On a class of Lie groups with a left invariant flat pseudo-metric

Abstract

The study of the Lie groups with a left invariant flat pseudo-metric is equivalent to the study of the left-symmetric algebras with a nondegenerate left invariant bilinear form. In this paper, we consider such a structure satisfying an additional condition that there is a decomposition into a direct sum of the underlying vector spaces of two isotropic subalgebras. Moreover, there is a new underlying algebraic structure, namely, a special L-dendriform algebra and then there is a bialgebra structure which is equivalent to the above structure. The study of coboundary cases leads to a construction from an analogue of the classical Yang–Baxter equation.