# Algebraic properties of the information geometry's fourth Frobenius manifold

@inproceedings{Combe2021AlgebraicPO, title={Algebraic properties of the information geometry's fourth Frobenius manifold}, author={Noemie C. Combe and Philippe Combe and Hanna Nencka}, year={2021} }

Recently, it has been shown that within the statistical manifold, related to exponential families, there exists a submanifold having a Frobenius manifold structure. This appears as the fourth class of Frobenius manifolds. It has a structure of a projective manifold over a rank two Frobenius algebra A, being the algebra of paracomplex numbers and generated by 1, ε such that ε = 1. This last result is a key step towards an algebraization of the results concerning the manifold of probability…

#### Tables from this paper

#### One Citation

A note on exponential varieties, statistical manifolds and Frobenius structures

- Mathematics
- 2021

In this paper we consider a class of manifolds corresponding to statistical models, related to exponential families. Exponential manifolds have been considered from the point of view of information…

#### References

SHOWING 1-10 OF 29 REFERENCES

Three constructions of Frobenius manifolds: A comparative study

- Mathematics
- 1998

The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent…

Weak Frobenius manifolds

- Mathematics
- 1998

We establish a new universal relation between the Lie bracket and $\circ$-multiplication of tangent fields on any Frobenius (super)manifold. We use this identity in order to introduce the notion of…

Spaces over Algebras and Their Applications

- Mathematics
- 2002

For Kazan research in geometry, the study of spaces over algebras is fairly traditional. Even in 1895, A. P. Kotel’nikov, developing Clifford’s ideas, showed that the screw theory of the…

A Taste of Jordan Algebras

- Mathematics
- 2003

In this book, Kevin McCrimmon describes the history of Jordan Algebras and he describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim…

Frobenius manifolds, quantum cohomology, and moduli spaces

- Mathematics
- 1999

Introduction: What is quantum cohomology? Introduction to Frobenius manifolds Frobenius manifolds and isomonodromic deformations Frobenius manifolds and moduli spaces of curves Operads, graphs, and…

The Foundations Of Differential Geometry

- Computer Science
- 2016

The the foundations of differential geometry is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can get it instantly.

Geometry of 2D topological field theories

- Physics, Mathematics
- 1994

These lecture notes are devoted to the theory of “equations of associativity” describing geometry of moduli spaces of 2D topological field theories.

Information and the Accuracy Attainable in the Estimation of Statistical Parameters

- Mathematics
- 1992

The earliest method of estimation of statistical parameters is the method of least squares due to Mark off. A set of observations whose expectations are linear functions of a number of unknown…

F‐Manifolds and geometry of information

- Mathematics
- 2020

The theory of $F$-manifolds, and more generally, manifolds endowed with commutative and associative multiplication of their tangent fields, was discovered and formalised in various models of quantum…