} on locally nilpotent derivations of $K[x,y]$ given in \cite[Ch. 5]{Es}, we prove that every automorphism of $K[x^d, x^{d-1}y,\ldots,xy^{d-1}, y^d]$ is induced by an automorphism of $K[x,y]$ if $K$ is an algebraically closed field of characteristic zero. We also show that the amalgamated free product structure of the automorphism group of $K[x,y]$ induces an amalgamated free product structure on the automorphism group of $K[x^d, x^{d-1}y,\ldots,xy^{d-1}, y^d]$.
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 $d$-roots is induced by an automorphism of $A$ and $P$, respectively.
Recommended Citation
Aitzhanova, Bakhyt, "Automorphisms Of Veronese Subalgebras Of Polynomial Algebras And Free Poisson Algebras" (2023). Wayne State University Dissertations. 3852.
https://digitalcommons.wayne.edu/oa_dissertations/3852