Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T01:57:35.411Z Has data issue: false hasContentIssue false
  • English
  • Français

Diagrams and discrete extensions for finitary 2-representations

Published online by Cambridge University Press:  14 December 2017

AARON CHAN  and
VOLODYMYR MAZORCHUK
AARON CHAN
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden. e-mail: [email protected], [email protected]
VOLODYMYR MAZORCHUK
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden. e-mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary 2-representations of finitary 2-categories.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 2 , March 2019 , pp. 325 - 352
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Alperin, J. Diagrams for modules. J. Pure Appl. Algebra 16 (1980), no. 2, 111–119.Google Scholar
[2] Bernstein, J., Frenkel, I. and Khovanov, M. A categorification of the Temperley–Lieb algebra and Schur quotients of U(2) via projective and Zuckerman functors. Selecta Math. (N.S.) 5 (1999), no. 2, 199–241.Google Scholar
[3] Chuang, J. and Rouquier, R. Derived equivalences for symmetric groups and 2 -categorification. Ann. of Math. (2) 167 (2008), no. 1, 245–298.Google Scholar
[4] Crane, L. Clock and category: is quantum gravity algebraic? J. Math. Phys. 36 (1995), no. 11, 6180–6193.Google Scholar
[5] Crane, L. and Frenkel, I. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Topology and physics. J. Math. Phys. 35 (1994), no. 10, 5136–5154.Google Scholar
[6] Elias, B. The two-colour Soergel calculus. Composito. Math. 152 (2016), no. 2, 327–398.Google Scholar
[7] Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V. Tensor categories. (American Mathematical Society, 2015).Google Scholar
[8] Ganyushkin, O., Mazorchuk, V. and Steinberg, B. On the irreducible representations of a finite semigroup. Proc. Amer. Math. Soc. 137 (2009), no. 11, 3585–3592.Google Scholar
[9] Grensing, A.-L. and Mazorchuk, V. Categorification of the Catalan monoid. Semigroup Forum 89 (2014), no. 1, 155–168.Google Scholar
[10] Grensing, A.-L. and Mazorchuk, V. Finitary 2-categories associated with dual projection functors. Commun. Contemp. Math. 19 (2017), no. 3, 1650016, 40 pp.Google Scholar
[11] Huerfano, R. and Khovanov, M. A category for the adjoint representation. J. Algebra 246 (2001), no. 2, 514–542.Google Scholar
[12] Kazhdan, D. and Lusztig, G. Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979), no. 2, 165–184.Google Scholar
[13] Khovanov, M. A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359–426.Google Scholar
[14] Kildetoft, T. and Mazorchuk, V. Parabolic projective functors in type A. Adv. Math. 301 (2016), 785–803.Google Scholar
[15] Kudryavtseva, G. and Mazorchuk, V. On multisemigroups. Port. Math. 72 (2015), no. 1, 47–80.Google Scholar
[16] Leinster, T. Basic bicategories. Preprint arXiv:math/9810017.Google Scholar
[17] Lusztig, G. On a theorem of Benson and Curtis. J. Algebra 71 (1981), no. 2, 490–498.Google Scholar
[18] Mac Lane, S. Categories for the Working Mathematician (Springer–Verlag, 1998).Google Scholar
[19] Mazorchuk, V. and Miemietz, V. Cell 2-representations of finitary 2-categories. Compositi. Math. 147 (2011), 1519–1545.Google Scholar
[20] Mazorchuk, V. and Miemietz, V. Additive versus abelian 2-representations of fiat 2-categories. Moscow Math. J. 14 (2014), no. 3, 595–615.Google Scholar
[21] Mazorchuk, V. and Miemietz, V. Endmorphisms of cell 2-representations. Int. Math. Res. Notes Vol. 2016, No. 24, 7471–7498.Google Scholar
[22] Mazorchuk, V. and Miemietz, V. Morita theory for finitary 2-categories. Quantum Topol. 7 (2016), no. 1, 1–28.Google Scholar
[23] Mazorchuk, V. and Miemietz, V. Transitive 2-representations of finitary 2-categories. Trans. Amer. Math. Soc. 368 (2016), no. 11, 7623–7644.Google Scholar
[24] Mazorchuk, V. and Miemietz, V. Isotypic faithful 2-representations of -simple fiat 2-categories. Math. Z. 282 (2016), no. 1-2, 411–434.Google Scholar
[25] Munn, W. Matrix representations of semigroups. Proc. Camb. Phil. Soc. 53 (1957), 5–12.Google Scholar
[26] Psaroudakis, C. and Vitória, J. Recollements of module categories. Appl. Categ. Structures 22 (2014), no. 4, 579–593.Google Scholar
[27] Ringel, C.M. Exceptional modules are tree modules. Linear Algebra Appl. 275/276 (1998), 471–493.Google Scholar
[28] Rouquier, R. 2-Kac-Moody algebras. Preprint arXiv:0812.5023.Google Scholar
[29] Rouquier, R. Quiver Hecke algebras and 2-Lie algebras. Algebra Colloquium 19 (2012), 359–410.Google Scholar
[30] Shan, P., Varagnolo, M. and Vasserot, E. On the center of quiver-Hecke algebras. Duke Math. J. 166 (2017), no. 6, 1005–1101.Google Scholar
[31] Stroppel, C. Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors. Duke Math. J. 126 (2005), no. 3, 547–596.Google Scholar
[32] Xantcha, Q. Gabriel 2-quivers for finitary 2-categories. J. Lond. Math. Soc. (2) 92 (2015), no. 3, 615–632.Google Scholar
[33] Zhang, X. Duflo involutions for 2-categories associated to tree quivers. J. Algebra Appl. 15 (2016), no. 3, 1650041, 25 pp.Google Scholar
[34] Zhang, X. Simple transitive 2-representations and Drinfeld center for some finitary 2-categories. J. Pure Appl. Algebra 222 (2018), no. 1, 97–130.Google Scholar
[35] Zimmermann, J. Simple transitive 2-representations of Soergel bimodules in type B 2. J. Pure Appl. Algebra 221 (2017), no. 3, 666–690.Google Scholar