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Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3)

Prosper Rosaire Mama Assandje, Michel Bertrand Djiadeu Ngaha, Romain Nimpa Pefoukeu, Salomon Joseph Mbatakou

TL;DR

The paper addresses how to generalize the Fisher information metric within Souriau's thermodynamic Lie group framework for the rotation groups $SO(2)$ and $SO(3)$, and investigates how central $2$-cocycles influence the resulting information geometry and the integrability of gradient flows on coadjoint orbits. It develops an explicit unique one-cocycle $\Theta$ for these groups, derives the distinguished density and potential functions, and constructs gradient and Hamiltonian systems that are completely integrable, including a Lax-pair representation in the $SO(2)$ case. The results connect invariant statistical models with the symplectic geometry of coadjoint orbits and provide concrete, computable formulas for the generalized Fisher metric, along with foliations by 1D torus leaves. Overall, the approach offers a geometric framework for information-geometry-inspired analysis on Lie groups with potential applications in geometric machine learning and neural-inspired optimization.

Abstract

We study the generalize Fisher metric on SO(2) and SO(3) via the thermodynamics Lie group theories of Souriau. Then we give the effect of 2-cocycle on the integrability of gradient systems due to the Fisher metric and Souriau-Fisher metric. In addition, we show how the cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups SO(2) and SO(3).

Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3)

TL;DR

The paper addresses how to generalize the Fisher information metric within Souriau's thermodynamic Lie group framework for the rotation groups and , and investigates how central -cocycles influence the resulting information geometry and the integrability of gradient flows on coadjoint orbits. It develops an explicit unique one-cocycle for these groups, derives the distinguished density and potential functions, and constructs gradient and Hamiltonian systems that are completely integrable, including a Lax-pair representation in the case. The results connect invariant statistical models with the symplectic geometry of coadjoint orbits and provide concrete, computable formulas for the generalized Fisher metric, along with foliations by 1D torus leaves. Overall, the approach offers a geometric framework for information-geometry-inspired analysis on Lie groups with potential applications in geometric machine learning and neural-inspired optimization.

Abstract

We study the generalize Fisher metric on SO(2) and SO(3) via the thermodynamics Lie group theories of Souriau. Then we give the effect of 2-cocycle on the integrability of gradient systems due to the Fisher metric and Souriau-Fisher metric. In addition, we show how the cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups SO(2) and SO(3).

Paper Structure

This paper contains 11 sections, 12 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie Algebra
  4. Coadjoint operator and Coadjoint Orbits (Kirillov Representation)
  5. Folliations
  6. Gradient system and completely integrable Hamiltonian system.
  7. Distinguished density function
  8. Gradient system and Hamiltonian system on Lie algebra $\mathfrak{so}(2)$
  9. Lax pair representation
  10. Generalized Fisher metric on Lie group $SO(3)$
  11. Conclusion

Key Result

Proposition 1

Lie The Lie algebras of $SL(n,\mathrm{I\! K}),\;O(n),\;SO(n)$ are is the Lie algebra of traceless $n\times n$ matrices with coefficients $n \times n\in\mathrm{I\! K}$, and $dim_{\mathrm{I\! R}}\mathfrak{sl}(n,\mathrm{I\! R}) =n^{2}-1,\; dim_{\mathbb{C}}\mathfrak{sl}(n,\mathbb{C})=n^{2}-1$ , $dim_{\mathrm{I\! R}}\mathfrak{sl}(n,\mathbb{C})=2(n^{2}-1)$. is the Lie algebra of antisymmetric real $n\

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 2
  • Definition 8
  • ...and 23 more