On the definition of Heisenberg category

Abstract

We introduce a diagrammatic monoidal category Heis k (z, t) which we call the quantum Heisenberg category; here, k ∈ Z is "central charge" and z and t are invertible parameters. Special cases were known before: for central charge k = −1 and parameters z = q − q −1 and t = −z −1 our quantum Heisenberg category may be obtained from the deformed version of Khovanov's Heisenberg category introduced by Licata and the second author by inverting its polynomial generator, while Heis 0 (z, t) is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in Heis k (z, t).

A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory

“…So that this also makes sense in the case r = 0, it is natural to adopt the convention that •−1 := 1 1 and • Proof. This is similar to Lemma 1.8 of [Bru3]. We first show by induction on r that I f,f ′ contains 2r−1 • − δ r 1 1 for all r ≥ 0.…”

Heisenberg and Kac–Moody categorification

“…In the special case k = −1, the Heisenberg category was defined originally in the degenerate case by Khovanov [25] and in the quantum case by Licata and the second author [28]. The appropriate extension of the definition to arbitrary central charge was worked out much more recently; see [4,30] in the degenerate case and [13] in the quantum case. A categorical Heisenberg action on a category R is the data of a strict monoidal functor Heis k → End(R), where End(R) is the strict monoidal category consisting of endofunctors and natural transformations.…”

Frobenius Heisenberg categorification

“…In [2], Brundan introduced a new approach to Heisenberg categorification, proving that the higher level Heisenberg categories of [15], which include Khovanov's original category, can be defined using a smaller set of relations, including an "inversion relation". This approach also shows that the affine oriented Brauer category of [4] can be viewed as the level zero Heisenberg category.…”

“…In this way, the categories introduced in the current paper unify and generalize these previous constructions. Our approach is inspired by the inversion relation method of [2]. As a consequence, even when we specialize to the setting of [16], which corresponds to the choice k = −1, we obtain two new presentations of the Heisenberg categories defined there.…”

“…The proofs of these and other theorems stated in this introduction will be given in Section 3. In the special case that F = k, Theorems 1.2 and 1.3 are due to Brundan [2,Th. 1.2,Th.…”