Lifts of longest elements to braid groups acting on derived categories

Grant, Joseph

doi:10.1090/S0002-9947-2014-06104-7

Lifts of longest elements to braid groups acting on derived categories
HTML articles powered by AMS MathViewer

by Joseph Grant PDF

Trans. Amer. Math. Soc. 367 (2015), 1631-1669 Request permission

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.

References

Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
A. I. Bondal and M. M. Kapranov, Framed triangulated categories, Mat. Sb. 181 (1990), no. 5, 669–683 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 1, 93–107. MR 1055981, DOI 10.1070/SM1991v070n01ABEH001253
Alexander Braverman and Dennis Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (1996), no. 2, 315–328. MR 1383469, DOI 10.1006/jabr.1996.0122
M. C. R. Butler and A. D. King, Minimal resolutions of algebras, J. Algebra 212 (1999), no. 1, 323–362. MR 1670674, DOI 10.1006/jabr.1998.7599
Sheila Brenner, Michael C. R. Butler, and Alastair D. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory 5 (2002), no. 4, 331–367. MR 1930968, DOI 10.1023/A:1020146502185
Samuel Eilenberg, Alex Rosenberg, and Daniel Zelinsky, On the dimension of modules and algebras. VIII. Dimension of tensor products, Nagoya Math. J. 12 (1957), 71–93. MR 98774
Joseph Grant, Derived autoequivalences from periodic algebras, Proc. Lond. Math. Soc. (3) 106 (2013), no. 2, 375–409. MR 3021466, DOI 10.1112/plms/pds043
Joseph Grant, Derived autoequivalences and braid relations, Proceedings of the 44th Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent. Theory Organ. Comm., Nagoya, 2012, pp. 50–54. MR 2975605
Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389. MR 910167, DOI 10.1007/BF02564452
Ruth Stella Huerfano and Mikhail Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), no. 2, 514–542. MR 1872113, DOI 10.1006/jabr.2001.8962
Birger Iversen, Octahedra and braids, Bull. Soc. Math. France 114 (1986), no. 2, 197–213 (English, with French summary). MR 860816
Bernhard Keller and Amnon Neeman, The connection between May’s axioms for a triangulated tensor product and Happel’s description of the derived category of the quiver $D_4$, Doc. Math. 7 (2002), 535–560. MR 2015053
Hideya Matsumoto, Générateurs et relations des groupes de Weyl généralisés, C. R. Acad. Sci. Paris 258 (1964), 3419–3422 (French). MR 183818
J. P. May, The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1, 34–73. MR 1867203, DOI 10.1006/aima.2001.1995
Volodymyr Mazorchuk, Serge Ovsienko, and Catharina Stroppel, Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1129–1172. MR 2457393, DOI 10.1090/S0002-9947-08-04539-X
Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. MR 265437, DOI 10.1090/S0002-9947-1970-0265437-8
Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48. MR 1099084, DOI 10.1112/jlms/s2-43.1.37
Raphaël Rouquier and Alexander Zimmermann, Picard groups for derived module categories, Proc. London Math. Soc. (3) 87 (2003), no. 1, 197–225. MR 1978574, DOI 10.1112/S0024611503014059
P. Seidel, Lagrangian homology spheres in $(A_m)$ Milnor fibres via $\mathbb {C}^*$-equivariant $A_\infty$-modules, arXiv:1202.1955v3 [math.SG], preprint 2012.
Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, DOI 10.1215/S0012-7094-01-10812-0