Automorphisms of the endomorphism semigroup of a free commutative algebra

Abstract

We describe the automorphism group of the endomorphism semigroup $\End(K[x_1,...,x_n])$ of ring $K[x_1,...,x_n]$ of polynomials over an {\it arbitrary} field $K$. A similar result is obtained for automorphism group of the category of finitely generated free commutative-associative algebras of the variety $\mathcal{C}\mathcal{A}$ commutative algebras. This solves two problems posed by B. Plotkin (\cite{24}, Problems 12 and 15). More precisely, we prove that if $φ\in \Aut\End(K[x_1,...,x_n])$ then there exists a semi-linear automorphism $s:K[x_1,...,x_n]\to K[x_1,...,x_n]$ such that $\phi (g)=s\circ g\circ s^{-1}$ for any $g\in\End(K[x_1,...,x_n])$. This extends the result by A. Berzins obtained for an infinite field $K$.

15 pages

References in corpus (3)

• Automorphisms of categories of free modules and free Lie algebras
• The group of automorphisms of the category of free associative algebras
• On automorphisms of categories of free algebras of some varieties

Cited by in corpus (6)

• The Jacobian Conjecture, together with Specht and Burnside-type problems
• On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups
• Independence of the B-KK Isomorphism of Infinite Prime
• The Largest Subsemilattices of the Endomorphism Monoid of an Independence Algebra
• Polynomial automorphisms, quantization and Jacobian conjecture related problems
• Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry