On the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits

Abstract

In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitiansymmetric space of compact type, which is a particular example of an Hilbert manifold, is transitively acted upon by a Hilbert Lie group of isometries. In this paper we give the classification of infinitedimensional irreducible Hermitian-symmetric affine coadjoint orbits of L * -groups of compact type using the notion of simple roots of non-compact type. The key step is, given an infinite-dimensional symmetric pair (g, k), where g is a simple L * -algebra and k a subalgebra of g, to construct an increasing sequence of finite-dimensional subalgebras g n of g together with an increasing sequence of finite-dimensional subalgebras k n of k such that g = ∪g n , k = ∪k n , and such that the pairs (g n , k n ) are symmetric. Comparing with the classification of Hermitian-symmetric spaces given by W. Kaup, it follows that any Hermitian-symmetric space of compact type is an affine-coadjoint orbit of an Hilbert Lie group.

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

“…Then, the operator that transform the right algebra to its dual algebra is given by: (190) As there is an operator to change the view of algebra, there is one that did the same on the dual algebra, the co-adjoint operator that is the conjugate action of Lie group on its dual algebra: (191) We can then develop this expression for our use case of affine sup-group, we find: (192) and we can also observed that: (193) And the following relation between the left and the right algebras:…”

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

“…This last property allows to determine all homogeneous spaces of a Lie group admitting an invariant symplectic structure by the action of this group: there are the orbits of the coadjoint representation of this group or of a central extension of this group (the central extension allowing to suppress the cocycle). For affine coadjoint orbits, we give reference to Alice Tumpach PhD [189,190,191] that has developed previous works of K.H. Neeb, O. Biquard and P. Gauduchon.…”

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

“…This last property allows us to determine all homogeneous spaces of a Lie group admitting an invariant symplectic structure by the action of this group: for example, there are the orbits of the coadjoint representation of this group or of a central extension of this group (the central extension allowing suppressing the cocycle). For affine coadjoint orbits, we make reference to Alice Tumpach Ph.D. [197][198][199] who has developed previous works of Neeb [200], Biquard and Gauduchon [201][202][203][204].…”