An official website of the United States government
The trace (or zeroth Hochschild homology) of Khovanov’s Heisenberg category is identified with a quotient of the algebra $$W_{1+\infty }$$. This induces an action of $$W_{1+\infty }$$ on the center of the categorified Fock space representation, which can be identified with the action of $$W_{1+\infty }$$ on symmetric functions.
more »
« less
The degenerate Heisenberg category and its Grothendieck ring
The degenerate Heisenberg category Heis_k is a strict monoidal category which was originally introduced in the special case k=-1 by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi envelope of his category is isomorphic to a certain \Z-form for the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra specialized at central charge -1. We prove this conjecture and extend it to arbitrary central charge k. We also explain how to categorify the comultiplication (generically).
more »
« less
- Award ID(s):
- 2101783
- PAR ID:
- 10511882
- Publisher / Repository:
- Centre Mersenne
- Date Published:
- Journal Name:
- Annales Scientifiques de l'École Normale Supérieure
- ISSN:
- 0012-9593
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $$\mathbb{CP}^{n}$$ , for $$k\geqslant n$$ , with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $$\mathbb{C}P^{n}$$ ( $$n$$ -dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $$x_{1}x_{2}\ldots x_{n+1}=0$$ . By localizing, we deduce that the (fully) wrapped Fukaya category of the $$n$$ -dimensional pair of pants is equivalent to the derived category of $$x_{1}x_{2}\ldots x_{n+1}=0$$ . We also prove similar equivalences for finite abelian covers of the $$n$$ -dimensional pair of pants.more » « less
-
We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with [Formula: see text]-equivariant cohomology of [Formula: see text]. Motivated by the relationship between the Temperley–Lieb algebra and annular Khovanov homology, we also introduce an equivariant analog of the Temperley–Lieb algebra.more » « less
-
We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we define a Kirby-colored Khovanov homology that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu–Neithalath 2-handle formula, Kirby-colored Khovanov homology agrees with the\mathfrak{gl}_{2}skein lasagna module, hence is an invariant of 4-dimensional 2-handlebodies.more » « less
-
Abstract We describe the universal target of annular Khovanov–Rozansky link homology functors as the homotopy category of a free symmetric monoidal linear category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov–Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.24033/asens.2560
