Algebraic properties of the information geometry's fourth Frobenius manifold

TL;DR

This paper explores the algebraic and geometric structure of the fourth Frobenius manifold related to statistical manifolds, revealing its decomposition into symmetric submanifolds and providing new insights into the algebraic nature of probability distribution manifolds.

Contribution

It proves the decomposition of the fourth Frobenius manifold into symmetric submanifolds associated with ideals of the paracomplex algebra, advancing the algebraization of statistical manifold theory.

Findings

Decomposition of the manifold into symmetric totally geodesic submanifolds

Identification of submanifolds with modules over ideals of the algebra

Symmetry established via the Peirce mirror

Abstract

Recently, it has been shown that the statistical manifold, related to exponential families, has a Frobenius manifold structure and appears as the fourth class of Frobenius manifolds. It has a structure of a projective manifold over a rank two Frobenius algebra $\frak{A}$, being the algebra of paracomplex numbers and generated by $1, \epsilon$ such that $\epsilon^2=1$. This result is a key step towards an algebraization of the results concerning the manifold of probability distributions and thus offers a new perspective on it. In this paper, we prove that the fourth Frobenius manifold is decomposed into a pair of symmetric totally geodesic pseudo-Riemannian submanifolds, each of which correspond to a module over an ideal of $\frak{A}$. This pair of ideals are othogonal idempotents. The symmetry is obtained under the Peirce mirror.