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Lifts of longest elements to braid groups acting on derived categories

Author: Joseph Grant
Journal: Trans. Amer. Math. Soc. 367 (2015), 1631-1669
MSC (2010): Primary 18E30, 16E35, 16D50; Secondary 16E45, 20F36
Published electronically: September 5, 2014
MathSciNet review: 3286495
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Abstract: If we have a braid group acting on a derived category by spherical twists, how does a lift of the longest element of the symmetric group act? We give an answer to this question, using periodic twists, for the derived category of modules over a symmetric algebra. The question has already been answered by Rouquier and Zimmermann in a special case. We prove a lifting theorem for periodic twists, which allows us to apply their answer to the general case.

Along the way we study tensor products in derived categories of bimodules. We also use the lifting theorem to give new proofs of two known results: the existence of braid relations and, using the theory of almost Koszul duality due to Brenner, Butler, and King, the result of Rouquier and Zimmermann mentioned above.

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Additional Information

Joseph Grant
Affiliation: School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

Keywords: Symmetric algebra, braid group, longest element, derived equivalence, spherical twist, derived Picard group, almost Koszul duality
Received by editor(s): August 6, 2012
Received by editor(s) in revised form: December 30, 2012
Published electronically: September 5, 2014
Additional Notes: This work was first supported by the Japan Society for the Promotion of Science and then by the Engineering and Physical Sciences Research Council [grant number EP/G007947/1].
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.