Some computational results on Koszul–Vinberg cochain complexes

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 left-symmetric algebra of which the second cohomology group does not vanish.