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Uniqueness up to Inner Automorphism of Regular Exact Borel Subalgebras

Anna Rodriguez Rasmussen

Abstract

Külshammer, König 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 Külshammer 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 $g\in G$.

Uniqueness up to Inner Automorphism of Regular Exact Borel Subalgebras

Abstract

Külshammer, König and Ovsienko proved that for any quasi-hereditary algebra there exists a Morita equivalent quasi-hereditary algebra containing a basic exact Borel subalgebra . The obtained Borel subalgebra is in fact a regular exact Borel subalgebra. Later, Conde showed that given a quasi-hereditary algebra with a basic regular exact Borel subalgebra and a Morita equivalent quasi-hereditary algebra with a basic regular exact Borel subalgebra , the algebras and are isomorphic, and Külshammer and Miemietz showed that there is even an isomorphism such that . In this article, we show that if , 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 is a finite-dimensional algebra and is a finite group acting on via automorphisms, then under some natural compatibility conditions, there is a Morita equivalent quasi-hereditary algebra with a basic regular exact Borel subalgebra such that for every .
Paper Structure (22 sections, 36 theorems, 299 equations)

This paper contains 22 sections, 36 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Twisted Modules over dg-algebras
  5. Twisted Modules over A-infinity Algebras
  6. A-infinity algebras
  7. A-infinity categories
  8. Twisted Modules
  9. Truncation of A-infinity algebras
  10. Proof of the main result
  11. Strong Regular Exact Borel Subalgebras
  12. G-equivariance
  13. Preliminaries of Group Actions
  14. The proof of the main theorem of the section
  15. Example
  16. ...and 7 more sections

Key Result

Theorem 1

(Theorem theorem_conjugation) Let $A$ be a finite-dimensional algebra and let $B, B'$ be two basic regular exact subalgebras of $A$ with simple modules $\{L_1^B, \dots , L_n^B\}=\mathop{\mathrm{Sim}}\nolimits(B)$ and $\{L_1^{B'}, \dots , L_n^{B'}\}=\mathop{\mathrm{Sim}}\nolimits(B')$ such that for e Then there is an invertible element $a\in A$ such that $aBa^{-1}=B'$.

Theorems & Definitions (101)

  • Theorem
  • Theorem
  • Theorem
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.6
  • proof
  • ...and 91 more