Affine Brauer category and parabolic category $${\mathcal {O}}$$ O in types B, C, D

Abstract

A strict monoidal category referred to as affine Brauer category AB is introduced over a commutative ring κ containing multiplicative identity 1 and invertible element 2. We prove that morphism spaces in AB are free over κ. The cyclotomic (or level k) Brauer category CB f (ω) is a quotient category of AB. We prove that any morphism space in CB f (ω) is free over κ with maximal rank if and only if the u-admissible condition holds in the sense of (1.32). Affine Nazarov-Wenzl algebras [24] and cyclotomic Nazarov-Wenzl algebras [2] will be realized as certain endomorphism algebras in AB and CB f (ω), respectively. We will establish higher Schur-Weyl duality between cyclotomic Nazarov-Wenzl algebras and parabolic BGG categories O associated to symplectic and orthogonal Lie algebras over the complex field C. This enables us to use standard arguments in [1,26,27] to compute decomposition matrices of cyclotomic Nazarov-Wenzl algebras. The level two case was considered by Ehrig and Stroppel in [14].

Invariants of the orthosymplectic Lie superalgebra and super Pfaffians

“…We point out directions for further research, and make more comments on relationship of the present paper to other work. In contrast to the categories in [28,1], both of which are monoidal categories, our category AB(δ) was designed to only admit a tensor product functor ⊗ : AB(δ) × B(δ) −→ AB(δ) with the usual Brauer category B(δ), but not a natural monoidal structure. The reason for this is that we want to use AB(δ) to study the category T M (V ) of g-modules with objects M ⊗ V ⊗r , which only has a natural tensor product ⊗ : T M (V ) × T (V ) −→ T M (V ) with the category T (V ) of the modules V ⊗r .…”

“…In order to compare our category with that of [28], we consider the respective endomorphism algebras. The endomorphism algebras of the category in [28] are the Nazarov-Wenzl algebra at different degrees defined in [8, Definition 2.1] with generators and relations (V W.1) -(V W.8), which involve an infinite family of arbitrary parameters w k for all non-negative integers k, see (V W.3) and (V W.4) in [8, Definition 2.1] .…”

“…We note that Schur-Weyl dualities for modules M ⊗V ⊗r with M being some types of projective parabolic Verma modules were established in [8,28].…”

“…There exist an affine Brauer category in [28, Definition 1.2] and a related affine VW supercategory [1, §1.4], whose endomorphism algebras are the affine Nazarov-Wenzl algebras [8, Definition 2.1] and some of their variants [1, Definition 39] respectively. The categories of [1,28] are presented in terms of dotted Brauer diagrams (analogous to the dotted Temperley-Lieb diagrams in [12]), which makes use of a different type of diagrammatics from that of the polar Brauer diagrams of AB(δ). However, these categories were designed for the purpose of studying modules of the type M ⊗V ⊗r for the orthogonal and symplectic Lie algebras, and the periplectic Lie superalgebra p(n) respectively, where V is the natural module, thus have a high degree of similarity to our category in their general features.…”

“…However, these categories were designed for the purpose of studying modules of the type M ⊗V ⊗r for the orthogonal and symplectic Lie algebras, and the periplectic Lie superalgebra p(n) respectively, where V is the natural module, thus have a high degree of similarity to our category in their general features. We will make a comparison of our category with the categories of [1,28] in precise terms in Section 7.1.…”