The equitable presentation for the quantum group associated with a symmetrizable Kac-Moody algebra. (English) Zbl 1106.17021
In the paper [T. Ito, P. Terwilliger, C. Weng, ”The quantum algebra \(U_q(sl_2)\) and its equitable presentation”, J. Algebra 298, No. 1, 284–301 (2006; Zbl 1090.17004)] an interesting and very symmetric presentation of the quantum algebra \(U_q(\text{sl}_2)\) was found. In the present paper the author rewrites the standard presentation of the quantum group associated with a symmetrizable Kac-Moody algebra in terms of these new “equitable” \(U_q(\text{sl}_2)\)-generators. The paper is very technical, and the main technical difficulty is to find an appropriate replacement for quantum Serre relations.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Citations:
Zbl 1090.17004References:
| [1] | Brown, K.; Goodearl, K., Lectures on Algebraic Quantum Groups, Adv. Courses Math. CRM Barcelona (2002), Birkhäuser: Birkhäuser Basel · Zbl 1027.17010 |
| [2] | Chari, V.; Pressley, A., A Guide to Quantum Groups (1994), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0839.17009 |
| [3] | Gasper, G.; Rahman, M., Basic Hypergeometric Series, Encyclopedia Math. Appl., vol. 35 (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0695.33001 |
| [4] | Kac, V., Infinite Dimensional Lie Algebras (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0716.17022 |
| [5] | T. Ito, P. Terwilliger, Tridiagonal pairs and the quantum affine algebra \(U_q( \hat{s l}_2)\) math.QA/0310042; T. Ito, P. Terwilliger, Tridiagonal pairs and the quantum affine algebra \(U_q( \hat{s l}_2)\) math.QA/0310042 |
| [6] | Ito, T.; Terwilliger, P.; Weng, C., The quantum algebra \(U_q(sl_2)\) and its equitable presentation, J. Algebra, 298, 1, 284-301 (2006) · Zbl 1090.17004 |
| [7] | Hong, J.; Kang, S., Introduction to Quantum Groups and Crystal Bases, Grad. Stud. Math., vol. 42 (2002), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1134.17007 |
| [8] | Jantzen, J. C., Lectures on Quantum Groups, Grad. Stud. Math., vol. 6 (1996), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0842.17012 |
| [9] | Joseph, A., Quantum Groups and Their Primitive Ideals (1995), Springer-Verlag: Springer-Verlag Berlin · Zbl 0808.17004 |
| [10] | Kassel, C., Quantum Groups (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0808.17003 |
| [11] | Klimyk, A.; Schmüdgen, K., Quantum Groups and Their Representations (1997), Springer-Verlag: Springer-Verlag Berlin · Zbl 0891.17010 |
| [12] | Koelink, E., Askey-Wilson polynomials and the quantum \(SU(2)\) group: Survey and applications, Acta Appl. Math., 44, 295-352 (1996) · Zbl 0865.33013 |
| [13] | Koelink, E., Eight lectures on quantum groups and \(q\)-special functions, Rev. Colombiana Mat., 30, 93-180 (1996) · Zbl 0902.17004 |
| [14] | Koornwinder, T. H., Orthogonal polynomials in connection with quantum groups, (Nevai, P., Orthogonal Polynomials: Theory and Practice. Orthogonal Polynomials: Theory and Practice, NATO Adv. Study Inst. Ser. C, vol. 294 (1990), Kluwer: Kluwer Norwell, MA), 257-292 · Zbl 0697.42019 |
| [15] | Koornwinder, T. H., Askey-Wilson polynomials as zonal spherical functions on the \(SU(2)\) quantum group, SIAM J. Math. Anal., 24, 795-813 (1993) · Zbl 0799.33015 |
| [16] | Lusztig, G., Introduction to Quantum Groups, Progr. Math., vol. 110 (1993), Birkhäuser: Birkhäuser Boston · Zbl 0788.17010 |
| [17] | Majid, S., Foundations of Quantum Group Theory (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0857.17009 |
| [18] | Masuda, T.; Mimachi, K.; Nakagami, Y.; Noumi, M.; Saburi, Y.; Ueno, K., Unitary representations of the quantum group \(SU_q(1, 1)\): Structure of the dual space of \(U_q(s l(2))\), Lett. Math. Phys., 19, 187-194 (1990) · Zbl 0704.17007 |
| [19] | Noumi, M., Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math., 123, 16-77 (1996) · Zbl 0874.33011 |
| [20] | Noumi, M.; Dijkhuizen, M. S.; Sugitani, T., Multivariable Askey-Wilson polynomials and quantum complex Grassmannians, (Special Functions, \(q\)-Series and Related Topics. Special Functions, \(q\)-Series and Related Topics, Toronto, ON, 1995. Special Functions, \(q\)-Series and Related Topics. Special Functions, \(q\)-Series and Related Topics, Toronto, ON, 1995, Fields Inst. Commun., vol. 14 (1997), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 167-177 · Zbl 0877.33012 |
| [21] | Noumi, M.; Mimachi, K., Askey-Wilson polynomials and the quantum group \(SU_q(2)\), Proc. Japan Acad. Ser. A Math. Sci., 66, 146-149 (1990) · Zbl 0707.33010 |
| [22] | Noumi, M.; Mimachi, K., Askey-Wilson polynomials as spherical functions on \(SU_q(2)\), (Quantum Groups. Quantum Groups, Leningrad, 1990. Quantum Groups. Quantum Groups, Leningrad, 1990, Lecture Notes in Math., vol. 1510 (1992), Springer-Verlag: Springer-Verlag Berlin), 98-103 · Zbl 0743.33011 |
| [23] | Rosengren, H., A new quantum algebraic interpretation of the Askey-Wilson polynomials, \((q\)-Series from a Contemporary Perspective. \(q\)-Series from a Contemporary Perspective, South Hadley, MA, \(1998. q\)-Series from a Contemporary Perspective. \(q\)-Series from a Contemporary Perspective, South Hadley, MA, 1998, Contemp. Math., vol. 254 (2000), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 371-394 · Zbl 1058.33503 |
| [24] | Stokman, J., Vertex-IRF transformations, dynamical quantum groups and harmonic analysis, Indag. Math., 14, 545-570 (2003) · Zbl 1052.46059 |
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.
