Soergel Bimodules, the Steenrod Algebra and Triply Graded Link homology

TL;DR

This paper connects Soergel bimodules with spectra, enabling the definition of their integral forms and Steenrod algebra actions, and extends triply graded link homology to new algebraic contexts.

Contribution

It introduces a spectral perspective on Soergel bimodules, allowing for integral forms and Steenrod algebra actions, and extends triply graded link homology over various rings.

Findings

Soergel bimodules can be realized as cohomology of spectra.

Steenrod algebra acts on Hochschild homology of bimodules over F_p.

Triply graded link homology is defined over rings where 2 is invertible.

Abstract

Soergel bimodules and their Hochschild homology are known to be important in the context of link homology. In this article we observe that Soergel bimodules may be naturally identified as the cohomology of well-defined objects in the category of spectra. This allows us to define forms of Soergel bimodules over the integers after inverting 2, indexed by elements in Braid groups associated to compact Lie groups of adjoint type. Reducing the Soergel bimodules modulo an odd prime, we endow the bimodules with an action of the reduced Steenrod algebra. This action extends to an action of the reduced Steenrod algebra on the Hochschild homology of Soergel bimodules over the primary field F_p for odd primes p. In the special case of the n-stranded Braid group, our results allow us to define the triply graded link homology over any ring where 2 is invertible, and deduce that the reduced Steenrod algebra acts on triply graded link homology over the field F_p for odd primes p.