Author:
PANYUSHEV DMITRI I.,YAKIMOVA OKSANA S.
Abstract
AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Algebra and Number Theory
Reference28 articles.
1. A. Chervov, G. Falqui, L. Rybnikov, Limits of Gaudin algebras, quantization of bending ows, Jucys–Murphy elements and Gelfand–Tsetlin bases, Lett. Math. Phys. 91 (2010), no. 2, 129–150.
2. J.-P. Dufour, N.T. Zung, Poisson Structures and their Normal Forms, Progress in Mathematics, Vol. 242, Birkhäuser Verlag, Basel, 2005.
3. B. Feigin, E. Frenkel, Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras, Int. J. Mod. Phys. A7, Suppl. 1A (1992), 197–215.
4. B. Feigin, E. Frenkel, N. Reshetikhin, Gaudin model, Bethe Ansatz and critical level, Comm. Math. Phys. 166 (1994), no. 1, 27–62.
5. E. Frenkel, Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics, Vol. 103, Cambridge University Press, Cambridge, 2007.
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