Associative Structures in Pseudo-Riemannian Lie Algebras

This paper investigates the algebraic and geometric consequences of the associativity of the symmetric part $U$ of the Levi-Civita connection on a pseudo-Riemannian Lie algebra $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. We demonstrate that in the Riemannian (positive-definite) setting, the associativity of $U$ is an extremely restrictive condition that forces the tensor to vanish identically, thereby recovering the class of bi-invariant metrics. In contrast, in the pseudo-Riemannian setting, we focus on the subclass where $(\mathfrak{g}, U)$ is associative and unimodular. As a primary result, we establish that every connected Lie group endowed with a left-invariant pseudo-Riemannian metric whose $U$-tensor is associative and unimodular is geodesically complete. Finally, we explore the families of 2-step nilpotent and almost-abelian Lie algebras. For the latter, we obtain some rigid structural classifications, showing that the paradigmatic models are the $3$-dimensional Heisenberg algebra with certain (anti)-Lorentzian metrics or a semi-direct extension involving a nondegenerate infinitesimal $\beta$-transformation on the canonical neutral space $W \oplus W^*$.