# zbMATH — the first resource for mathematics

An equivalence between truncations of categorified quantum groups and Heisenberg categories. (Une équivalence entre des troncations de groupes quantiques catégorifiés et des catégories de Heisenberg.) (English. French summary) Zbl 1433.17011
Summary: We introduce a simple diagrammatic 2-category $$\mathcal{A}$$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $$\mathfrak{sl}_\infty$$. We show that $$\mathcal{A}$$ is equivalent to a truncation of the Khovanov-Lauda categorified quantum group $$\mathcal{U}$$ of type $$A_\infty$$, and also to a truncation of Khovanov’s Heisenberg 2-category $$\mathcal{H}$$. This equivalence is a categorification of the principal realization of the basic representation of $$\mathfrak{sl}_\infty$$. As a result of the categorical equivalences described above, certain actions of $$\mathcal{H}$$ induce actions of $$\mathcal{U}$$, and vice versa. In particular, we obtain an explicit action of $$\mathcal{U}$$ on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of $$\mathcal{H}$$. The 2-category $$\mathcal{A}$$ can be viewed as a graphical calculus describing the functors of $$i$$-induction and $$i$$-restriction for symmetric groups, together with the natural transformations between their compositions. The resulting computational tool is used to give simple diagrammatic proofs of (apparently new) representation theoretic identities.

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B65 Infinite-dimensional Lie (super)algebras 16D90 Module categories in associative algebras
Full Text:
##### References:
 [1] A. Beliakova, K. Habiro, A. D. Lauda & B. Webster, “Current algebras and categorified quantum groups”, J. London Math. Soc. (2)95 (2017) no. 1, p. 248-276 · Zbl 1407.17027 [2] J. Brundan & A. Kleshchev, “Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras”, Invent. Math.178 (2009) no. 3, p. 451-484 · Zbl 1201.20004 [3] J. Brundan & A. Kleshchev, “Graded decomposition numbers for cyclotomic Hecke algebras”, Adv. Math.222 (2009) no. 6, p. 1883-1942 · Zbl 1241.20003 [4] S. Cautis, J. Kamnitzer & A. Licata, “Coherent sheaves on quiver varieties and categorification”, Math. Ann.357 (2013) no. 3, p. 805-854 · Zbl 1284.14016 [5] S. Cautis & A. Licata, “Vertex operators and 2-representations of quantum affine algebras”, 2011 [6] S. Cautis & A. Licata, “Heisenberg categorification and Hilbert schemes”, Duke Math. J.161 (2012) no. 13, p. 2469-2547 · Zbl 1263.14020 [7] S. Cautis & A. D. Lauda, “Implicit structure in 2-representations of quantum groups”, Selecta Math. (N.S.)21 (2015) no. 1, p. 201-244 · Zbl 1370.17017 [8] S. Cautis, A. D. Lauda, A. Licata & J. Sussan, “W-algebras from Heisenberg categories”, J. Inst. Math. Jussieu (2016), online, · Zbl 1405.81045 [9] J. Chuang & R. Rouquier, “Derived equivalences for symmetric groups and $$\operatorname{\mathfrak{s}\mathfrak{l}}_2$$-categorification”, Ann. of Math. (2)167 (2008) no. 1, p. 245-298 · Zbl 1144.20001 [10] W. Fulton & J. Harris, Representation theory, Graduate Texts in Math. 129, Springer-Verlag, New York, 1991 [11] V. G. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990 · Zbl 0716.17022 [12] M. Khovanov, “Heisenberg algebra and a graphical calculus”, Fund. Math.225 (2014) no. 1, p. 169-210 · Zbl 1304.18019 [13] M. Khovanov & A. D. Lauda, “A categorification of quantum $$\operatorname{ sl }(n)$$”, Quantum Topol.1 (2010) no. 1, p. 1-92 · Zbl 1206.17015 [14] A. Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics 163, Cambridge University Press, Cambridge, 2005 · Zbl 1080.20011 [15] A. Kleshchev, “Modular representation theory of symmetric groups”, 2014 · Zbl 1373.20010 [16] J. Lemay, “Geometric realizations of the basic representation of $$\hat{\operatorname{\mathfrak{g}\mathfrak{l}}}_r$$”, Selecta Math. (N.S.)22 (2016) no. 1, p. 341-387 · Zbl 1395.17050 [17] A. Licata, D. Rosso & A. Savage, “Categorification and Heisenberg doubles arising from towers of algebras”, J. Combinatorial Theory Ser. A , to appear, · Zbl 1302.05204 [18] A. Licata & A. Savage, “Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves”, Selecta Math. (N.S.)16 (2010) no. 2, p. 201-240 · Zbl 1213.14011 [19] A. Licata & A. Savage, “Hecke algebras, finite general linear groups, and Heisenberg categorification”, Quantum Topol.4 (2013) no. 2, p. 125-185 · Zbl 1279.20006 [20] M. Mackaay & A. Savage, “Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification”, 2017 · Zbl 1437.20004 [21] M. Mackaay, M. Stošić & P. Vaz, “A diagrammatic categorification of the $$q$$-Schur algebra”, Quantum Topol.4 (2013) no. 1, p. 1-75 · Zbl 1272.81098 [22] K. Nagao, “Quiver varieties and Frenkel-Kac construction”, J. Algebra321 (2009) no. 12, p. 3764-3789 · Zbl 1196.17021 [23] H. Nakajima, “Quiver varieties and Kac-Moody algebras”, Duke Math. J.91 (1998) no. 3, p. 515-560 · Zbl 0970.17017 [24] H. Queffelec & D. E. V. Rose, “The $$\operatorname{\mathfrak{s}\mathfrak{l}}_n$$ foam 2-category: A combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality”, Adv. Math.302 (2016), p. 1251-1339 · Zbl 1360.57025 [25] R. Rouquier, “2-Kac-Moody algebras”, 2008 [26] D. Rosso & A. Savage, “A general approach to Heisenberg categorification via wreath product algebras”, Math. Z.286 (2017) no. 1-2, p. 603-655 · Zbl 1366.18006 [27] A. Savage, “A geometric boson-fermion correspondence”, C. R. Math. Rep. Acad. Sci. Canada28 (2006) no. 3, p. 65-84 · Zbl 1137.17025 [28] P. Shan, M. Varagnolo & E. Vasserot, “On the center of quiver Hecke algebras”, Duke Math. J.166 (2017) no. 6, p. 1005-1101 · Zbl 1380.20005 [29] M. Varagnolo & E. Vasserot, “Canonical bases and KLR-algebras”, J. reine angew. Math.659 (2011), p. 67-100 · Zbl 1229.17019 [30] B. Webster, “A categorical action on quantized quiver varieties”, 2012 [31] H. Zheng, “Categorification of integrable representations of quantum groups”, Acta Mech. Sinica (English Ed.)30 (2014) no. 6, p. 899-932     ##img## Creative Commons License BY-ND     ISSN : 2429-7100 - e-ISSN : 2270-518X · Zbl 1343.17012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.