Affine oriented Frobenius Brauer categories

Abstract

We define the affine Frobenius Brauer category AB(A, − ⋆ ) associated to each symmetric involutive Frobenius superalgebra A. We then define an action of these categories on the categories of finite-dimensional supermodules for orthosymplectic Lie superalgebras defined over A. The case where A is the base field recovers the known action of the affine Brauer category on categories of supermodules for orthogonal and symplectic Lie algebras. The definition and associated action of AB(A, − ⋆ ) are both novel when A is e.g. the quaternions H, a finite group algebra, a zigzag superalgebra, or a truncated polynomial algebra. Finally, we state a conjecture for bases of hom-spaces in AB(A, − ⋆ ) and outline a potential proof strategy.

Diagrammatics for real supergroups

“…When A is a Frobenius superalgebra, OB k (A) was called the oriented Frobenius Brauer supercategory in [27,Definition 4.1]. The Frobenius structure on A allows one to enlarge it to the affine oriented Frobenius Brauer category of [27,Definition 4.3], which is the central charge zero special case of the Frobenius Heisenberg supercategory introduced in [32], and further studied in [5,27]. We refer the reader to these papers for proofs omitted here, none of which use the Frobenius structure on A.…”

“…We refer the reader to these papers for proofs omitted here, none of which use the Frobenius structure on A. Our presentation of OB k (A) is slightly different from the one given in [27,Definition 4.1]. Precisely, the relations (5.4) are the reflections in the vertical axis of the ones in [27, (4.4)].…”

“…[5,Theorem 7.2], ignoring the dots. To prove linear independence, we consider the affine Frobenius Brauer supercategory AOB(A) defined in [27,Definition 4.3]. This is the central charge k = 0 case of the Frobenius Heisenberg supercategory introduced in [32] and further studied in [5].…”

“…To any associative superalgebra A over a field k, we define a diagrammatic supercategory OB k (A), which is the free rigid symmetric monoidal supercategory on an object with endomorphism superalgebra A. Imposing a condition on the categorical dimension yields a quotient category OB k (A; d), for d ∈ k. This category has essentially appeared in [5,27,32], although our definition is slightly more general. When A = k, OB k (k; d) is the oriented Brauer category (over a general field k) mentioned above.…”

“…The supercategories introduced here have affine analogues [27,31], generalizing the affine oriented Brauer category of [3] and the affine Brauer category of [30]. These affine supercategories act naturally on categories of supermodules over supergroups.…”