Hopfological Algebra for Infinite Dimensional Hopf Algebras

Abstract

We consider “Hopfological” techniques as in Khovanov, M., J. Knot Theory Ramificat 25(3), 26 (2016) but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, H = k[ℤ]#k[x]/x2 is the first example, whose corepresentations category is d.g. vector spaces. Motivated by this example we define the “Homology functor” (we prove it is homological) for any co-Frobenius algebra, with coefficients in H-comodules, that recover usual homology of a complex when H = k[ℤ]#k[x]/x2. Another easy example of co-Frobenius Hopf algebra gives the category of “mixed complexes” and we see (by computing an example) that this homology theory differs from cyclic homology, although there exists a long exact sequence analogous to the SBI-sequence. Finally, because we have a tensor triangulated category, its K0 is a ring, and we prove a “last part of a localization exact sequence” for K0 that allows us to compute -or describe- K0 of some family of examples, giving light of what kind of rings can be categorified using this techniques.