Uniqueness up to inner automorphism of regular exact Borel subalgebras (Q6663134)

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 there is an isomorphism \(\varphi: R\rightarrow R'\) such that \(\varphi(B)=B'\). If \(R=R'\), then \(\varphi\) can be chosen to be an inner automorphism. Instead of proving this for regular exact Borel subalgebras of quasi-hereditary algebras, A. R. Rasmussen generalizes this to an appropriate class of subalgebras of arbitrary finite-dimensional algebras. As an application, the author shows 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 \(g \in G\).