Super solutions of the dynamical Yang-Baxter equation

Karaali, Gizem

doi:10.1090/S0002-9939-06-08495-4

Super solutions of the dynamical Yang-Baxter equation
HTML articles powered by AMS MathViewer

by Gizem Karaali

Proc. Amer. Math. Soc. 134 (2006), 2521-2531

DOI: https://doi.org/10.1090/S0002-9939-06-08495-4

Published electronically: March 22, 2006

PDF | Request permission

Abstract:

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical $r-$matrices. A super dynamical $r-$matrix $r$ satisfies the zero weight condition if \[ [h\otimes 1 + 1 \otimes h, r(\lambda )] = 0 \text { for all } h \in \mathfrak {h}, \lambda \in \mathfrak {h}^*. \] In this paper we classify super dynamical $r-$matrices with zero weight.

References

A. A. Belavin and V. G. Drinfel′d, Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funktsional. Anal. i Prilozhen. 16 (1982), no. 3, 1–29, 96 (Russian). MR 674005
A. A. Belavin and V. G. Drinfel′d, Triangle equations and simple Lie algebras, Mathematical physics reviews, Vol. 4, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., vol. 4, Harwood Academic Publ., Chur, 1984, pp. 93–165. Translated from the Russian. MR 768939
Pavel Etingof, On the dynamical Yang-Baxter equation, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 555–570. MR 1957065
Pavel Etingof, Travis Schedler, and Olivier Schiffmann, Explicit quantization of dynamical $r$-matrices for finite dimensional semisimple Lie algebras, J. Amer. Math. Soc. 13 (2000), no. 3, 595–609. MR 1758755, DOI 10.1090/S0894-0347-00-00333-7
Pavel Etingof and Olivier Schiffmann, Lectures on quantum groups, Lectures in Mathematical Physics, International Press, Boston, MA, 1998. MR 1698405
Pavel Etingof and Alexander Varchenko, Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys. 192 (1998), no. 1, 77–120. MR 1612160, DOI 10.1007/s002200050292
Geer, N.; “Etingof-Kazhdan Quantization of Lie Superbialgebras”; arXiv:math.QA/0409563
Gizem Karaali, Constructing $r$-matrices on simple Lie superalgebras, J. Algebra 282 (2004), no. 1, 83–102. MR 2095573, DOI 10.1016/j.jalgebra.2004.07.005
Karaali, G.; “A New Lie Bialgebra Structure on $sl(2,1)$”, Contemp. Math. (to appear).
Olivier Schiffmann, On classification of dynamical r-matrices, Math. Res. Lett. 5 (1998), no. 1-2, 13–30. MR 1618367, DOI 10.4310/MRL.1998.v5.n1.a2