Geometry and Topology Seminar

Many quantum invariants of links are controlled by the representation
theory of quantum groups at generic value of the parameter q,
while quantum invariants of 3-manifolds require specializing q to a root
of unity. Categorification of link invariants and quantum groups
is well-understood when q is generic. We will consider categorification
of some of these structures when q is a root of unity of prime order p.
This categorification happens over a field of characteristic p and
requires working with p-complexes, where the
p-th power rather than the second power of the differential is zero.
Homology groups of complexes generalize to objects of the stable
category of modules over certain finite-dimensional Hopf algebras, and
the notion of a differential graded algebra is substituted by a p-dg
algebra, an algebra with a nilpotent p-derivation. Categorical
liftings of the defining relations in quantum groups will be explained
in this setup.