Deformation of affine structures and the cohomology of Koszul-Vinberg algebras on the lie groups SO(2), H3(r) and Galilei group SGal(3)

Abstract: In this work, we compare the De Rham cohomology and the Koszul-Vinberg cohomology groups on the Lie groups SO(2), H3(R) and SGal(3). We model their interactions by constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups. By computing the exact dimensions of these complexes, we evaluate their algebraic quotient, which measures the gap separating global topological invariants from left-invariant flat affine structures. Extending this geometric framework to the coadjoint orbits of the Heisenberg group H3(R) and the Galilei group SGal(3), we investigate their properties under an invariant Lagrangian foliation inherited from a constant rank Nijenhuis endomorphism preserving the Boyom complex. Finally, we establish a vanishing theorem for the second KV-cohomology group. We demonstrate that any infinitesimal deformation of the affine structure governed by the polarized Maurer Cartan equation is trivial, thereby proving the structural rigidity of these orbits.