Uniqueness up to Inner Automorphism of Regular Exact Borel Subalgebras

Abstract

Külshammer, König and Ovsienko proved that for any quasi-hereditary algebra $(A,{\le }_A)$ there exists a Morita equivalent quasi-hereditary algebra $(R,{\le }_R)$ containing a basic exact Borel subalgebra $B$. The obtained Borel subalgebra is in fact a regular exact Borel subalgebra. Later, Conde showed that given a quasi-hereditary algebra $(R,{\le }_R)$ with a basic regular exact Borel subalgebra $B$ and a Morita equivalent quasi-hereditary algebra $(R^{\prime },{\le }_{R^{\prime }})$ with a basic regular exact Borel subalgebra $B^{\prime }$, the algebras $R$ and $R^{\prime }$ are isomorphic, and Külshammer and Miemietz showed that there is even an isomorphism $\phi :R\to R^{\prime }$ such that $\phi (B)=B^{\prime }$. In this article, we show that if $R=R^{\prime }$, then $\phi$ can be chosen to be an inner automorphism. Moreover, instead of just proving this for regular exact Borel subalgebras of quasi-hereditary algebras, we generalize this to an appropriate class of subalgebras of arbitrary finite-dimensional algebras. As an application, we show that if $(A,{\le }_A)$ is a finite-dimensional algebra and $G$ is a finite group acting on $A$ via automorphisms, then under some natural compatibility conditions, there is a Morita equivalent quasi-hereditary algebra $(R,{\le }_R)$ with a basic regular exact Borel subalgebra $B$ such that $g(B)=B$ for every $g\in G$.

53 pages