Some Computational Results on Koszul-Vinberg Cochain Complexes

TL;DR

This paper computes explicit differentials in Koszul-Vinberg cohomology related to flat affine connections and explores their connections to information geometry, providing new examples of non-vanishing second cohomology groups.

Contribution

It provides explicit calculations of KV cochain differentials and links them to classical information geometry objects, also presenting a novel example of a KV algebra with non-zero second cohomology.

Findings

Explicit differentials of KV cochains computed.

Connections established between KV cohomology and information geometry.

Example of a KV algebra with non-vanishing second cohomology provided.

Abstract

An affine connection is said to be flat if its curvature tensor vanishes identically. Koszul-Vinberg (KV for abbreviation) cohomology has been invoked to study the deformation theory of flat and torsion-free affine connections on tangent bundle. In this Note, we compute explicitly the differentials of various specific KV cochains, and study their relation to classical objects in information geometry, including deformations associated with projective and dual-projective transformations of a flat and torsion-free affine connection. As an application, we also give a simple yet non-trivial example of a KV algebra of which second cohomology group does not vanish.