Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Transitive $ 2$-representations of finitary $ 2$-categories


Authors: Volodymyr Mazorchuk and Vanessa Miemietz
Journal: Trans. Amer. Math. Soc. 368 (2016), 7623-7644
MSC (2010): Primary 18D05; Secondary 16D20, 17B10, 16G10
DOI: https://doi.org/10.1090/tran/6583
Published electronically: December 22, 2015
MathSciNet review: 3546777
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we define and study the class of simple transitive $ 2$-representations of finitary $ 2$-categories. We prove a weak version of the classical Jordan-Hölder Theorem where the weak composition subquotients are given by simple transitive $ 2$-representations. For a large class of finitary $ 2$-categories we prove that simple transitive $ 2$-representations are exhausted by cell $ 2$-representations. Finally, we show that this large class contains finitary quotients of $ 2$-Kac-Moody algebras.


References [Enhancements On Off] (What's this?)

  • [AM] Troels Agerholm and Volodymyr Mazorchuk, On selfadjoint functors satisfying polynomial relations, J. Algebra 330 (2011), 448-467. MR 2774639 (2012e:16041), https://doi.org/10.1016/j.jalgebra.2011.01.004
  • [BFK] Joseph Bernstein, Igor Frenkel, and Mikhail Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $ U(\mathfrak{sl}_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199-241. MR 1714141 (2000i:17009), https://doi.org/10.1007/s000290050047
  • [BG] J. N. Bernstein and S. I. Gelfand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245-285. MR 581584 (82c:17003)
  • [CL] Sabin Cautis and Aaron D. Lauda, Implicit structure in 2-representations of quantum groups, Selecta Math. (N.S.) 21 (2015), no. 1, 201-244. MR 3300416, https://doi.org/10.1007/s00029-014-0162-x
  • [CR] Joseph Chuang and Raphaël Rouquier, Derived equivalences for symmetric groups and $ \mathfrak{sl}_2$-categorification, Ann. of Math. (2) 167 (2008), no. 1, 245-298. MR 2373155 (2008m:20011), https://doi.org/10.4007/annals.2008.167.245
  • [DG] S. Doty and A. Giaquinto, Cellular bases of generalized $ q$-Schur algebras, Preprint arXiv:1012.5983v3.
  • [EW] Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089-1136. MR 3245013, https://doi.org/10.4007/annals.2014.180.3.6
  • [Fr] Peter Freyd, Representations in abelian categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 95-120. MR 0209333 (35 #231)
  • [Fro1] G. Frobenius, Über Matrizen aus positiven Elementen, 1, Sitzungsber. Königl. Preuss. Akad. Wiss. (1908), 471-476.
  • [Fro2] G. Frobenius, Über Matrizen aus positiven Elementen, 2, Sitzungsber. Königl. Preuss. Akad. Wiss. (1909), 514-518.
  • [GM] Olexandr Ganyushkin and Volodymyr Mazorchuk, Classical finite transformation semigroups: An introduction, Algebra and Applications, vol. 9, Springer-Verlag London, Ltd., London, 2009. MR 2460611 (2009i:20123)
  • [KK] Seok-Jin Kang and Masaki Kashiwara, Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras, Invent. Math. 190 (2012), no. 3, 699-742. MR 2995184, https://doi.org/10.1007/s00222-012-0388-1
  • [Ka] Masaki Kashiwara, Biadjointness in cyclotomic Khovanov-Lauda-Rouquier algebras, Publ. Res. Inst. Math. Sci. 48 (2012), no. 3, 501-524. MR 2973390, https://doi.org/10.2977/PRIMS/78
  • [KL] Mikhail Khovanov and Aaron D. Lauda, A categorification of quantum $ {\rm sl}(n)$, Quantum Topol. 1 (2010), no. 1, 1-92. MR 2628852 (2011g:17028), https://doi.org/10.4171/QT/1
  • [Le] T. Leinster, Basic bicategories, Preprint arXiv:math/9810017.
  • [McL] Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872 (2001j:18001)
  • [MM1] Volodymyr Mazorchuk and Vanessa Miemietz, Cell 2-representations of finitary 2-categories, Compos. Math. 147 (2011), no. 5, 1519-1545. MR 2834731, https://doi.org/10.1112/S0010437X11005586
  • [MM2] Volodymyr Mazorchuk and Vanessa Miemietz, Additive versus abelian $ 2$-representations of fiat $ 2$-categories, Mosc. Math. J. 14 (2014), no. 3, 595-615, 642 (English, with English and Russian summaries). MR 3241761
  • [MM3] Volodymyr Mazorchuk and Vanessa Miemietz, Endomorphisms of cell $ 2$-representations, Preprint arXiv:1207.6236.
  • [MM4] Volodymyr Mazorchuk and Vanessa Miemietz, Morita theory for finitary $ 2$-categories, Preprint arXiv:1304.4698. To appear in Quantum Topol.
  • [Me] Carl Meyer, Matrix analysis and applied linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. With 1 CD-ROM (Windows, Macintosh and UNIX) and a solutions manual (iv+171 pp.). MR 1777382
  • [Pe] Oskar Perron, Zur Theorie der Matrices, Math. Ann. 64 (1907), no. 2, 248-263 (German). MR 1511438, https://doi.org/10.1007/BF01449896
  • [Ro1] Raphaël Rouquier, $ 2$-Kac-Moody algebras, Preprint arXiv:0812.5023.
  • [Ro2] Raphaël Rouquier, Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq. 19 (2012), no. 2, 359-410. MR 2908731, https://doi.org/10.1142/S1005386712000247
  • [So] Wolfgang Soergel, The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math. 429 (1992), 49-74. MR 1173115 (94b:17011), https://doi.org/10.1515/crll.1992.429.49
  • [VV] M. Varagnolo and E. Vasserot, Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67-100. MR 2837011, https://doi.org/10.1515/CRELLE.2011.068
  • [We] B. Webster, Knot invariants and higher representation theory, Preprint arXiv:1309.3796
  • [Xa] Q. Xantcha, Gabriel $ 2$-quivers for finitary $ 2$-categories, Preprint arXiv:1310.1586.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 18D05, 16D20, 17B10, 16G10

Retrieve articles in all journals with MSC (2010): 18D05, 16D20, 17B10, 16G10


Additional Information

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, Box 480, 751 06, Uppsala, Sweden
Email: mazor@math.uu.se

Vanessa Miemietz
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
Email: v.miemietz@uea.ac.uk

DOI: https://doi.org/10.1090/tran/6583
Received by editor(s): May 14, 2014
Received by editor(s) in revised form: September 18, 2014
Published electronically: December 22, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society