Construction of invariants of the coadjoint representation of Lie groups using linear algebra methods

2016 ◽  
Vol 188 (1) ◽  
pp. 965-979 ◽  
Author(s):  
O. L. Kurnyavko ◽  
I. V. Shirokov
2021 ◽  
Vol 61 ◽  
pp. 79-104
Author(s):  
Tuyen Nguyen ◽  
◽  
Vu Le

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.


Author(s):  
O.G. Styrt

The paper studies stationary subalgebras in general position of compact linear groups. We prove that, except for several specific cases, a stationary subalgebra in general position of a tensor product of real or complex compact group representations acts as a scalar on all tensor factors but possibly one. In the real case, it means that this stationary subalgebra in general position is contained in one of the direct summand subalgebras. We used the following concepts to solve this problem: conventional linear algebra arguments; theory of Lie groups, Lie algebras and their representations; and methods similar to those of solving similar problems for complex reductive linear groups.


Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 642
Author(s):  
Frédéric Barbaresco

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.


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