Lifts of longest elements to braid groups acting on derived categories
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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
- Email: j.s.grant@leeds.ac.uk
- 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].
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 1631-1669
- MSC (2010): Primary 18E30, 16E35, 16D50; Secondary 16E45, 20F36
- DOI: https://doi.org/10.1090/S0002-9947-2014-06104-7
- MathSciNet review: 3286495