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A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups

Eugene Karolinsky (2000)

Banach Center Publications

Let G be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous G-spaces with connected isotropy subgroups is given. This result is based on Drinfeld's correspondence between Poisson homogeneous G-spaces and Lagrangian subalgebras in the double D𝖌 (here 𝖌 = Lie G). A geometric interpretation of some Poisson homogeneous G-spaces is also proposed.

Abelian ideals of a Borel subalgebra and root systems

Dmitri I. Panyushev (2014)

Journal of the European Mathematical Society

Let 𝔤 be a simple Lie algebra and 𝔄𝔟 o the poset of non-trivial abelian ideals of a fixed Borel subalgebra of 𝔤 . In [8], we constructed a partition 𝔄𝔟 o = μ 𝔄𝔟 μ parameterised by the long positive roots of 𝔤 and studied the subposets 𝔄𝔟 μ . In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of 𝔤 is a join-semilattice.

Affine braid group actions on derived categories of Springer resolutions

Roman Bezrukavnikov, Simon Riche (2012)

Annales scientifiques de l'École Normale Supérieure

In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a “categorical version” of Kazhdan-Lusztig-Ginzburg’s construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course...

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