Complete Description of Invariant, Associative Pseudo-Euclidean Metrics on Left Leibniz Algebras via Quadratic Lie Algebras

Abstract

A pseudo-Euclidean non-associative algebra $(\mathfrak{g}, \bullet)$ is a real algebra of finite dimension that has a metric, i.e., a bilinear, symmetric, and non-degenerate form $\langle\;\rangle$. The metric is considered $\mathrm{L}$-invariant (resp. $\mathrm{R}$-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if $\langle u\bullet v,w\rangle= \langle u,v\bullet w\rangle$ for all $u, v, w \in \mathfrak{g}$. These three notions coincide when $\mathfrak{g}$ is a Lie algebra and in this case $\mathfrak{g}$ endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of $\mathrm{L}$-invariant, $\mathrm{R}$-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a $\mathrm{L}$-invariant or $\mathrm{R}$-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from these complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an $\mathrm{L}$-invariant metric, up to dimension 4, and $\mathrm{R}$-invariant metric up to dimension 5.