Abstract
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.
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Notes
- 1.
From now on, the notation \(\mathbb {K}\) stands for the ground field, which is \({\mathbb {C}}\) or \(\mathbb {R}\).
- 2.
If R is a Hecke symmetry we should additionally require q to be generic, that is \(q^n\not =1\) for any integer n.
- 3.
They differ from our braided Yangians by the middle terms, which are also current R-matrices. Observe that there are known many versions of such RE algebras.
- 4.
In order to get this limit we first change the basis in the Yangian, or in other words, we pass to the shifted form of this Yangian.
- 5.
Note that the QM algebras as introduced in [12], are defined in a similar way, but with the help of the second braiding F, in a sense compatible with R: \(L_{\overline{k}}=F_{k-1} L_{\overline{k-1}}F_{k-1}^{-1}\).
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Acknowledgments
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.
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Gurevich, D., Saponov, P. (2018). From Reflection Equation Algebra to Braided Yangians. In: Buchstaber, V., Konstantinou-Rizos, S., Mikhailov, A. (eds) Recent Developments in Integrable Systems and Related Topics of Mathematical Physics. MP 2016. Springer Proceedings in Mathematics & Statistics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-030-04807-5_7
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