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- W214400007 abstract "Given a Frobenius category $mathcal{F}$ satisfying certain finiteness conditions, we consider the localization of its Hall algebra $mathcal{H(F)}$ at the classes of all projective-injective objects. We call it the {it Hall algebra} $mathcal{SDH(F, P(F))}.$ We discuss its functoriality properties and show that it is a free module over a twisted group algebra of the Grothendieck group $K_0(mathcal{P(F)})$ of the full subcategory of projective-injective objects, with a basis parametrized by the isomorphism classes of objects in the stable category $underline{mathcal{F}}$. We prove that it is isomorphic to an appropriately twisted tensor product of $mathbb{Q}K_0(mathcal{P(F)})$ with the derived Hall algebra (in the sense of Toen and Xiao-Xu) of $underline{mathcal{F}},$ when both of them are well-defined. We discuss some situations where the semi-derived Hall algebra is defined while the derived Hall algebra is not. The main example is the case of $2-$periodic derived category of an abelian category with enough projectives, where the semi-derived Hall algebra was first considered by Bridgeland who used it to categorify quantum groups." @default.
- W214400007 created "2016-06-24" @default.
- W214400007 creator A5049743105 @default.
- W214400007 date "2014-09-24" @default.
- W214400007 modified "2023-09-25" @default.
- W214400007 title "Semi-derived and derived Hall algebras for stable categories" @default.
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