Uniqueness up to Inner Automorphism of Regular Exact Borel Subalgebras

K\"ulshammer, K\"onig and Ovsienko proved that for any quasi-hereditary algebra (A, A) there exists a Morita equivalent quasi-hereditary algebra (R, 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, R) with a basic regular exact Borel subalgebra B and a Morita equivalent quasi-hereditary algebra (R', ₑ') 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: R R' such that (B) =B'. In this article, we show that if R=R', then 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, 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, R) with a basic regular exact Borel subalgebra B such that g (B) =B for every g G.