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Type C blocks of super category \(\mathcal{O}\). (English) Zbl 1446.17029
Authors’ abstract: “We show that the blocks of category \(\mathcal O\) for the Lie superalgebra \(\mathfrak q_n(\mathbb C)\) associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type \(C_\infty\). This allows us to prove two conjectures formulated by Cheng, Kwon and Wang. We then focus on the full subcategory consisting of finite-dimensional representations, which we show is a highest weight category with blocks that are Morita equivalent to certain generalized Khovanov arc algebras.”
The Lie superalgebra \(\mathfrak q_n(\mathbb C)\) is the subalgebra of the general linear Lie superalgebra \(\mathfrak{gl}_{n|n}(\mathbb C)\), consisting of matrices of the form \(\begin{pmatrix} A & B \\ B & A \end{pmatrix}\). The main ingredient is some involved combinatorics related to tensor product modules over the Lie algebras \(\mathfrak{sl}_{+\infty}\) and \(\mathfrak{sp}_{2\infty}\).

17B45 Lie algebras of linear algebraic groups
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Zbl 1406.17014
Full Text: DOI
[1] Brundan, J., Losev, I., Webster, B.: Tensor product categorifications and the super Kazhdan-Lusztig Conjecture. Int. Math. Res. Notices 20, 6329-6410 (2017) · Zbl 1405.17045
[2] Brundan, J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra \[{{\mathfrak{q}}}(n)\] q(n). Adv. Math. 182, 28-77 (2004) · Zbl 1048.17003
[3] Brundan, J.: Tilting modules for Lie superalgebras. Commun. Algebra 32, 2251-2268 (2004) · Zbl 1077.17006
[4] Brundan, J.; Mason, G. (ed.); etal., Representations of the general linear Lie superalgebra in the BGG category \[{\cal{O}} O, No. 38, 71-98 (2014)\], Berlin
[5] Brundan, J., Davidson, N.: Categorical actions and crystals. Contemp. Math. 684, 116-159 (2017) · Zbl 1418.17053
[6] Brundan, J., Davidson, N.: Type A blocks of super category \[{\cal{O}}O\]. J. Algebra 473, 447-480 (2017) · Zbl 1396.17005
[7] Brundan, J., Ellis, A.: Monoidal supercategories. Commun. Math. Phys. 351, 1045-1089 (2017) · Zbl 1396.17012
[8] Brundan, J., Kleshchev, A.: Hecke-Clifford superalgebras, crystals of type \[A_{2\ell }^{(2)}\] A2ℓ(2) and modular branching rules for \[{\widehat{S}}_nS\]^n. Represent. Theory 5, 317-403 (2001) · Zbl 1005.17010
[9] Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebras I: cellularity. Mosc. Math. J. 11, 685-722 (2011) · Zbl 1275.17012
[10] Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebras IV: the general linear supergroup. JEMS 14, 373-419 (2012) · Zbl 1243.17004
[11] Chen, C.-W.: Reduction method for representations of queer Lie superalgebras. J. Math. Phys. 57(5), 051703-12 (2016) · Zbl 1386.17011
[12] Cheng, S.-J., Kwon, J.-H., Wang, W.: Character formulae for queer Lie superalgebras and canonical bases of types A/C. Commun. Math. Phys. 352, 1091-1119 (2017) · Zbl 1406.17014
[13] Cheng, S.-J., Kwon, J.-H.: Finite-dimensional half-integer weight modules over queer Lie superalgebras. Commun. Math. Phys. 346, 945-965 (2016) · Zbl 1406.17017
[14] Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85-99 (1988) · Zbl 0657.18005
[15] Davidson, N.: Type B blocks of super category \[{\cal{O}}O\] (in preparation)
[16] Hill, D., Kujawa, J., Sussan, J.: Degenerate affine Hecke-Clifford algebras and type Q Lie superalgebras. Math. Z. 268, 1091-1158 (2011) · Zbl 1231.20005
[17] Jantzen, J.C.: Lectures on Quantum Groups, AMS (1995) · Zbl 0842.17012
[18] Kac, V.: Characters of typical representations of classical Lie superalgebras. Commun. Algebra 5, 889-897 (1977) · Zbl 0359.17010
[19] Kang, S.-J., Kashiwara, M., Tsuchioka, S.: Quiver Hecke superalgebras. J. Reine Angew. Math. 711, 1-54 (2016) · Zbl 1360.17018
[20] Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups I. Represent. Theory 13, 309-347 (2009) · Zbl 1188.81117
[21] Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups II. Trans. Am. Math. Soc. 363, 2685-2700 (2011) · Zbl 1214.81113
[22] Losev, I., Webster, B.: On uniqueness of tensor products of irreducible categorifications. Sel. Math. 21, 345-377 (2015) · Zbl 1359.17013
[23] Lusztig, G.: Introduction to Quantum Groups. Birkhäuser, Basel (1993) · Zbl 0788.17010
[24] Nazarov, M.: Young’s symmetrizers for projective representations of the symmetric group. Adv. Math. 127, 190-257 (1997) · Zbl 0930.20011
[25] Penkov, I.: Characters of typical irreducible finite-dimensional \[\mathfrak{q}(n)\] q(n)-modules. Funct. Anal. Appl. 20, 30-37 (1986) · Zbl 0595.17003
[26] Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023 · Zbl 1247.20002
[27] Webster, B.: Canonical bases and higher representation theory. Compos. Math. 151, 121-166 (2015) · Zbl 1393.17029
[28] Webster, B.: Knot invariants and higher representation theory. Mem. Am. Math. Soc. 1191, 133 (2017) · Zbl 1446.57001
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