Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T15:59:26.311Z Has data issue: false hasContentIssue false
  • English
  • Français

A CATEGORIFICATION OF A QUANTUM FROBENIUS MAP

Part of: Homological methods Groups and algebras in quantum theory

Published online by Cambridge University Press:  20 July 2017

You Qi Open the ORCID record for You Qi [Opens in a new window]
You Qi*
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06511, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

A quantum Frobenius map a la Lusztig for $\mathfrak{s}\mathfrak{l}_{2}$ is categorified at a prime root of unity.

Keywords

Categorificationquantum 𝔰𝔩2Frobenius maproots of unityhopfological algebra

MSC classification

Primary: 81R50: Quantum groups and related algebraic methods 16E20: Grothendieck groups, $K$-theory, etc. 16E35: Derived categories
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 5 , September 2019 , pp. 899 - 939
Copyright
© Cambridge University Press 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

Arkhipov, S., Bezrukavnikov, R. and Ginzburg, V., Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17(3) (2004), 595–678.Google Scholar
Arkhipov, S. and Gaitsgory, D., Another realization of the category of modules over the small quantum group, Adv. Math. 173(1) (2003), 114–143.Google Scholar
Brundan, J., On the definition of Kac–Moody 2-category, Math. Ann. 364(1–2) (2016), 353–372.Google Scholar
Cautis, S. and Lauda, A. D., Implicit structure in 2-representations of quantum groups, Selecta Math. (N.S.) 21(1) (2015), 201–244.Google Scholar
Drinfeld, V., DG quotients of DG categories, J. Algebra 272(2) (2004), 643–691.Google Scholar
Elias, B. and Qi, Y., A categorification of quantum sl(2) at prime roots of unity, Adv. Math. 299 (2016), 863–930.Google Scholar
Elias, B. and Qi, Y., A categorification of some small quantum groups II, Adv. Math. 288 (2016), 81–151.Google Scholar
Ellis, A. P. and Qi, Y., The differential graded odd nilHecke algebra, Comm. Math. Phys. 344(1) (2016), 275–331.Google Scholar
Kazhdan, D. and Lusztig, G., Tensor structures arising from affine Lie algebras IV, J. Amer. Math. Soc. 7(2) (1994), 383–453.Google Scholar
Khovanov, M., Hopfological algebra and categorification at a root of unity: the first steps, J. Knot Theory Ramifications 25(3) (2016).Google Scholar
Khovanov, M. and Lauda, A. D., A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309–347.Google Scholar
Khovanov, M. and Lauda, A. D., A categorification of quantum sl(n), Quantum Topol. 2(1) (2010), 1–92.Google Scholar
Khovanov, M. and Lauda, A. D., A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363(5) (2011), 2685–2700.Google Scholar
Khovanov, M., Lauda, A. D., Mackaay, M. and Stošić, M., Extended graphical calculus for categorified quantum sl(2), Mem. Amer. Math. Soc. 219 (2012), vi + 87.Google Scholar
Khovanov, M. and Qi, Y., An approach to categorification of some small quantum groups, Quantum Topol. 6(2) (2015), 185–311.Google Scholar
Lauda, A. D., A categorification of quantum sl(2), Adv. Math. 225(6) (2010), 3327–3424.Google Scholar
Lusztig, G., Modular representations and quantum groups, in Classical Groups and Related Topics (Beijing, 1987), Contemporary Mathematics, Volume 82, pp. 59–77 (American Mathematical Society, Providence, RI, 1989).Google Scholar
Lusztig, G., Introduction to Quantum Groups, Progress in Mathematics, Volume 110 (Birkhäuser Boston Inc., Boston, MA, 1993).Google Scholar
McGerty, K., Hall algebras and quantum Frobenius, Duke Math. J. 154(1) (2010), 181–206.Google Scholar
Qi, Y., Hopfological algebra, Compos. Math. 150(01) (2014), 1–45.Google Scholar
Rouquier, R., 2-Kac-Moody algebras, Preprint, 2008, arXiv:0812.5023.Google Scholar
Stošić, M., Indecomposable objects and Lusztig’s canonical basis, Math. Res. Lett. 22(1) (2015), 245–278.Google Scholar