Pseudocompact Algebras and Highest Weight Categories

Abstract

We develop a new approach to highest weight categories C with good (and cogood) posets of weights via pseudocompact algebras by introducing ascending (and descending) quasihereditary pseudocompact algebras. For C admitting a Chevalley duality, we define and investigate tilting modules and Ringel duals of the corresponding pseudocompact algebras. Finally, we illustrate all these concepts on an explicit example of the general linear supergroup GL(1|1).

The Center of Dist (GL(m|n)) in Positive Characteristic

“…Thus M can be embedded into an injective envelope I(λ) of L(λ). By Lemma 7.1 (c) of [14], we infer that I(λ) = L(λ). By Chevalley duality, if Ext 1 G (L(λ), L(µ) = 0, then Ext 1 G (L(µ), L(λ)) = 0 and we conclude that B(λ) = {λ}.…”

“…Finally, assume p||λ|. If Ext 1 G (L(λ), L(µ)) = 0 or Ext 1 G (L(µ), L(λ)) = 0, then by Lemma 7.1 (c) of [14] we obtain that λ = µ ± α. Conversely, if λ = µ ± α, then Ext 1 G (L(λ), L(µ)) = 0 and Ext 1 G (L(µ), L(λ)) = 0 using Lemma 7.1 (c) of [14] again. Therefore B(λ) = λ + Zα.…”

Donkin-Koppinen filtration for GL(m|n) and generalized Schur superalgebras

“…Recall Definition 3.8 of [13] stating that a weight µ is a predecessor of λ if µ ✁ λ, and there is no weight π such that µ ✁ π ✁ λ. Since any weight λ ∈ X(T ) + has only finitely many predecessors, the poset X(T ) + is good in the sense of Definition 3.9 of [13] (cf. [9], Example 5.1).…”

“…Otherwise, λ ′ k is incompatible with all weights µ such that H 0 + (−µ)⊗H 0 − (µ) is a quotient of a good filtration of U/W k . In this case, Lemma 4.2 of [13] implies that the embedding W k → U splits, that is, U = R ⊕ W k and R ≃ U/W k . Since the projection U → W k maps M to zero, we have M ⊆ R, which implies N ⊆ R. Thus N ∩ V ′ k = 0, contradicting the minimality of k. For a dominant weight λ, let W (λ) denote a finite-dimensional G-supermodule that satisfies the following conditions:…”

“…The second part of the paper is devoted to results about generalized Schur superalgebras S Γ for an arbitrary ideal Γ of weights. In Proposition 9.2, we show that if Γ is finitely generated, then S Γ is an ascending quasi-hereditary algebra in the sense of [13]. There is a natural superalgebra morphism π Γ : Dist(G) → S Γ given by the restriction.…”