scholarly journals Compatible Poisson brackets, quadratic Poisson algebras and classical r-matrices

2009 ◽  
pp. 311-333 ◽  
Author(s):  
V. Roubtsov ◽  
T. Skrypnyk
Keyword(s):  
Poisson Brackets ◽  
Poisson Algebras ◽  
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).

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

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.

scholarly journals Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite Derivations

2003 ◽  
Vol 55 (4) ◽  
pp. 856-896 ◽  
Author(s):  
Yucai Su
Keyword(s):  
Poisson Bracket ◽  
Lie Algebras ◽  
Poisson Brackets ◽  
Cohomology Groups ◽  
Poisson Algebras ◽  
Locally Finite ◽  
Isomorphism Classes ◽  
Sandwich Method

AbstractXu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a “sandwich” method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.

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