Homogeneous statistical manifolds

TL;DR

This paper develops the theory of homogeneous statistical manifolds, providing explicit low-dimensional examples and classifying certain 3D Lie groups with specific statistical structures, advancing the mathematical foundation of information geometry.

Contribution

It introduces a general theory of homogeneous statistical manifolds and classifies 3D Lie groups with non-trivial conjugate symmetric structures, offering new explicit examples.

Findings

Constructed explicit examples of low-dimensional homogeneous statistical manifolds

Classified 3D Lie groups with conjugate symmetric statistical structures

Enhanced understanding of geometric structures in information geometry

Abstract

The methods of Information geometry have been glowing up to develop various subjects of theoretical physics, including quantum information systems. The present article has two purposes. The first one is to develop general theory of homogeneous statistical manifolds. In particular we construct explicit examples of homogeneous statistical manifolds of low dimension. The second purpose is to classify $3$-dimensional Lie groups admitting non-trivial conjugate symmetric left invariant statistical structure.