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A general approach to Heisenberg categorification via wreath product algebras. (English) Zbl 1366.18006
Categorification of (quantum) Heisenberg algebras using graphical calculus has been an active area of research since the work of Khovanov in [M. Khovanov, Fund. Math. 225, No. 1, 169–210 (2014; Zbl 1304.18019)]. In the paper under review, the authors establish a general framework for categorifying Heisenberg algebras that can be specialized to obtain previous constructions in a more conceptual, and sometimes simpler, way.
Specifically, given an $$\mathbb{N}$$-graded Frobenius superalgebra $$B$$ over an algebraically closed field $$\mathbb{F}$$ of characteristic $$0$$, the authors construct a quantum Heisenberg algebra $$\mathfrak{h}_B$$ which can be viewed as either a lattice Heisenberg algebra associated to the Grothendieck group $$K_0(B)$$ or the projective Heisenberg double associated to the tower of wreath product algebras $$A_n=B^{\otimes n}\rtimes\mathbb{F}[S_n]$$, $$n\in\mathbb{N}$$. They also construct a monoidal category $$\mathcal{H}_B$$ which is the Karoubi envelope, also called the idempotent completion, of a category $$\mathcal{H}_B'$$ generated as monoidal category by objects $$\{ n,\epsilon\} P$$ and $$\{ n,\epsilon\} Q$$ for $$n\in\mathbb{Z}$$, $$\epsilon\in\mathbb{Z}/2\mathbb{Z}$$ (thought of as grading-shifted versions of two objects $$P$$ and $$Q$$). The morphisms in $$\mathcal{H}_B'$$ are given by planar diagrams with strands labelled by elements of $$B$$.
The main result of the paper is that if $$B$$ does not equal its $$\mathbb{N}$$-degree $$0$$ component, $$\mathcal{H}_B$$ categorifies $$\mathfrak{h}_B$$ in the sense that there is an algebra isomorphism $$p$$ from $$\mathfrak{h}_B$$ to the Grothendieck ring of $$\mathcal{H}_B$$. To show the existence of $$p$$, the authors derive a presentation of $$\mathfrak{h}_B$$ and show that there are isomorphisms between suitable objects in $$\mathcal{H}_B$$ corresponding to the relations in the presentation. The injectivity of $$p$$ follows from an action of $$\mathcal{H}_B$$ as functors on the category $$\bigoplus_{n\in\mathbb{N}} A_n$$-mod and the faithfulness of the Fock space representation of a Heisenberg double over $$\mathbb{Q}$$. The assumption on the $$\mathbb{N}$$-grading of $$B$$ is only needed for showing that $$p$$ is surjective.
Additionally, the paper explores the rich structure of endomorphism rings in $$\mathcal{H}_B$$. For instance, even in the simplest case $$B=\mathbb{F}$$, the authors obtain Jucys-Murphy elements of symmetric group algebras and degenerate affine Hecke algebras. Using another specialization of $$B$$, they prove a conjecture of Cautis and Lauda from [S. Cautis and A. Licata, Duke Math. J. 161, No. 13, 2469–2547 (2012; Zbl 1263.14020)].

##### MSC:
 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B65 Infinite-dimensional Lie (super)algebras 19A22 Frobenius induction, Burnside and representation rings
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