Algebraic Properties of the Semi-direct Product of Kac–Moody and Virasoro Lie Algebras and Associated Bi-Hamiltonian Systems

2016 ◽  
pp. 1-7
Author(s):  
Alexander Zuevsky
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Direct Product ◽  
Algebraic Properties

scholarly journals Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras

2018 ◽  
Vol 16 (1) ◽  
pp. 1-8
Author(s):  
A. Zuevsky
Keyword(s):  
Shallow Water ◽  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Direct Product ◽  
Coadjoint Orbits ◽  
Verma Modules ◽  
Casimir Functions ◽  
Algebraic Proofs

AbstractWe prove new theorems related to the construction of the shallow water bi-Hamiltonian systems associated to the semi-direct product of Virasoro and affine Kac–Moody Lie algebras. We discuss associated Verma modules, coadjoint orbits, Casimir functions, and bi-Hamiltonian systems.

scholarly journals Homomorphisms on infinite direct product algebras, especially Lie algebras

2011 ◽  
Vol 333 (1) ◽  
pp. 67-104 ◽  
Author(s):  
George M. Bergman ◽  
Nazih Nahlus
Keyword(s):  
Lie Algebras ◽  
Direct Product

DEFORMATIONS OF LOOP ALGEBRAS AND CLASSICAL INTEGRABLE SYSTEMS: FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

2004 ◽  
Vol 16 (07) ◽  
pp. 823-849 ◽  
Author(s):  
T. SKRYPNYK
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Finite Dimensional ◽  
Quasigraded Lie Algebras ◽  
Classical Integrable

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

scholarly journals Complete commutative subalgebras in polynomial poisson algebras: A proof of the Mischenko-Fomenko conjecture

2016 ◽  
Vol 43 (2) ◽  
pp. 145-168 ◽  
Author(s):  
Alexey Bolsinov
Keyword(s):  
Poisson Bracket ◽  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Dual Space ◽  
State University ◽  
Integrable Hamiltonian Systems ◽  
Finite Dimensional ◽  
Complete Set ◽  
Commutative Subalgebras ◽  
Moscow State University

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on g* and consider some examples. (This text is a revised version of my paper published in Russian: A. V. Bolsinov, Complete commutative families of polynomials in Poisson?Lie algebras: A proof of the Mischenko?Fomenko conjecture in book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, 87?109.)

Reduction of Hamiltonian systems, affine Lie algebras and Lax equations

1979 ◽  
Vol 54 (1) ◽  
pp. 81-100 ◽  
Author(s):  
A. G. Reyman ◽  
M. A. Semenov-Tian-Shansky
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Affine Lie Algebras ◽  
Lax Equations

2-Nilpotent multipliers of a direct product of Lie algebras

2016 ◽  
Vol 65 (3) ◽  
pp. 519-523 ◽  
Author(s):  
Peyman Niroomand ◽  
Mohsen Parvizi
Keyword(s):  
Lie Algebras ◽  
Direct Product

scholarly journals Some Topological and Algebraic Features of Symmetric Spaces

2020 ◽  
pp. 144-155
Author(s):  
Um Salama ◽  
Ahmed Abd Alla ◽  
A. Elemam
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Lie Groups ◽  
Lie Algebras ◽  
Symmetric Spaces ◽  
Algebraic Approach ◽  
Root Systems ◽  
Algebraic Properties ◽  
Important Field

In this study, we introduce some approaches, geometrical and algebraic, which help to give further understanding of symmetric spaces. Symmetric space is a very important field for understanding abstract and applied features of spaces. We have introduced Riemannian Manifold, Lie groups and Lie algebras, and some of their topological and algebraic properties, with some concentration on Lie algebras and root systems , which help classification and many applications of symmetric spaces. The paper is an attempt to explain some algebraic features of symmetric spaces and how to get some of their properties using algebraic approach, concluded with some results.

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