Highest weight theory for finite-dimensional graded algebras with triangular decomposition

Abstract

We show that the category of graded modules over a finite-dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show furthermore that this highest weight category has tilting modules in the sense of Ringel. This provides a new perspective on the representation theory of such algebras, and leads to several new structures attached to them. There are a wide variety of examples in algebraic Lie theory to which this applies: restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras.

To appear in Adv. Math

References in corpus (3)

• Counting resolutions of symplectic quotient singularities
• Cuspidal Calogero-Moser and Lusztig families for Coxeter groups
• Cores of graded algebras with triangular decomposition

Cited by in corpus (4)

• Semi-infinite highest weight categories
• Representations of the oriented skein category
• Wreath Macdonald polynomials at q=t as characters of rational Cherednik algebras
• Simple modules over finite quantum groups and their Drinfel'd doubles