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

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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Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning

Entropy ◽

10.3390/e20110840 ◽

2018 ◽

Vol 20 (11) ◽

pp. 840 ◽

Cited By ~ 6

Author(s):

Frédéric Barbaresco

Keyword(s):

Machine Learning ◽

Maximum Entropy ◽

Lie Groups ◽

Lie Group ◽

Data Analytics ◽

Geometric Theory ◽

Higher Order ◽

Small Data ◽

Algebra Structure ◽

Fisher Metric

We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted to describe density of probability (maximum entropy Gibbs density) of data living on groups or on homogeneous manifolds. For small data analytics (rarified gases, sparse statistical surveys, …), the density of maximum entropy should consider higher order moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by Ingarden. We use a poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e., no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. This Lie groups thermodynamics is the bedrock for Lie group machine learning providing a full covariant maximum entropy Gibbs density based on representation theory (symplectic structure of coadjoint orbits for Souriau non-equivariant model associated to a class of co-homology).

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The moment map of a Lie group representation

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1992-1040046-6 ◽

1992 ◽

Vol 330 (1) ◽

pp. 257-268 ◽

Cited By ~ 11

Author(s):

N. J. Wildberger

Keyword(s):

Lie Group ◽

Group Representation ◽

Moment Map ◽

The Moment

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On collective complete integrability according to the method of Thimm

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700001930 ◽

1983 ◽

Vol 3 (2) ◽

pp. 219-230 ◽

Cited By ~ 27

Author(s):

Victor Guillemin ◽

Shlomo Sternberg

Keyword(s):

Integrable System ◽

Symplectic Manifold ◽

Lie Group ◽

Geodesic Flow ◽

Necessary Conditions ◽

Complete Integrability ◽

Moment Map ◽

The Real ◽

Completely Integrable System ◽

The Moment

AbstractLet G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.

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Equivariant prequantization and the moment map

Forum Mathematicum ◽

10.1515/forum-2020-0287 ◽

2021 ◽

Vol 33 (3) ◽

pp. 593-600

Author(s):

Roberto Ferreiro Pérez

Keyword(s):

Lie Algebra ◽

Lie Group ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Moment Map ◽

The Moment ◽

Necessary And Sufficient ◽

Connected Lie Group

Abstract If ω is a closed G-invariant 2-form and μ is a moment map, we obtain necessary and sufficient conditions for equivariant prequantizability that can be computed in terms of the moment map μ. Our main result is that G-equivariant prequantizability is related to the fact that the moment map μ should be quantized for certain vectors on the Lie algebra of G. We also compute the obstructions to lift the action of G to a prequantization bundle of ω. Our results are valid for any compact and connected Lie group G.

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Some groups with T1 primitive ideal spaces

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870002259x ◽

1985 ◽

Vol 38 (1) ◽

pp. 55-64 ◽

Cited By ~ 2

Author(s):

A. L. Carey ◽

W. Moran

Keyword(s):

Normal Subgroup ◽

Lie Groups ◽

Lie Group ◽

Locally Compact Group ◽

Primitive Ideal ◽

Ideal Space ◽

Solvable Lie Groups ◽

Locally Compact ◽

Simply Connected ◽

Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

Journal of Function Spaces and Applications ◽

10.1155/2013/483951 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Yu Liu ◽

Jianfeng Dong

Keyword(s):

Lie Groups ◽

Lie Group ◽

Riesz Transform ◽

Integral Operators ◽

Higher Order ◽

Reverse Hölder Class ◽

Stratified Lie Group ◽

Homogeneous Dimension ◽

Stratified Lie Groups ◽

Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

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Toric Structures on the Moduli Space of Flat Connections on a Riemann Surface: Volumes and the Moment Map

Advances in Mathematics ◽

10.1006/aima.1994.1054 ◽

1994 ◽

Vol 106 (2) ◽

pp. 151-168 ◽

Cited By ~ 15

Author(s):

L.C. Jeffrey ◽

J. Weitsman

Keyword(s):

Riemann Surface ◽

Moduli Space ◽

Moment Map ◽

Flat Connections ◽

The Moment

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ON THE MODEL THEORY OF THE LOGARITHMIC FUNCTION IN COMPACT LIE GROUPS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500552 ◽

2013 ◽

Vol 12 (08) ◽

pp. 1350055

Author(s):

SONIA L'INNOCENTE ◽

FRANÇOISE POINT ◽

CARLO TOFFALORI

Keyword(s):

Lie Groups ◽

Lie Group ◽

Model Theory ◽

Effective Model ◽

Logarithmic Function ◽

Compact Lie Groups ◽

Model Completeness ◽

Schanuel's Conjecture ◽

Logarithm Function ◽

Natural Expansion

Given a compact linear Lie group G, we form a natural expansion of the theory of the reals where G and the graph of a logarithm function on G live. We prove its effective model-completeness and decidability modulo a suitable variant of Schanuel's Conjecture.

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