Year: 2024
Type: article
Abstract: We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient $$A \rightarrow B$$ A → B satisfying some finiteness conditions, the Frobenius tensor category $${\mathcal {D}}$$ D of graded B-comodules with its stable model structure induces a monoidal model structure on $${\mathcal {C}}$$ C . We consider the corresponding homotopy quotient $$\gamma : {\mathcal {C}} \rightarrow Ho {\mathcal {C}}$$ γ : C → H o C and the induced quotient $${\mathcal {T}} \rightarrow Ho {\mathcal {T}}$$ T → H o T for the tensor category $${\mathcal {T}}$$ T of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in $$Ho {\mathcal {T}}$$ H o T . We apply these results in the Rep(GL(m|n))-case and study its homotopy category $$Ho {\mathcal {T}}$$ H o T associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of $$Ho{\mathcal {T}}$$ H o T by the negligible morphisms is again the representation category of a supergroup scheme.
Source: Applied Categorical Structures
Authors: Thorsten Heidersdorf, Rainer Weissauer
Institutions: Newcastle University, Heidelberg University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4401498881