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THE HALL ALGEBRAS OF SURFACES I

Part of: Lie algebras and Lie superalgebras Symplectic geometry, contact geometry

Published online by Cambridge University Press:  22 October 2018

Benjamin Cooper  and
Peter Samuelson
Benjamin Cooper
Affiliation:
University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa City, IA 52242-1419, USA ([email protected])
Peter Samuelson
Affiliation:
University of California, Riverside, Department of Mathematics, 900 University Ave., Riverside 92521, USA ([email protected])
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Abstract

We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with $m$ marked intervals and we give a conjectural description of the Hall algebras of all surfaces with enough marked intervals. Then we use a functoriality result to show that a graded version of the HOMFLY-PT skein relation holds among certain arcs in the Hall algebras of general surfaces.

Keywords

Fukaya categoriesHall algebrastopological field theory

MSC classification

Primary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 971 - 1028
Copyright
© Cambridge University Press 2018

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References

Abouzaid, M., On the Fukaya categories of higher genus surfaces, Adv. Math. 217(3) (2008), 1192–1235.Google Scholar
Antieau, B., Gepner, D. and Heller, J., K-theoretic obstructions to bounded t-structures, Preprint, 2016, arXiv:1610.07207v3.Google Scholar
Bergeron, F., Garsia, A., Sergel Leven, E. and Xin, G., Compositional (km, kn)-shuffle conjectures, Int. Math. Res. Not. IMRN (14) (2016), 4229–4270.Google Scholar
Blanchet, C., Habegger, N., Masbaum, G. and Vogel, P., Topological quantum field theories derived from the Kauffman bracket, Topology 34(4) (1995), 883–927.Google Scholar
Bocklandt, R., Noncommutative mirror symmetry for punctured surfaces, Trans. Amer. Math. Soc. 368(1) (2016), 429–469.Google Scholar
Bridgeland, T., Stability conditions on triangulated categories, Ann. of Math. (2) 166(2) (2007), 317–345.Google Scholar
Bridgeland, T., Quantum groups via Hall algebras of complexes, Ann. of Math. (2) 177(2) (2013), 739–759.Google Scholar
Brundan, J., Modular branching rules and the Mullineux map for Hecke algebras of type A, Proc. Lond. Math. Soc. (3) 77(3) (1998), 551–581.Google Scholar
Burban, I. and Schiffmann, O., On the Hall algebra of an elliptic curve, I, Duke Math. J. 161(7) (2012), 1171–1231.Google Scholar
Cherednik, I., Jones polynomials of torus knots via DAHA, Int. Math. Res. Not. IMRN 2013(23) (2013), 5366–5425.Google Scholar
Cooper, B. and Samuelson, P., The Hall algebras of surfaces II (in preparation).Google Scholar
Crane, L. and Frenkel, I. B., Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35(10) (1994), 5136–5154.Google Scholar
Dyckerhoff, T. and Kapranov, M., J. Eur. Math. Soc. (JEMS) 20(6) (2018), 1473–1524.Google Scholar
Feigin, B. L. and Tsymbaliuk, A. I., Equivariant K-theory of Hilbert schemes via shuffle algebra, Kyoto J. Math. 51(4) (2011), 831–854.Google Scholar
Goldman, W. M., Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85(2) (1986), 263–302.Google Scholar
Gorsky, E. and Negut, A., Refined knot invariants and Hilbert schemes, J. Math. Pures Appl. (9) 104(3) (2015), 403–435.Google Scholar
Haiden, F., Katzarkov, L. and Kontsevich, M., Flat surfaces and stability structures, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 247–318.Google Scholar
Henning, K., Krull–Schmidt categories and projective covers, Preprint, 2014, arXiv:1410.2822.Google Scholar
Hernandez, D. and Leclerc, B., Quantum grothendieck rings and derived hall algebras, J. Reine Angew. Math. 701 (2015), 77–126.Google Scholar
Kapranov, M., Heisenberg doubles and derived categories, J. Algebra 202(2) (1998), 712–744.Google Scholar
Kapustin, A., Katzarkov, L., Orlov, D. and Yotov, M., Homological mirror symmetry for manifolds of general type, Cent. Eur. J. Math. 7(4) (2009), 571–605.Google Scholar
Khovanov, M. and Lauda, A. D., A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309–347.Google Scholar
Lekili, Y. and Polishchuk, A., Auslander orders over nodal stacky curves and partially wrapped fukaya categories, J. Topol. 11(3) (2018), 615–644.Google Scholar
Lusztig, G., Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3(1) (1990), 257–296.Google Scholar
Lusztig, G., Introduction to Quantum Groups, Modern Birkhäuser Classics, (Birkhäuser/Springer, New York, 2010).Google Scholar
Morton, H. and Samuelson, P., The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra, Duke Math. J. 166(5) (2017), 801–854.Google Scholar
Nadler, D., Cyclic symmetries of A n-quiver representations, Adv. Math. 269 (2015), 346–363.Google Scholar
Negut, A., The shuffle algebra revisited, Int. Math. Res. Not. IMRN 2014(22) (2014), 6242–6275.Google Scholar
Polishchuk, A. and Zaslow, E., Categorical mirror symmetry in the elliptic curve, in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Studies in Advanced Mathematics, Volume 23, pp. 275–295 (Amer. Math. Soc., Providence, RI, 2001).Google Scholar
Ringel, C. M., Hall algebras and quantum groups, Invent. Math. 101 (1990), 583–592.Google Scholar
Rosso, M., An analogue of P.B.W. theorem and the universal R-matrix for U hsl(N + 1), Commun. Math. Phys. 124(2) (1989), 307–318.Google Scholar
Rouquier, R., 2-Kac-Moody algebras, Preprint, 2008, arXiv:0812.5023.Google Scholar
Schiffmann, O., Indecomposable vector bundles and stable Higgs bundles over smooth projective curves, Ann. of Math. (2) 183(1) (2016), 297–362.Google Scholar
Schiffmann, O. and Vasserot, E., Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A2, Publ. Math. Inst. Hautes Études Sci. 118 (2013), 213–342.Google Scholar
Schiffmann, O. and Vasserot, E., The elliptic Hall algebra and the K-theory of the Hilbert scheme of 𝔸2, Duke Math. J. 162(2) (2013), 279–366.Google Scholar
Seidel, P., Fukaya Categories and Picard–Lefschetz Theory, Zurich Lectures in Advanced Mathematics, (European Mathematical Society (EMS), Zürich, 2008).Google Scholar
Sibilla, N., Treumann, D. and Zaslow, E., Ribbon graphs and mirror symmetry, Selecta Math. (N.S.) 20(4) (2014), 979–1002.Google Scholar
Toën, B., Derived Hall algebras, Duke Math. J. 135(3) (2006), 587–615.Google Scholar
Turaev, V. G., Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Éc. Norm. Supér. (4) 24(6) (1991), 635–704.Google Scholar
Turaev, V. G., Quantum Invariants of Knots and 3-manifolds, De Gruyter Studies in Mathematics, Volume 18 (De Gruyter, Berlin, 2016).Google Scholar
Varagnolo, M. and Vasserot, E., Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67–100.Google Scholar
Witten, E., Quantum field theory and the jones polynomial, Commun. Math. Phys. 121 (1989), 351–399.Google Scholar
Xiao, J. and Xu, F., Hall algebras associated to triangulated categories, Duke Math. J. 143(2) (2008), 357–373.Google Scholar