On a class of pseudo-Euclidean left-symmetric algebras

Abstract: A pseudo-Euclidean left-symmetric algebra $(A, .,< , >)$ is a real left-symmetric algebra $(A,.)$ 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 talk, We will give an inductive classification of pseudo-Euclidean left-symmetric algebras $(A, .,< , >)$ such that commutators of allelements of A are contained in the left annihilator of $(A, .),$ these algebras will be called pseudo-Euclidean left-symmetric L−algebras of any signature. To do this, we will 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.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar