On the definition of quantum Heisenberg category (Q2190836)

The paper under review defines a diagrammatic, monoidal, \(\Bbbk\)-linear (where \(\Bbbk\) is any commutative ground ring) category \(\operatorname{Heis}_{k}(z,t)\), for parameters \(z,t \in \Bbbk^{\times}\) and central charge \(k \in \mathbb{Z}\). It is referred to as a quantum Heisenberg category. The obtained category includes many previously studied diagrammatic categories as its specializations. For \(k=-1\), a two parameter deformation of the Heisenberg category introduced in [\textit{M. Khovanov}, Fundam. Math. 225, 169--210 (2014; Zbl 1304.18019)] is obtained. More generally, \(\operatorname{Heis}_{k}(z,t)\) provides a quantum analog of the category \(\operatorname{Heis}_{k}(\delta)\) studied in [\textit{J. Brundan}, Algebr. Comb. 1, No. 4, 523--544 (2018; Zbl 1457.17009)]. Three diagrammatic presentations of \(\operatorname{Heis}_{k}(z,t)\) are given, together with explicit isomorphisms between them. Using these, it is shown that \(\operatorname{Heis}_{k}(z,t)\) is strictly pivotal, and further, a basis theorem for the morphism spaces in \(\operatorname{Heis}_{k}(z,t)\) is proven. Some interesting \(\operatorname{Heis}_{k}(z,t)\)-module categories are described. It is shown that \(\operatorname{Heis}_{0}(q-q^{-1}, q^{n})\) acts on \(U_{q}(\mathfrak{gl}_{n})\!\operatorname{-mod}\) by tensoring with the natural module and its dual. Using this action, a new way to obtain a familiar set of generators for the center of \(U_{q}(\mathfrak{gl}_{n})\) is given. It is also shown that certain quantum Heisenberg categories act by induction and restriction functors on (the direct sum of) categories of modules over cyclotomic Hecke algebras. Using the Knizhnik-Zamolodchikov functor, this action is extended to an action by Bezrukavnikov-Etingof induction and restriction functors on the category \(\mathcal{O}\) of an associated rational Cherednik algebra. Another family of module categories studied in the article gives a quantum version of generalized cyclotomic quotients of Heisenberg categories described in [loc. cit.]. Finally, a quantum analog of the categorical comultiplication of [\textit{J. Brundan} et al., ''The degenerate Heisenberg category and its Grothendieck ring'', Preprint, \url{arXiv:1812.03255}] is constructed. However, due to the absence of an explicit description of the (split) Grothendieck ring of \(\operatorname{Heis}_{k}(z,t)\) (such a description is given in [loc. cit.] in the non-quantum case), the central result regarding the comultiplication is not as strong as that in the non-quantum case.