Khovanov homology associates to a link L in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex defined combinatorially from a link diagram. It detects the unknot (Kronheimer-Mrowka) and gives a sharp lower bound (Rasmussen, using a deformation of E.S. Lee) on the 4-ball genus of torus knots.

When L is realized as the closure of a braid (or more generally, of a "balanced" tangle), one can use a variant of Khovanov´s construction due to Asaeda- Przytycki-Sikora and L. Roberts to define its sutured Khovanov homology, an invariant of the tangle closure in the solid torus. Sutured Khovanov homology distinguishes braids from other tangles (joint with Ni) and detects the trivial braid conjugacy class (joint with Baldwin).

In this talk, I will describe some of the representation theory of the sutured Khovanov homology of a tangle closure. It (perhaps unsurprisingly) carries an action of the Lie algebra sl(2). More surprisingly, this action extends to the action of a slightly larger Lie superalgebra whose structure hints at a unification with the Lee deformation. This is joint work with Tony Licata and Stephan Wehrli.