Exact Borel subalgebras of tensor algebras of quasi-hereditary algebras

TL;DR

The paper studies how exact Borel subalgebras and their regularity behave under two natural constructions: tensor products of quasi-hereditary algebras and tensor algebras of generalized species. It proves that the tensor product of exact Borel subalgebras $B$ and $B'$ yields an exact Borel subalgebra of $A\otimes A'$, and it characterizes when this product is regular, showing regularity is typically constrained to very strong conditions. It then extends the framework to tensor algebras of species, establishing a quasi-hereditary criterion (Theorem thm_qh_species) and a construction for exact Borel subalgebras (Theorem thm_borel_species) from vertex algebras and bimodules, with special simplifying cases for triangular matrix rings. While providing broad utility for assembling quasi-hereditary structures across composed algebras, the authors demonstrate that regularity is not generally preserved, offering explicit degenerate and directedness-based scenarios where regularity holds or fails, thus clarifying the limits of these constructions in homological terms.

Abstract

Given two quasi-hereditary algebras, their tensor product is quasi-hereditary. In this article, we show that given two exact Borel subalgebras for these quasi-hereditary algebras, their tensor product is an exact Borel subalgebra. Moreover, we describe in which cases the tensor product of two regular exact Borel subalgebras is again regular. Additionally, we investigate tensor algebras of generalised species of quasi-hereditary algebras and exact Borel subalgebras thereof.

Theorems & Definitions (32)

• Theorem
• Theorem
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 3.1
• Theorem 3.2
• proof
• Example 3.3