Hopfological Algebra for Infinite Dimensional Hopf Algebras
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Hopfological Algebra for Infinite Dimensional Hopf Algebras Marco A. Farinati1 Received: 5 February 2020 / Accepted: 14 September 2020 / © Springer Nature B.V. 2020
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]/x 2 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]/x 2 . 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. Keywords Co-Frobenius Hopf algebras · Tensor triangulated categories · Homology theories · K0 · Categorification Mathematics Subject Classification (2010) 16T05 · 16E35 · 18G99 · 18D99 · 19A49 · 81R50
1 Introduction This paper has mainly 3 contributions: (1)
The “Hopfological algebra” can be developed not only for finite dimensional Hopf algebras but also for infinite dimensional ones, provided they are co-Frobenius. The language of comodules is better addapted than the language of modules.
Presented by: Alistair Savage Partially supported by UBACyT 2018-2021 “K-teor´ıa y bi´algebras en a´ lgebra, geometr´ıa y topolog´ıa” and PICT 2018-00858 “Aspectos algebraicos y anal´ıticos de grupos cu´anticos”. Marco A. Farinati
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Dpto de Matem´atica FCEyN UBA - IMAS (Conicet), Buenos Aires, Argentina
M.A. Farinati
(2) (3)
The formula “Ker d/Im d” can be written in Hopf-co-Frobenius language. Some K-theoretical results allow us to compute K0 of the stable categories associated to co-Frobenius Hopf algebras of the form H0 #B, with H0 cosemisimple and B finite dimensional.
The paper is organized as follows: In Sections 2 and 3 we show points (1) and (2) respectively. In Section 4 we develop some tools to understand the triangulated structure. In Section 5 we exhibit the first examples. Section 6 deals with K0 . Section 7 illustrate the first step on how to develop -in the setting of co-Frobenius Hopf algebras- the direction taken in [8] for finite dimensional Hopf algebras.
2 Integrals, Co-Frobenius and Triangulated Structure k will be a field, H a Hopf algebra over k, all comodules will be right comodules. The category of H -comodules is denoted MH and the subcategory of finite dimensional comodules is denoted mH .
2.1 Integrals Definition 2.1 (Hochschild, 1965; G. I. Kac, 1961; L
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