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On the definition of Heisenberg category. (English) Zbl 06963903
Summary: We revisit the definition of the Heisenberg category of central charge $$k \in \mathbb{Z}$$. For central charge $$-1$$, this category was introduced originally by Khovanov, but with some additional cyclicity relations which we show here are unnecessary. For other negative central charges, the definition is due to Mackaay and Savage, also with some redundant relations, while central charge zero recovers the affine oriented Brauer category of Brundan, Comes, Nash and Reynolds. We also discuss cyclotomic quotients.

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
##### Keywords:
Heisenberg category; string calculus
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##### References:
 [1] Ariki, Susumu, On the decomposition numbers of the Hecke algebra of $$G(m,1,n)$$, J. Math. Kyoto Univ., 36, 4, 789-808, (1996) · Zbl 0888.20011 [2] Brundan, Jonathan, On the definition of Kac-Moody 2-category, Math. Ann., 364, 1-2, 353-372, (2016) · Zbl 1395.17014 [3] Brundan, Jonathan, Representations of oriented skein categories, (2017) [4] Brundan, Jonathan; Comes, Jonathan; Nash, David; Reynolds, Andrew, A basis theorem for the affine oriented Brauer category and its cyclotomic quotients, Quantum Topol., 8, 1, 75-112, (2017) · Zbl 1419.18011 [5] Brundan, Jonathan; Davidson, Nicholas, Categorification and higher representation theory, 684, Categorical actions and crystals, 116-159, (2017), American Mathematical Society · Zbl 1418.17053 [6] Brundan, Jonathan; Kleshchev, Alexander, Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math., 222, 6, 1883-1942, (2009) · Zbl 1241.20003 [7] Brundan, Jonathan; Savage, Alistair, On the definition of quantum Heisenberg category · Zbl 06963903 [8] Cautis, Sabin; Lauda, Aaron; Licata, Anthony; Samuelson, Peter; Sussan, Joshua, The elliptic Hall algebra and the deformed Khovanov Heisenberg category, (2016) · Zbl 1405.81045 [9] Cautis, Sabin; Lauda, Aaron; Licata, Anthony; Sussan, Joshua, $$W$$-algebras from Heisenberg categories, J. Inst. Math. Jussieu, 1-37, (2016) · Zbl 1405.81045 [10] Cautis, Sabin; Licata, Anthony, Heisenberg categorification and Hilbert schemes, Duke Math. J., 161, 13, 2469-2547, (2012) · Zbl 1263.14020 [11] Comes, Jonathan; Kujawa, Jonathan, Higher level twisted Heisenberg supercategories [12] Hill, David; Sussan, Joshua, A categorification of twisted Heisenberg algebras, Adv. Math., 295, 368-420, (2016) · Zbl 1405.17017 [13] Khovanov, Mikhail, Heisenberg algebra and a graphical calculus, Fundam. Math., 225, 169-210, (2014) · Zbl 1304.18019 [14] Khovanov, Mikhail; Lauda, Aaron, A categorification of quantum $$\mathfrak{sl}(n)$$, Quantum Topol., 1, 1, 1-92, (2010) · Zbl 1206.17015 [15] Kleshchev, Alexander, Linear and projective representations of symmetric groups, xiv+277 pp., (2005), Cambridge University Press · Zbl 1080.20011 [16] Licata, Anthony; Savage, Alistair, Hecke algebras, finite general linear groups, and Heisenberg categorification, Quantum Topol., 4, 2, 125-185, (2013) · Zbl 1279.20006 [17] Mackaay, Marco; Savage, Alistair, Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification, J. Algebra, 505, 150-193, (2018) · Zbl 1437.20004 [18] Queffelec, Hervé; Savage, Alistair; Yacobi, Oded, An equivalence between truncations of categorified quantum groups and Heisenberg categories, J. Éc. Polytech., Math., 5, 192-238, (2018) [19] Rosso, Daniele; Savage, Alistair, A general approach to Heisenberg categorification via wreath product algebras, Math. Z., 286, 1-2, 603-655, (2017) · Zbl 1366.18006 [20] Rouquier, Raphael, 2-Kac-Moody algebras, (2008) [21] Rouquier, Raphael, Quiver Hecke algebras and $$2$$-Lie algebras, Algebra Colloq., 19, 2, 359-410, (2012) · Zbl 1247.20002 [22] Rui, Hebing; Su, Yucai, Affine walled Brauer algebras and super Schur-Weyl duality, Adv. Math., 285, 28-71, (2015) · Zbl 1356.17012 [23] Savage, Alistair, Frobenius Heisenberg categorification, (2018) [24] Webster, Ben, Canonical bases and higher representation theory, Compos. Math., 151, 1, 121-166, (2015) · Zbl 1393.17029
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