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.
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Acknowledgments
I wish to thank Juan Cuadra for answering questions and pointing useful references on co-Frobenius coalgebras. I also thanks Gastón A. García for helping me with coradical problems. This work was partially supported by UBACyT 2018-2021, 256BA.
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Presented by: Alistair Savage
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Partially supported by UBACyT 2018-2021 “K-teoría y biálgebras en álgebra, geometría y topología” and PICT 2018-00858 “Aspectos algebraicos y analíticos de grupos cuánticos”.
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Farinati, M.A. Hopfological Algebra for Infinite Dimensional Hopf Algebras. Algebr Represent Theor 24, 1325–1357 (2021). https://doi.org/10.1007/s10468-020-09993-7
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DOI: https://doi.org/10.1007/s10468-020-09993-7