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W-algebras from Heisenberg categories. (English) Zbl 1405.81045
In this paper it is shown that the trace (corresponding to the zeroth Hochschild homology) of the Khovanov Heisenberg category is isomorphic to a quotient of the algebra $$W_{1+\infty}$$. This fact enables the authors to define an action of the latter algebra on the centre of the so-called categorified Fock space representation that results in an identification of the $$W_{1+\infty}$$-algebra on symmetric functions. The results are used to deduce interesting isomorphisms between representations of the corresponding categories.

##### MSC:
 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 20C08 Hecke algebras and their representations 17B65 Infinite-dimensional Lie (super)algebras 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 30H20 Bergman spaces and Fock spaces
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