On functors associated to a simple root

By Volodymyr Mazorchuk, C. Stroppel. 2004

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Abstract Associated to a simple root of a finite-dimensional complex semisimple Lie algebra, there are several endofunctors (defined by Arkhipov, Enright, Frenkel, Irving, Jantzen, Joseph, Mathieu, Vogan and Zuckerman) on the BGG category O . We study their relations, compute cohomologies of their derived functors and describe the monoid generated by Arkhipov's and Joseph's functors and the monoid generated by Irving's functors. It turns out that the endomorphism rings of all elements in these monoids are isomorphic. We prove that the functors give rise to an action of the singular braid monoid on the bounded derived category of O 0 . We also use Arkhipov's, Joseph's and Irving's functors to produce new generalized tilting modules.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7c921252642931fcd79f6b0354cb8f891002200

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Citations in the corpus

Lectures on algebraic categorification (2010, unknown): “ σi= 1, for all i; σiσi+1σi= σi+1σiσi+1, for all i; σiσj= σjσi, if |i−j| >1; τiσjσi= σjσiτj, if |i−j| = 1; σiτj= τjσi, if |i−j| 6= 1; τiτj= τjτi, if |i−j| >1. The following theorem is proved in [MS2]: Theorem 5.12 (Weak categoriﬁcationof SBn). Let λbe regular, integral and dominant. Then, mapp”
A combinatorial approach to functorial quantum slk knot invariants (2007, background): “By Soergel’s Struktursatz this means that we multiply the natural transformation with a certain element of the centre of one of the involved categories ([25], [23]).”
Projective-injective modules, Serre functors and symmetric algebras (2005, background): “In [53], Theorem 1(3), it is proved that the kernel of cans is isomorphic to L1Zs.”