Uniqueness up to Inner Automorphism of Regular Exact Borel Subalgebras

Abstract

K\"ulshammer, K\"onig and Ovsienko proved that for any quasi-hereditary algebra $(A,\leq_A)$ there exists a Morita equivalent quasi-hereditary algebra $(R, \leq_R)$ 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 $(R,\leq_R)$ with a basic regular exact Borel subalgebra $B$ and a Morita equivalent quasi-hereditary algebra $(R',\leq_{R'})$ with a basic regular exact Borel subalgebra $B'$, the algebras $R$ and $R'$ are isomorphic, and K\"ulshammer and Miemietz showed that there is even an isomorphism $\varphi:R\rightarrow R'$ such that $\varphi(B)=B'$. In this article, we show that if $R=R'$, then $\varphi$ 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 $(A, \leq_A)$ 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 $(R, \leq_R)$ with a basic regular exact Borel subalgebra $B$ such that $g(B)=B$ for every $g\in G$.