Quasi-Frobenius-Lusztig kernels for simple Lie algebras

Liu, Gongxiang; Van Oystaeyen, Fred; Zhang, Yinhuo

doi:10.1090/tran/6731

Quasi-Frobenius-Lusztig kernels for simple Lie algebras

Authors: Gongxiang Liu, Fred Van Oystaeyen and Yinhuo Zhang
Journal: Trans. Amer. Math. Soc. 369 (2017), 2049-2086
MSC (2010): Primary 17B37; Secondary 16T05
DOI: https://doi.org/10.1090/tran/6731
Published electronically: August 22, 2016
MathSciNet review: 3581227
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the first author’s Math. Res. Lett. paper (2014), the quasi-Frobenius-Lusztig kernel associated with $\mathfrak {sl}_{2}$ was constructed. In this paper we construct the quasi-Frobenius-Lusztig kernels associated with any simple Lie algebra $\mathfrak {g}$.

References

Nicolás Andruskiewitsch and Hans-Jürgen Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math. (2) 171 (2010), no. 1, 375–417. MR 2630042, DOI https://doi.org/10.4007/annals.2010.171.375
Iván Ezequiel Angiono, Basic quasi-Hopf algebras over cyclic groups, Adv. Math. 225 (2010), no. 6, 3545–3575. MR 2729015, DOI https://doi.org/10.1016/j.aim.2010.06.013
Damien Calaque and Pavel Etingof, Lectures on tensor categories, Quantum groups, IRMA Lect. Math. Theor. Phys., vol. 12, Eur. Math. Soc., Zürich, 2008, pp. 1–38. MR 2432988, DOI https://doi.org/10.4171/047-1/1
Claude Cibils and Marc Rosso, Hopf quivers, J. Algebra 254 (2002), no. 2, 241–251. MR 1933868, DOI https://doi.org/10.1016/S0021-8693%2802%2900080-7
R. Dijkgraaf, V. Pasquier, and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys. B Proc. Suppl. 18B (1990), 60–72 (1991). Recent advances in field theory (Annecy-le-Vieux, 1990). MR 1128130, DOI https://doi.org/10.1016/0920-5632%2891%2990123-V
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
V. G. Drinfel′d, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060–1064 (Russian). MR 802128
V. G. Drinfel′d, Quasi-Hopf algebras, Algebra i Analiz 1 (1989), no. 6, 114–148 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 6, 1419–1457. MR 1047964
A. M. Gaĭnutdinov, A. M. Semikhatov, I. Yu. Tipunin, and B. L. Feĭgin, The Kazhdan-Lusztig correspondence for the representation category of the triplet $W$-algebra in logorithmic conformal field theories, Teoret. Mat. Fiz. 148 (2006), no. 3, 398–427 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 148 (2006), no. 3, 1210–1235. MR 2283660, DOI https://doi.org/10.1007/s11232-006-0113-6
Shlomo Gelaki, Basic quasi-Hopf algebras of dimension $n^3$, J. Pure Appl. Algebra 198 (2005), no. 1-3, 165–174. MR 2132880, DOI https://doi.org/10.1016/j.jpaa.2004.10.003
Pavel Etingof and Shlomo Gelaki, The small quantum group as a quantum double, J. Algebra 322 (2009), no. 7, 2580–2585. MR 2553696, DOI https://doi.org/10.1016/j.jalgebra.2009.05.046
Pavel Etingof and Shlomo Gelaki, On radically graded finite-dimensional quasi-Hopf algebras, Mosc. Math. J. 5 (2005), no. 2, 371–378, 494 (English, with English and Russian summaries). MR 2200756, DOI https://doi.org/10.17323/1609-4514-2005-5-2-371-378
Frank Hausser and Florian Nill, Doubles of quasi-quantum groups, Comm. Math. Phys. 199 (1999), no. 3, 547–589. MR 1669685, DOI https://doi.org/10.1007/s002200050512
HuaLin Huang, From projective representations to quasi-quantum groups, Sci. China Math. 55 (2012), no. 10, 2067–2080. MR 2972630, DOI https://doi.org/10.1007/s11425-012-4437-4
Hua-Lin Huang, Quiver approaches to quasi-Hopf algebras, J. Math. Phys. 50 (2009), no. 4, 043501, 9. MR 2513989, DOI https://doi.org/10.1063/1.3103569
Hua-Lin Huang, Gongxiang Liu, and Yu Ye, Quivers, quasi-quantum groups and finite tensor categories, Comm. Math. Phys. 303 (2011), no. 3, 595–612. MR 2786212, DOI https://doi.org/10.1007/s00220-011-1229-6
Michio Jimbo, A $q$-difference analogue of $U({\mathfrak g})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, DOI https://doi.org/10.1007/BF00704588
Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145
Gongxiang Liu, The quasi-Hopf analogue of ${\bf u}_q(\mathfrak {sl}_2)$, Math. Res. Lett. 21 (2014), no. 3, 585–603. MR 3272031, DOI https://doi.org/10.4310/MRL.2014.v21.n3.a12
George Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3 (1990), no. 1, 257–296. MR 1013053, DOI https://doi.org/10.1090/S0894-0347-1990-1013053-9
George Lusztig, Quantum groups at roots of $1$, Geom. Dedicata 35 (1990), no. 1-3, 89–113. MR 1066560, DOI https://doi.org/10.1007/BF00147341
S. Majid, Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998), no. 1, 1–9. MR 1631648, DOI https://doi.org/10.1023/A%3A1007450123281
Shahn Majid, Quasi-quantum groups as internal symmetries of topological quantum field theories, Lett. Math. Phys. 22 (1991), no. 2, 83–90. MR 1122044, DOI https://doi.org/10.1007/BF00405171
Peter Schauenburg, Hopf modules and the double of a quasi-Hopf algebra, Trans. Amer. Math. Soc. 354 (2002), no. 8, 3349–3378. MR 1897403, DOI https://doi.org/10.1090/S0002-9947-02-02980-X
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324