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.

From Diffeomorphisms to Dark Energy?

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2467-2474 ◽  
Author(s):  
Vincent G. J. Rodgers ◽  
Takeshi Yasuda
Keyword(s):  
Dark Energy ◽  
Lie Algebras ◽  
Virasoro Algebra ◽  
Two Dimensions ◽  
Coadjoint Orbits ◽  
Coadjoint Representation ◽  
Natural Setting ◽  
Affine Lie Algebras ◽  
Yang Mills ◽  
Geometric Action

There are two physical actions that have a natural setting in terms of the coadjoint representation of the algebra of diffeomorphisms and of affine Lie algebras. One is the usual geometric action that comes from coadjoint orbits. The other action lives on the phase space that is transverse to the orbits and are called transverse actions, where Yang-Mills theory in two dimensions is an example. Here we show that the transverse action associated with the Virasoro algebra might contain clues for a theory for dark energy. These actions might also suggests a mechanism for symmetry changing.

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.

INFORMATION GEOMETRY FOR SOME LIE ALGEBRAS

1999 ◽  
Vol 02 (03) ◽  
pp. 441-460 ◽  
Author(s):  
H. NENCKA ◽  
R. F. STREATER
Keyword(s):  
Lie Algebras ◽  
Discrete Series ◽  
Complex Structure ◽  
Solvable Model ◽  
Coadjoint Orbits ◽  
Exactly Solvable Model ◽  
Metaplectic Representation ◽  
Canonical Algebra ◽  
Model Dynamics ◽  
Exactly Solvable

For certain unitary representations of a Lie algebra [Formula: see text] we define the statistical manifold ℳ of states as the convex cone of [Formula: see text] for which the partition function Z= Tr exp {-X} is finite. The Hessian of Ψ= log Z defines a Riemannian metric g on [Formula: see text], (the Bogoliubov–Kubo–Mori metric); [Formula: see text] foliates into the union of coadjoint orbits, each of which can be given a complex structure (that of Kostant). The program is carried out for so(3), and for sl(2,R) in the discrete series. We show that ℳ=R+× CP 1 and R+×H respectively. We show that for the metaplectic representation of the quadratic canonical algebra, ℳ=R+× CP 2/Z2. Exactly solvable model dynamics is constructed in each case.

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