Semi-infinite highest weight categories

Abstract

Dedicated to Jens Carsten Jantzen on the occasion of his 70th birthday.

Representations of cyclotomic oriented Brauer categories

“…When ℓ = 1, (I, A − , A • , A + ) is the same as that for the oriented Brauer category OB(δ 1 ) in [17]. In fact, it gives a triangular decomposition in the sense of [8].…”

“…such that ρ(λ) = a for any λ ∈ Λ a , where I is given in Definition 2.5. Following [8], define In other words, for each λ ∈ Λ, there exists a projective object P λ admitting a finite ∆ ε -flag with ∆ ε (λ) at the top and other sections ∆ ε (µ) for µ ∈ Λ with ρ(µ) ρ(λ). Let A-mod ∆ be the category of all left A-modules with a finite ∆-flag.…”

“…Thanks to Corollaries 2.16(1) and 3.19 (2), P(λ) has a finite ∆-flag such that ∆(λ) appears as the top section and other sections ∆(µ) with wt(λ) wt(µ). So, A-lfdmod is an upper finite +-stratified category in the sense of [8,Definition 3.36] with respect to the stratification wt. It is fully stratified since ∆(µ) has a finite ∆-flag with sections ∆(ν) such that wt(ν) = wt(µ).…”

“…(TPC3) The category C is an upper finite fully stratified category in the sense of Brundan and Stroppel [8] with poset P ×P . Moreover, the poset structure of P ×P is given by the "inverse dominance order" on P × P in (3.27).…”

“…Assume that B is the locally unital algebra associated to either an oriented skein category or an oriented Brauer category. The triangular decomposition property is the key in the proof of that the category B-lfdmod of locally finite dimensional left B-modules is an upper finite fully stratified category in the sense of Brundan and Stroppel [2,8]. Based on [17], Brundan also constructed certain endofunctors of B-lfdmod and its associated graded categories so as to give a tensor product categorification of an integrable lowest weight with an integrable highest weight module of the same level 1 for g.…”