Algebraic properties of the information geometry's fourth Frobenius manifold

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 $\mathfrak{A}$, 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 $\mathfrak{A}$. This pair of ideals are othogonal idempotents. The symmetry is obtained under the Peirce mirror.