Algebraic properties of the information geometry's fourth Frobenius manifold

N. C. Combe; P. G. Combe; H. K. Nencka

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arXiv · 2109.13605

Algebraic properties of the information geometry's fourth Frobenius manifold

N. C. Combe·P. G. Combe·H. K. Nencka

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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.

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BibTeX RIS

N. C. Combe, P. G. Combe, H. K. Nencka. 2021-09-28. Algebraic properties of the information geometry's fourth Frobenius manifold. https://arxiv.org/abs/2109.13605

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