Automorphisms of the semigroup of endomorphisms of free associative algebras

Abstract

Let $A=A(x_1,...,x_n)$ be a free associative algebra in $\mathcal{A}$ freely generated over $K$ by a set $X=\{x_1,...,x_n\}$, $EndA$ be the semigroup of endomorphisms of $A$, and $AutEndA$ be the group of automorphisms of the semigroup $EndA$. We investigate the structure of the groups $AutEndA$ and $Aut{\mathcal{A}}^{\circ }$, where ${\mathcal{A}}^{\circ }$ is the category of finitely generated free algebras from $\mathcal{A}$. We prove that the group $AutEndA$ is generated by semi-inner and mirror automorphisms of $EndF$ and the group $Aut{\mathcal{A}}^{\circ }$ is generated by semi-inner and mirror automorphisms of the category ${\mathcal{A}}^{\circ }$. This result solves an open Problem formulated in \cite{22}

Cited by in corpus (3)

• The Jacobian Conjecture, together with Specht and Burnside-type problems
• Automorphisms of the endomorphism semigroup of a free commutative algebra
• On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups