Highest weight categories arising from Khovanov’s diagram algebra. IV: The general linear supergroup. (English) Zbl 1243.17004
In the first three papers of the series the authors studied representation-theoretic properties of a special class of infinite dimensional diagram algebras inspired by and obtained via a limiting procedure from algebras used by Khovanov in his categorification of the Jones polynomial. In the paper under review all this hard work pays off in a spectacular application to the representation theory of the general linear supergroup. The main result of the paper proves that blocks of the general linear supergroup are Morita equivalent to these generalized Khovanov algebras.
As an application one gets that blocks of the same atypicality are equivalent (this recovers a result by Serganova), that these blocks have a natural Koszul grading, that Kac modules are rigid (in the sense that they have a unique Loewy filtration) and many other interesting results.
To compare the two categories the authors use categorification approach and essentially realize both categories as underlying categories for certain 2-representations of a 2-Kac-Moody algebra.
Parts I–III, see Mosc. Math. J. 11, 685–722 (2011), http://arxiv.org/abs/0806.1532; Transform. Groups 15, No. 1, 1–45 (2010; Zbl 1205.17010); Represent. Theory 15, 170–243 (2011; Zbl 1261.17006).
As an application one gets that blocks of the same atypicality are equivalent (this recovers a result by Serganova), that these blocks have a natural Koszul grading, that Kac modules are rigid (in the sense that they have a unique Loewy filtration) and many other interesting results.
To compare the two categories the authors use categorification approach and essentially realize both categories as underlying categories for certain 2-representations of a 2-Kac-Moody algebra.
Parts I–III, see Mosc. Math. J. 11, 685–722 (2011), http://arxiv.org/abs/0806.1532; Transform. Groups 15, No. 1, 1–45 (2010; Zbl 1205.17010); Represent. Theory 15, 170–243 (2011; Zbl 1261.17006).
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 16S37 | Quadratic and Koszul algebras |
References:
| [1] | Bass, H.: Algebraic K-Theory. Benjamin, New York (1968) · Zbl 0174.30302 |
| [2] | Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9, 473-527 (1996) · Zbl 0864.17006 · doi:10.1090/S0894-0347-96-00192-0 |
| [3] | Boe, B., Kujawa, J., Nakano, D.: Complexity and module varieties for classical Lie super- algebras. Int. Math. Res. Notices 2011, no. 3, 696-724 · Zbl 1236.17031 · doi:10.1093/imrn/rnq090 |
| [4] | Brundan, J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalge- bra gl(m|n). J. Amer. Math. Soc. 16, 185-231 (2003) · Zbl 1050.17004 · doi:10.1090/S0894-0347-02-00408-3 |
| [5] | Brundan, J.: Tilting modules for Lie superalgebras. Comm. Algebra 32, 2251-2268 (2004) · Zbl 1077.17006 · doi:10.1081/AGB-120037218 |
| [6] | Brundan, J., Kleshchev, A.: Schur-Weyl duality for higher levels. Selecta Math. 14, 1-57 (2008) · Zbl 1211.17012 · doi:10.1007/s00029-008-0059-7 |
| [7] | Brundan, J., Kleshchev, A.: Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras. Invent. Math. 178, 451-484 (2009) · Zbl 1201.20004 · doi:10.1007/s00222-009-0204-8 |
| [8] | Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra I: cellularity. Moscow Math. J. 11, 685-722 (2011) · Zbl 1275.17012 |
| [9] | Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra II: Koszulity. Transform. Groups 15, 1-45 (2010) · Zbl 1205.17010 · doi:10.1007/s00031-010-9079-4 |
| [10] | Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra III: Category O. Represent. Theory 15, 170-243 (2011) · Zbl 1261.17006 · doi:10.1090/S1088-4165-2011-00389-7 |
| [11] | Cheng, S.-J., Kwon, J.-H., Lam, N.: A BGG-type resolution for tensor modules over general linear superalgebra. Lett. Math. Phys. 84, 75-87 (2008) · Zbl 1172.17005 · doi:10.1007/s11005-008-0231-1 |
| [12] | Cheng, S.-J., Lam, N.: Irreducible characters of general linear superalgebra and super du- ality. Comm. Math. Phys. 298, 645-672 (2010) · Zbl 1217.17004 · doi:10.1007/s00220-010-1087-7 |
| [13] | Cheng, S.-J., Wang, W.: Brundan-Kazhdan-Lusztig and super duality conjectures. Publ. Res. Inst. Math. Sci. 44, 1219-1272 (2008) · Zbl 1210.17009 · doi:10.2977/prims/1231263785 |
| [14] | Cheng, S.-J., Wang, W., Zhang, R. B.: Super duality and Kazhdan-Lusztig polynomials. Trans. Amer. Math. Soc. 360, 5883-5924 (2008) · Zbl 1234.17004 · doi:10.1090/S0002-9947-08-04447-4 |
| [15] | Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85-99 (1988) · Zbl 0657.18005 |
| [16] | Drinfeld, V.: Degenerate affine Hecke algebras and Yangians. Funct. Anal. Appl. 20, 69-70 (1986) · Zbl 0599.20049 · doi:10.1007/BF01077318 |
| [17] | Drouot, F.: Quelques propriétés des représentations de la super-alg‘ebre de Lie gl(m, n). PhD thesis, Université Henri Poincaré, Nancy (2008) |
| [18] | Kac, V.: Characters of typical representations of classical Lie superalgebras. Comm. Alge- bra 5, 889-897 (1977) · Zbl 0359.17010 · doi:10.1080/00927877708822201 |
| [19] | Khovanov, M.: A functor-valued invariant of tangles. Algebr. Geom. Topol. 2, 665-741 (2002) · Zbl 1002.57006 · doi:10.2140/agt.2002.2.665 |
| [20] | Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups I. Represent. Theory 13, 309-347 (2009) · Zbl 1188.81117 · doi:10.1090/S1088-4165-09-00346-X |
| [21] | Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023 |
| [22] | Serganova, V.: Kazhdan-Lusztig polynomials and character formula for the Lie superalge- bra gl(m|n). Selecta Math. 2, 607-651 (1996) · Zbl 0881.17005 · doi:10.1007/BF02433452 |
| [23] | Serganova, V.: Characters of irreducible representations of simple Lie superalgebras. In: Proc. Int. Congress of Mathematicians (Berlin, 1998), Vol. II, 583-593 · Zbl 0898.17002 |
| [24] | Stroppel, C.: Category O: quivers and endomorphism rings of projectives. Represent. Theory 7, 322-345 (2003) · Zbl 1050.17005 · doi:10.1090/S1088-4165-03-00152-3 |
| [25] | Stroppel, C.: Parabolic category O, perverse sheaves on Grassmannians, Springer fi- bres and Khovanov homology. Compos. Math. 145, 954-992 (2009) · Zbl 1187.17004 · doi:10.1112/S0010437X09004035 |
| [26] | Stroppel, C.: Schur-Weyl duality and link homology. In: Proc. Int. Congress of Mathe- maticians (Hyderabad, 2010), Vol. III, 1344-1365 · Zbl 1244.17006 |
| [27] | Zou, Y. M.: Categories of finite dimensional weight modules over type I classical Lie superalgebras. J. Algebra 180, 459-482 (1996) · Zbl 0843.17001 · doi:10.1006/jabr.1996.0077 |
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