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The elliptic Hall algebra and the deformed Khovanov Heisenberg category. (English) Zbl 1402.81172
Summary: We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined in [A. Licata and A. Savage, Quantum Topol. 4, No. 2, 125–185 (2013; Zbl 1279.20006)]. We also show that as an algebra, it is isomorphic to “half” of a central extension of the elliptic Hall algebra of I. Burban and O. Schiffmann [Duke Math. J. 161, No. 7, 1171–1231 (2012; Zbl 1286.16029)], specialized at \(\sigma = \bar{\sigma}^{-1} = q\). A key step in the proof may be of independent interest: we show that the sum (over \(n\)) of the Hochschild homologies of the positive affine Hecke algebras \(\mathrm{AH}_n^+\) is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the \(q\)-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.

81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
20C08 Hecke algebras and their representations
17B65 Infinite-dimensional Lie (super)algebras
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
Full Text: DOI
[1] Beliakova, A.; Guliyev, Z.; Habiro, K.; Lauda, AD, Trace as an alternative decategorification functor, Acta Math. Vietnam., 39, 425-480, (2014) · Zbl 1331.81153
[2] Beliakova, A.; Habiro, K.; Lauda, AD; Webster, B., Current algebras and categorified quantum groups, J. Lond. Math. Soc., 95, 248-276, (2017) · Zbl 1407.17027
[3] Beliakova, A.; Habiro, K.; Lauda, AD; Zivkovic, M., Trace decategorification of categorified quantum \(\mathfrak{sl}_2\), Math. Ann., 367, 397-440, (2017) · Zbl 1361.81075
[4] Burban, I.; Schiffmann, O., On the Hall algebra of an elliptic curve, I, Duke Math. J., 161, 1171-1231, (2012) · Zbl 1286.16029
[5] Cautis, Sabin; Lauda, Aaron D.; Licata, Anthony M.; Sussan, Joshua, W-ALGEBRAS FROM HEISENBERG CATEGORIES, Journal of the Institute of Mathematics of Jussieu, 17, 981-1017, (2016) · Zbl 1405.81045
[6] Ding, J.; Iohara, K., Generalization of Drinfeld quantum affine algebras, Lett. Math. Phys., 41, 181-193, (1997) · Zbl 0889.17011
[7] Dipper, R.; James, G., Blocks and idempotents of Hecke algebras of general linear groups, Proc. Lond. Math. Soc. (3), 54, 57-82, (1987) · Zbl 0615.20009
[8] Elias, B.; Lauda, AD, Trace decategorification of the Hecke category, J. Algebra, 449, 615-634, (2016) · Zbl 1368.18003
[9] Feigin, B.; Feigin, E.; Jimbo, M.; Miwa, T.; Mukhin, E., Quantum continuous \(\mathfrak{gl}_\infty \): semiinfinite construction of representations, Kyoto J. Math., 51, 337-364, (2011) · Zbl 1278.17012
[10] Feigin, BL; Tsymbaliuk, AI, Equivariant \(K\)-theory of Hilbert schemes via shuffle algebra, Kyoto J. Math., 51, 831-854, (2011) · Zbl 1242.14006
[11] Geck, M., Pfeiffer, G.: Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras. London Mathematical Society Monographs, New Series, vol. 21. The Clarendon Press, Oxford University Press, New York (2000) · Zbl 0996.20004
[12] Khovanov, M., Heisenberg algebra and a graphical calculus, Fund. Math., 225, 169-210, (2014) · Zbl 1304.18019
[13] Licata, A.; Savage, A., Hecke algebras, finite general linear groups, and Heisenberg categorification, Quantum Topol., 4, 125-185, (2013) · Zbl 1279.20006
[14] Lukac, SG, Idempotents of the Hecke algebra become Schur functions in the skein of the annulus, Math. Proc. Camb. Philos. Soc., 138, 79-96, (2005) · Zbl 1083.20002
[15] Miki, K., A \((q,\gamma )\) analog of the \(W_{1+\infty }\) algebra, J. Math. Phys., 48, 123520, (2007) · Zbl 1153.81405
[16] Morton, HR; Manchón, PMG, Geometrical relations and plethysms in the Homfly skein of the annulus, J. Lond. Math. Soc. (2), 78, 305-328, (2008) · Zbl 1153.57011
[17] Morton, H.R.: Power sums and Homfly skein theory. Invariants of Knots and 3-Manifolds (Kyoto, 2001). Geometry & Topology Monographs, vol. 4, pp. 235-244. Geometry and Topology Publication, Coventry (2002). arXiv:math/0111101(electronic) · Zbl 1035.57007
[18] Morton, H.; Samuelson, P., The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra, Duke Math. J., 166, 801-854, (2017) · Zbl 1369.16034
[19] Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces. University Lecture Series, vol. 18. American Mathematical Society, Providence (1999) · Zbl 0949.14001
[20] Negut, Andrei, The Shuffle Algebra Revisited, International Mathematics Research Notices, 2014, 6242-6275, (2013) · Zbl 1310.16030
[21] Negut, A., Moduli of flags of sheaves and their \(K\)-theory, Algebr. Geom., 2, 19-43, (2015) · Zbl 1322.14029
[22] Przytycki, J.: Skein module of links in a handlebody. Topology ’90 (Columbus, OH, 1990). Ohio State University Mathematics Research Institute Publication, vol. 1, pp. 315-342. de Gruyter, Berlin (1992) · Zbl 0770.57005
[23] Ringel, CM, Hall algebras and quantum groups, Invent. Math., 101, 583-591, (1990) · Zbl 0735.16009
[24] Schiffmann, O.; Vasserot, E., The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compos. Math., 147, 188-234, (2011) · Zbl 1234.20005
[25] Schiffmann, O.; Vasserot, E., Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on \( {A}^2\), Publ. Math. Inst. Hautes Études Sci., 118, 213-342, (2013) · Zbl 1284.14008
[26] Schiffmann, O.; Vasserot, E., The elliptic Hall algebra and the \(K\)-theory of the Hilbert scheme of \(\mathbb{A}^2\), Duke Math. J., 162, 279-366, (2013) · Zbl 1290.19001
[27] Shan, P.; Varagnolo, M.; Vasserot, E., On the center of quiver Hecke algebras, Duke Math. J., 166, 1005-1101, (2017) · Zbl 1380.20005
[28] Wan, J.; Wang, W., Frobenius map for the centers of Hecke algebras, Trans. Am. Math. Soc., 367, 5507-5520, (2015) · Zbl 1323.20009
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