Uniqueness up to inner automorphism of regular exact Borel subalgebras

Abstract

In [18], 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 Borel subalgebra B constructed in [18] is in fact a regular exact Borel subalgebra as defined in [7]. Later, Conde [9] 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 [20] showed that there is even an isomorphism $\phi :R\to R^{\prime }$ such that $\phi \left(B\right)=B^{\prime }$.

In this article, we show that if $R=R^{\prime }$, then φ 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 $g\left(B\right)=B$ for every $g\in G$.