Complete descriptions of pseudo-Euclidean left-symmetric L-algebras and their pseudo-Euclidean modules

Abstract

A pseudo-Euclidean left-symmetric algebra (A,.,〈,〉) is a left-symmetric algebra endowed with a non-degenerate symmetric bilinear form 〈,〉 such that left multiplications by any element of A are skew-symmetric with respect to 〈,〉. We recall that a pseudo-Euclidean Lie algebra (g,[,],〈,〉) is flat if and only if (g,.,〈,〉), its underlying vector space endowed with the Levi-Civita product associated with 〈,〉, is a pseudo-Euclidean left-symmetric algebra. In this paper, we study pseudo-Euclidean left-symmetric algebras (A,.,〈,〉) such that commutators of all elements of A are contained in the left annihilator of (A,.), these algebras will be called pseudo-Euclidean left-symmetric L-algebras. Next, we introduce and study pseudo-Euclidean modules of pseudo-Euclidean left-symmetric L-algebras. We develop double extension processes that allow us to have inductive descriptions of all pseudo-Euclidean left-symmetric L-algebras and of all its pseudo-Euclidean modules of any signature. This, in particular, improves some of the results obtained in [11].