Affine highest weight categories and affine quasihereditary algebras

Abstract

Abstract. Koenig and Xi introduced affine cellular algebras. Kleshchev and Loubert showed that an important class of infinite dimensional algebras, the KLR algebras R(Γ) of finite Lie type Γ, are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of quasihereditary algebras of Cline-Parshall-Scott in a finite dimensional situation. A fundamental result of Cline-Parshall-Scott says that a finite dimensional algebra A is quasihereditary if and only if the category of finite dimensional A-modules is a highest weight category. On the other hand, S. Kato and Brundan-Kleshchev-McNamara proved that the category of finitely generated graded R(Γ)-modules has many features reminiscent of those of a highest weight category. The goal of this paper is to axiomatize and study the notions of an affine quasihereditary algebra and an affine highest weight category. In particular, we prove an affine analogue of the Cline-Parshall-Scott Theorem. We also develop stratified versions of these notions.

Affine zigzag algebras and imaginary strata for KLR algebras

“…Informally, a proper stratification of R θ yields a stratification of the category R θ -mod of finitely generated graded R θ -modules by the categories B ξ -mod for much simpler algebras B ξ , see [17] for details. Description of the algebras B ξ is easily reduced to the semicuspidal cases, which split into real and imaginary subcases.…”

Stratifying KLR algebras of affine ADE types

“…by [18,Corollary 4.9], the stratum category C ξ is graded equivalent to B ξ -mod. We have a natural exact projection functor R ξ : C ≤ξ → C ξ .…”

Some extension algebras for standard modules over KLR algebras of type A

“…Let R θ,F be a Khovanov-Lauda-Rouquier (KLR) algebra of finite Lie type over a field F corresponding to θ ∈ Q + . It is known that R θ,F is affine quasihereditary [1,4,7,8]. In particular, it has finite global dimension and comes with a family of standard modules {∆(π) F | π ∈ KP(θ)} and proper standard modules {∆(π) F | π ∈ KP(θ)}, where KP(θ) denotes the set of the Kostant partitions of θ.…”