From Reflection Equation Algebra to Braided Yangians
In general, quantum matrix algebras are associated with a couple of compatible braidings. A particular example of such an algebra is the so-called Reflection Equation algebra In this paper we analyze its specific properties, which distinguish it from other quantum matrix algebras (in first turn, from the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity for its generating matrix, which in a limit turns into the Cayley-Hamilton identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we consider some specific properties of the braided Yangians, recently introduced by the authors. In particular, we exhibit an analog of the Cayley-Hamilton identityfor the generating matrix of such a braided Yangian. Besides, by passing to a limit of this braided Yangian, we get a Lie algebra similar to that entering the construction of the rational Gaudin model. In its enveloping algebra we construct a Bethe subalgebra by the method due to D.Talalaev.
KeywordsReflection equation algebra Braided Lie algebra Affinization Braided Yangian Quantum symmetric polynomials Cayley-Hamilton identity
AMS Mathematics Subject Classification, 2010:81R50
D.G. is grateful to the Max Planck Institute for Mathematics (Bonn), where the paper was mainly written, for stimulating atmosphere during his scientific visit. The work of P.S. has been funded by the Russian Academic Excellence Project ‘5-100’ and was also partially supported by the RFBR grant 16-01-00562. The authors are also thankful to D.Talalaev for elucidating discussion.
- 1.Drinfeld, V.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2, pp. 798–820. Berkeley, California (1986). (Am. Math. Soc. Providence, RI, 1987)Google Scholar
- 2.Frappat, L., Jing, N., Molev, A., Ragoucy, E.: Higher Sugawara operators for the quantum affine algebras of type \(A\) (2015). arXiv:1505.03667
- 6.Gurevich, D., Saponov, P.: Braided Yangians (2016). arXiv:1612.05929
- 8.Gurevich, D., Saponov, P., Slinkin, A.: Bethe subalgebras and matrix identities in Braided Yangians. in progressGoogle Scholar
- 11.Isaev, A., Ogievetsky, O.: Half-quantum linear algebra. In: Symmetries and groups in contemporary physics. World Scientific Publishing, Hackensack, NJ (2013). (Nankai Ser. Pure Appl. Math. Theoret. Phys. 11, 479–486)Google Scholar
- 13.Majid, Sh.: Foundations of quantum group theory. Cambridge University Press (1995)Google Scholar
- 14.Molev, A.: Yangians and classical Lie algebras. Mathematical Surveys and Monographs, vol. 143. AMS Providence, RI (2007)Google Scholar
- 15.Ogievetsky, O.: Uses of Quantum Spaces. Lectures at Bariloche Summer School. AMS Contemporary, Argentina (2000)Google Scholar