×

Twisted deformations vs. cocycle deformations for quantum groups. (English) Zbl 1490.17021

This paper studies two deformation by twists procedures for quantum groups: “comultiplication twisting” and “multiplication twisting”. Let us recall that quantum groups are Hopf algebra deformations of the universal enveloping algebra \(U(g)\) of a Lie algebra \(g\). From this deformation, \(g\) inherits a Lie cobracket that makes it a Lie bialgebra. In this work, the authors first recall the relevant notions and results about Hopf algebras and their deformations. Then they deal with quantum groups and their comultiplication twistings. The authors present also the multiplication twisting. Finally, the authors compare the two dual notions and prove that they actually coincide, in a proper sense.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T20 Ring-theoretic aspects of quantum groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andruskiewitsch, N. and Angiono, I. E., On Nichols algebras with generic braiding, in Modules and Comodules, (Birkhäuser Verlag, Basel, 2008), pp. 47-64. · Zbl 1227.16022
[2] Andruskiewitsch, N., Radford, D. and Schneider, H.-J., Complete reducibility theorems for modules over pointed Hopf algebras, J. Algebra324 (2010) 2932-2970. · Zbl 1223.16010
[3] Angiono, I. E., Distinguished pre-Nichols algebras, Transform. Groups21(1) (2016) 1-33. · Zbl 1355.16028
[4] Angiono, I. E. and Yamane, H., The \(R\)-matrix of quantum doubles of Nichols algebras of diagonal type, J. Math. Phys.56(2) (2015) 021702, 19 pp. · Zbl 1394.17066
[5] Artin, M., Schelter, W. and Tate, J., Quantum deformations of \(\text{GL}_n\), Comm. Pure Appl. Math.44(8-9) (1991) 879-895. · Zbl 0753.17015
[6] Benkart, G. and Witherspoon, S., Restricted two-parameter quantum groups, in Representations of Finite-Dimensional Algebras and Related Topics in Lie Theory and Geometry, , Vol. 40 (American Mathematical Society, Providence, RI, 2004), pp. 293-318. · Zbl 1048.16020
[7] Benkart, G. and Witherspoon, S., Two-parameter quantum groups and Drinfel’d doubles, Algebr. Represent. Theory7(3) (2004) 261-286. · Zbl 1113.16041
[8] Bergeron, N., Gao, Y. and Hu, N., Drinfel’d doubles and Lusztig’s symmetries of two-parameter quantum groups, J. Algebra301(1) (2006) 378-405. · Zbl 1148.17007
[9] Chari, V. and Pressley, A., A Guide to Quantum Group (Cambridge University Press, Cambridge, 1995). · Zbl 0839.17010
[10] Chin, W. and Musson, I., Multiparameter quantum enveloping algebras, Contact Franco-Belge en Algèbre (Diepenbeek, 1993), J. Pure Appl. Algebra107(2-3) (1996) 171-191. · Zbl 0859.17004
[11] Costantini, M. and Varagnolo, M., Quantum double and multiparameter quantum groups, Comm. Algebra22(15) (1994) 6305-6321. · Zbl 0815.17011
[12] Costantini, M. and Varagnolo, M., A family of Azumaya algebras arising from quantum groups, C. R. Acad. Sci. Paris Sér. I Math.323(2) (1996) 127-132. · Zbl 0860.17015
[13] Costantini, M. and Varagnolo, M., Multiparameter quantum function algebra at roots of 1, Math. Ann.306(4) (1996) 759-780. · Zbl 0858.16035
[14] Doi, Y. and Takeuchi, M., Multiplication alteration by two-cocycles — the quantum version, Comm. Algebra22 (1994) 5715-5732. · Zbl 0821.16038
[15] Drinfeld, V. G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2(Berkeley, Calif., 1986), pp. 798-820, Amer. Math. Soc., Providence, RI, 1987. · Zbl 0667.16003
[16] García, G. A., Multiparameter quantum groups, bosonizations and cocycle deformations, Rev. Un. Mat. Argentina57(2) (2016) 1-23. · Zbl 1360.16029
[17] G. A. García and F. Gavarini, Multiparameter quantum groups at roots of unity, preprint (2017); arXiv:1708.05760.
[18] G. A. García and F. Gavarini, Multiparameter quantum groups: A uniform approach, work in progress.
[19] Gavarini, F., Quantization of Poisson groups, Pacific J. Math.186 (1998) 217-266. · Zbl 0921.17004
[20] Hayashi, T., Quantum groups and quantum determinants, J. Algebra301(1) (2006) 378-405.
[21] Heckenberger, I., Lusztig isomorphisms for Drinfel’d doubles of bosonizations of Nichols algebras of diagonal type, J. Algebra323 (2010) 2130-2180. · Zbl 1238.17010
[22] Hodges, T. J., Levasseur, T. and Toro, M., Algebraic structure of multiparameter quantum groups, Adv. Math.126(1) (1997) 52-92. · Zbl 0878.17009
[23] Hu, N., Pei, Y. and Rosso, M., Multi-parameter quantum groups and quantum shuffles, I, in Quantum Affine Algebras, Extended Affine Lie Algebras, and Their Applications, , Vol. 506 (American Mathematical Society, Providence, RI, 2010), pp. 145-171. · Zbl 1267.17015
[24] Jimbo, M., A \(q\)-difference analogue of \(U(\mathfrak{g})\) and the Yang-Baxter equation, Lett. Math. Phys.10(1) (1985) 63-69. · Zbl 0587.17004
[25] Kac, V. G., Infinite Dimensional Lie Algebras, 3rd edition (Cambridge University Press, Cambridge, 1990). · Zbl 0716.17022
[26] A. N. Koryukin, A generalization of a two-parameter quantization of the group \(\text{GL}_2(k)\), Algebra Logika42(6) (2003) 692-711, 764 (in Russian); Algebra Logic42(6) (2003) 387-397. · Zbl 1054.16029
[27] Kreimer, H. F. and Takeuchi, M., Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J.30 (1981) 675-692. · Zbl 0451.16005
[28] Lentner, S., A Frobenius homomorphism for Lusztig’s quantum groups for arbitrary roots of unity, Commun. Contemp. Math.18(3) (2016) 1550040, 42 pp. · Zbl 1394.17038
[29] Y. Li, N. Hu and M. Rosso, Multi-parameter quantum groups via quantum quasi-symmetric algebras, preprint (2013); arXiv:1307.1381.
[30] Lusztig, G., Quantum groups at roots of 1, Geom. Dedicata35 (1990) 89-113. · Zbl 0714.17013
[31] Manin, Y. I., Quantum Groups and Noncommutative Geometry (Université de Montral, Centre de Recherches Mathématiques, Montreal, QC, 1988), vi+91 pp. · Zbl 0724.17006
[32] Montgomery, S., Hopf Algebras and Their Actions on Rings, , Vol. 82 (American Mathematical Society, Providence, RI, 1993). · Zbl 0793.16029
[33] Okado, M. and Yamane, H., \(R\)-matrices with gauge parameters and multi-parameter quantized enveloping algebras, in ICM-90 Satellite Conf. Proc.: Special Functions (Okayama, 1990) (Springer, Tokyo, 1991), pp. 289-293. · Zbl 0774.17022
[34] Radford, D. E., Hopf Algebras, , Vol. 49 (World Scientific Publishing, Hackensack, NJ, 2012). · Zbl 1266.16036
[35] Reshetikhin, N., Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys.20(4) (1990) 331-335. · Zbl 0719.17006
[36] Sudbery, H. J., Consistent multiparameter quantisation of \(\text{GL}(n)\), J. Phys. A23(15) (1990) L697-L704. · Zbl 0722.17007
[37] Takeuchi, M., A two-parameter quantization of \(\text{GL}(n)\) (summary), Proc. Japan Acad. Ser. A Math. Sci.66(5) (1990) 112-114. · Zbl 0723.17012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.