Invariant function algebras on Lie groups

1979 ◽  
Vol 20 (1) ◽  
pp. 15-25 ◽  
Author(s):  
V. M. Gichev
Keyword(s):  
Lie Groups ◽  
Invariant Function ◽  
Function Algebras

scholarly journals Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 642
Author(s):  
Frédéric Barbaresco
Keyword(s):  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Lie Groups ◽  
Lie Group ◽  
Invariant Function ◽  
Moment Map ◽  
Coadjoint Representation ◽  
Coadjoint Action ◽  
Casimir Invariant ◽  
The Moment

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

Sign in / Sign up

Export Citation Format

Share Document