Hopfological algebra and categorification at a root of unity: The first steps

Any finite-dimensional Hopf algebra [Formula: see text] is Frobenius and the stable category of [Formula: see text]-modules is triangulated monoidal. To [Formula: see text]-comodule algebras we assign triangulated module-categories over the stable category of [Formula: see text]-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable [Formula: see text], our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.

Hopfological algebra

We develop some basic homological theory of hopfological algebra as defined by Khovanov [Hopfological algebra and categorification at a root of unity: the first steps, Preprint (2006), arXiv:math/0509083v2]. Several properties in hopfological algebra analogous to those of usual homological theory of DG algebras are obtained.