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

Abstract

We consider self-injective finite-dimensional graded algebras admitting a triangular decomposition. In the preceding paper [7] we have shown that the graded module category of such an algebra is a highest weight category and has tilting objects in the sense of Ringel. In this paper we focus on the degree zero part of the algebra, the core of the algebra. We show that the core captures essentially all relevant information about the graded representation theory. Using tilting theory, we show that the core is cellular. We then describe a canonical construction of a highest weight cover, in the sense of Rouquier, of this cellular algebra using a finite subquotient of the highest weight category. Thus, beginning with a self-injective graded algebra admitting a triangular decomposition, we canonically construct a quasi-hereditary algebra which encodes key information, such as the graded multiplicities, of the original algebra. Our results are general and apply to a wide variety of examples, including restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras. We expect that the cell modules and quasi-hereditary algebras introduced here will provide a new way of understanding these important examples.into graded subalgebras given by the multiplication map, where we assume that − is concentrated in negative degree, in degree zero, and + in positive degree.There are a variety of examples of such algebras:(1) Restricted enveloping algebras (g );The proof of this theorem is a modification of very recent work of Andersen-Stroppel-Tubbenhauer [3], who prove this in the case of quantum groups. Unfortunately, one essential ingredient in loc. cit. is the notion of weight spaces, which is

On Hopf algebras with triangular decomposition

“…I thank Simon Riche for his warm hospitality, for the interesting discussions on character formulas and for his willingness to answer all my questions. I also thank Ulrich Thiel for the discussion on [9]. I am grateful to the referee and Gastón García for the careful reading of the manuscript and their useful comments.…”

“…[39, §1]) that the axiom of highest weight categories which fails is the existence of a partial order > in Λ such that: if M(µ) occurs in a standard filtration of P(λ), then µ > λ; and M(λ) occurs precisely once. However, Bellamy-Thiel [9] realized that this trouble can be fixed by using the grading; that is working inside of the category of graded modules.…”

“…In this survey, we first review a general framework due to Bellamy-Thiel [9] and Bonnafé-Rouquier [12, §F.2] which ensures that ( * ) holds. Explicitly, this is guaranteed for any graded algebra with triangular decomposition (under some mild extra conditions).…”

“…In[9] the standard modules are denoted by ∆ but, as it is usual in Hopf algebra theory, we keep ∆ to denote the comultiplication.2 and in other important examples, cf. [9, §1].…”

On Projective Modules Over Finite Quantum Groups

“…After this work appeared, Bellamy and Thiel [8] introduced a highest weight theory for finite-dimensional graded algebras with triangular decomposition. They show that the category of graded modules over such an algebra is highest weight.…”