2024/03/22 by Anna Rodriguez Rasmussen, Rasmussen, Anna Rodriguez
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2403.15580
openalex publication_date 2024/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Külshammer, König 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',≤R') with a basic regular exact Borel subalgebra B', the algebras R and R' are isomorphic, and Külshammer and Miemietz showed that there is even an isomorphism φ:R→ R' such that φ(B)=B'. In this article, we show that if R=R', 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, ≤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.