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

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]. [ABSTRACT FROM AUTHOR]