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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.

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
Full Text: DOI arXiv
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