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Decomposition of reductive regular Prehomogeneous Vector Spaces

Hubert Rubenthaler (2011)

Annales de l’institut Fourier

Let ( G , V ) be a regular prehomogeneous vector space (abbreviated to P V ), where G is a reductive algebraic group over . If V = i = 1 n V i is a decomposition of V into irreducible representations, then, in general, the PV’s ( G , V i ) are no longer regular. In this paper we introduce the notion of quasi-irreducible P V (abbreviated to Q -irreducible), and show first that for completely Q -reducible P V ’s, the Q -isotypic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate...

Drinfeld doubles via derived Hall algebras and Bridgeland's Hall algebras

Fan Xu, Haicheng Zhang (2021)

Czechoslovak Mathematical Journal

Let 𝒜 be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev’s theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of 𝒜 via its derived Hall algebra and Bridgeland’s Hall algebra of m -cyclic complexes.

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