Some Computational Results on Koszul-Vinberg Cochain Complexes
Hanwen Liu, Jun Zhang
TL;DR
This work investigates the deformation theory of flat, torsion-free affine connections through Koszul-Vinberg (KV) cohomology. It provides explicit computations of the KV differential $d_{KV}$ on key cochains, connects certain 2-cochains to projective and dual-projective deformations, and situates these results within Hessian geometry by relating the conjugate connection and curvature via $d_{KV}$. A central contribution is the explicit construction of a nontrivial second KV cohomology class in dimension two, demonstrating that KV cohomology can be non-vanishing and thus can capture nontrivial deformations. The appendix clarifies how KV cohomology compares to twisted de Rham cohomology, underscoring the distinct algebraic-topological nature of KV theory in information-geometry settings.
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.