The elliptic Hall algebra and the deformed Khovanov Heisenberg category

Abstract

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined in [LS13]. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann [BS12], specialized at σ =σ −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 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.

A Categorical Quantum Toroidal Action on the Hilbert Schemes

“…The quantum toroidal algebra (Definition 1.3) is an affinisation of the quantum Heisenberg algebra which has been realised in several contexts: the elliptic Hall algebra in [2, 30], the double shuffle algebra in [8, 9, 23], the trace of the deformed Khovanov Heisenberg category in [4](when ). …”

Perverse Sheaves with Nilpotent Singular Support on the Stack of Coherent Sheaves on an Elliptic Curve

“…The combinatorial study of such a ring is part of the theory of multisymmetric functions, which attempts to generalize to an arbitrary number of sets of variables the classical theory of Macdonald of symmetric functions ( [22]). The elliptic Hall algebra has now appeared in a great diversity of problems: in the study of the K-theory of the Hilbert scheme of the affine plane ( [31]), in skein theory of tori ( [24]), in diagrammatic categorification ( [5]),... Motivated by a putative Lagrangian construction of a specialization of the elliptic Hall algebra in the spirit of Lusztig's semicanonical basis of quantum groups ( [21]), by the fact that in the context of quivers, Lusztig sheaves ( [20]) are in canonical one-to-one correspondence with irreducible components of Lusztig nilpotent variety ( [15]) and by a possible geometric interpretation of elliptic Kostka polynomials, which we leave for future investigations, we were led to the study of the characteristic cycle map from the category of spherical Eisenstein sheaves to Lagrangian cycles in the stack of Higgs bundles.…”