A new approach to the Koszul property in representation theory using graded subalgebras

TL;DR

This paper introduces a new method for establishing the Koszul property in graded quasi-hereditary algebras, linking algebraic structures with Lie-theoretic properties and applying to quantum and algebraic group algebras.

Contribution

It provides conditions under which the graded algebra inherits Koszul and quasi-hereditary properties, and explores the structure of modules and applications to quantum and algebraic group algebras.

Findings

Graded algebra $ ext{gr} B$ can be Koszul and quasi-hereditary under certain conditions.

Standard modules for $B$ have graded structures as modules over a subalgebra.

New insights into $q$-Schur algebras and algebraic groups in positive characteristic.

Abstract

Given a quasi-hereditary algebra $B$, we present conditions which guarantee that the algebra $\gr B$ obtained by grading $B$ by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that $B$ might possess. The method involves working with a pair $(A,{\mathfrak a})$ consisting of a quasi-hereditary algebra $A$ and a (positively) graded subalgebra $\mathfrak a$. The algebra $B$ arises as a quotient $B=A/J$ of $A$ by a defining ideal $J$ of $A$. Along the way, we also show that the standard (Weyl) modules for $B$ have a structure as graded modules for $\mathfrak a$. These results are applied to obtain new information about the finite dimensional algebras (e.g., the $q$-Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated to semisimple algebraic groups in positive characteristic $p$. These results require, at least presently, considerable restrictions on the size of $p$.