AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA

What is it about?

The Universal Algebraic Geometry (UAG) developed by Boris Plotkin and his school works in the environment of varieties $\Theta$ of universal algebras. For a fixed algebra $H\in \Theta$ the affine spaces are the homomorphisms from the finitely generated free algebras $F(X)$ in $\Theta$ to $H$, where $X=\{x\sb1,\ldots,x_{n}\}$ (equivalently, the $n$-tuples in $H^{n}$) and the equations defining algebraic varieties are the congruences in $F(X)$. Many problems in UAG depend on the structure of ${\rm Aut}(\Theta^{\circ})$, where $\Theta^{\circ}$ is the category of finitely generated free algebras in $\Theta$, and on ${\rm Aut}({\rm End}(F(X)))$. In the paper under review the authors consider the variety $\scr A$ of all unitary associative algebras over a field $K$. The main result is the description of ${\rm Aut}({\rm End}(K\langle X\rangle))$. It is generated by semi-inner and mirror automorphisms. By a result of the second author [Internat. J. Algebra Comput. 17 (2007), no. 5-6, 941–949; MR2355676], this means that ${\rm Aut}({\rm End}(K\langle X\rangle))$ is generated by quasi-inner automorphisms. The automorphism $\Phi$ of ${\rm End}(K\langle X\rangle)$ is quasi-inner if there exists a bijection $s\colon K\langle X\rangle\to K\langle X\rangle$ such that $\Phi(\nu)=s\nu s^{-1}$ for all $\nu\in{\rm End}(K\langle X\rangle)$. As applications the paper clarifies the relations between ${\rm Aut}({\rm End}(K\langle X\rangle))$ and ${\rm Aut}({\rm End}(K[X]))$ and describes ${\rm Aut}({\scr A}^{\circ})$. Reviewed by Vesselin Drensky

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