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The elliptic Hall algebra and the deformed Khovanov Heisenberg category. (English) Zbl 1402.81172
Summary: We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined in [A. Licata and A. Savage, Quantum Topol. 4, No. 2, 125–185 (2013; Zbl 1279.20006)]. We also show that as an algebra, it is isomorphic to “half” of a central extension of the elliptic Hall algebra of I. Burban and O. Schiffmann [Duke Math. J. 161, No. 7, 1171–1231 (2012; Zbl 1286.16029)], specialized at \(\sigma = \bar{\sigma}^{-1} = q\). A key step in the proof may be of independent interest: we show that the sum (over \(n\)) of the Hochschild homologies of the positive affine Hecke algebras \(\mathrm{AH}_n^+\) is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the \(q\)-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.

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)
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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