scholarly journals Poisson Brackets and Nijenhuis Tensor

1997 ◽  
Vol 52 (1-2) ◽  
pp. 76-78
Author(s):  
Susumu Okubo
Keyword(s):  
Toda Lattice ◽  
Discrete Systems ◽  
Integrable Models ◽  
Poisson Brackets ◽  
Nijenhuis Tensor ◽  
Infinite Dimensional ◽  
Dimensional Models

Abstract Many integrable models satisfy the zero Nijenhuis tensor condition. Although its application for discrete systems is then straightforward, there exist some complications to utilize the condition for continuous infinite dimensional models. A brief sketch of how we deal with the problem is explained with an application to a continuous Toda lattice.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals ompleteness of SoV Representation for SL(2,R) Spin Chains

2021 ◽  
Author(s):  
Sergey E. Derkachov ◽  
◽  
Karol K. Kozlowski ◽  
Alexander N. Manashov ◽  
◽  
...  
Keyword(s):  
Singular Integrals ◽  
Separation Of Variables ◽  
Spin Chains ◽  
Integrable Models ◽  
New Method ◽  
Infinite Dimensional ◽  
Rank One ◽  
Quantum Integrable Models

This work develops a new method, based on the use of Gustafson's integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the SL(2,R) spin chains.

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