Classical r-matrices and compatible Poisson brackets for coupled KdV systems

1989 ◽  
Vol 17 (1) ◽  
pp. 25-29 ◽  
Author(s):  
A. P. Fordy ◽  
A. G. Reyman ◽  
M. A. Semenov-Tian-Shansky
Keyword(s):  
Poisson Brackets ◽  
Compatible Poisson Brackets

scholarly journals PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

2021 ◽  
Author(s):  
DMITRI I. PANYUSHEV ◽  
OKSANA S. YAKIMOVA
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Invariant Theory ◽  
Cyclic Permutation ◽  
Poisson Brackets ◽  
Enveloping Algebra ◽  
Order Automorphism ◽  
Finite Dimensional ◽  
Compatible Poisson Brackets ◽  
Commutative Subalgebras

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).

BUNDLES OF LIE ALGEBRAS AND COMPATIBLE POISSON BRACKETS

2002 ◽  
Vol 17 (06n07) ◽  
pp. 946-950
Author(s):  
A. B. YANOVSKI
Keyword(s):  
Lie Algebras ◽  
Poisson Brackets ◽  
Algebraic Structures ◽  
Compatible Poisson Brackets ◽  
Lie Brackets

We consider the mechanism of obtaining compatible Poisson-Lie tensors, based on the existence of some special algebraic structures - families of Lie brackets defined on one and the same space.

Compatible Poisson brackets for Lax equations and classical ζ -matrices

1989 ◽  
Vol 47 (2) ◽  
pp. 2493-2502
Author(s):  
A. G. Reiman ◽  
M. A. Semenov-Tyan-Shanskii
Keyword(s):  
Poisson Brackets ◽  
Compatible Poisson Brackets ◽  
Lax Equations
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