Affine structures on abelian Lie groups Original Research Article

The Nagano–Yagi–Goldmann theorem states that on the torus image every affine (or projective) structure is invariant or is constructed on the basis of some Goldmann rings [Nagano, T., Yagi, K., Osaka J. Math. 11 (1974) 181]. It shows the interest to study the invariant affine structure on the torus image or on abelian Lie groups. Recently, the works of Kim [J. Differential Geom. 24 (1986) 373] and Dekimpe–Ongenae [P.A.M.S., 128 (11) (2000) 3191] precise the number of non-equivalent invariant affine structures on an abelian Lie group in the case these structures are complete. In this paper, we propose a study of complete and non-complete affine structures on abelian Lie groups based on the geometry of the algebraic variety of finite dimensional associative algebras.