Exact Borel subalgebras, idempotent quotients and idempotent subalgebras

TL;DR

This work analyzes how exact Borel subalgebras interact with idempotent subalgebras and quotients in standardly stratified and quasi-hereditary algebras. It establishes conditions under which both $A/AeA$ and $eAe$ inherit standardly stratified (and in favorable cases exact Borel) structures from $A$, and shows that the corner subalgebras $eBe$ and the quotients $B/BeB$ preserve exact Borel properties when $e$ is compatible. A Morita-representative matrix framework $V_{[(A,Q_0,\unlhd)]}$ is developed to track decomposition multiplicities, revealing a block triangular decomposition that splits into quotient and corner contributions. The results yield explicit lower bounds on projective multiplicities in principal blocks of BGG category $\mathcal{O}$ for non-$\mathfrak{sl}_2$ types, highlighting the quantitative impact on representation theory of semisimple Lie algebras. Overall, the paper clarifies the stability of exact Borel and standardly stratified structures under idempotent operations and provides concrete multiplicity bounds in important categorical contexts.

Abstract

This article studies the compatibility of Koenig's notion of an exact Borel subalgebra of a quasi-hereditary or, more generally, standardly stratified algebra with taking idempotent subalgebras or quotients. As an application, we provide bounds for the multiplicities of indecomposable projectives in the principal blocks of BGG category $\mathcal{O}$ having basic regular exact Borel subalgebras.

Theorems & Definitions (78)

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• Definition 3.2