A new invariant for finite dimensional Leibniz/Lie algebras

TL;DR

This paper introduces a universal algebra invariant for finite-dimensional Leibniz and Lie algebras, which helps classify automorphisms and gradings, and establishes a universal commutative Hopf algebra for such algebras.

Contribution

The paper defines a new invariant called the universal algebra, providing tools to solve open problems in automorphisms, gradings, and Hopf algebra associations for Leibniz and Lie algebras.

Findings

Automorphism group characterized via invertible group-like elements.

Classification and parameterization of all G-gradings.

Existence of a universal commutative Hopf algebra for finite-dimensional Leibniz algebras.

Abstract

For an $n$-dimensional Leibniz/Lie algebra $\mathfrak{h}$ over a field $k$ we introduce a new invariant ${\mathcal A}(\mathfrak{h})$, called the \emph{universal algebra} of $\mathfrak{h}$, as a quotient of the polynomial algebra $k[X_{ij} \, | \, i, j = 1, \cdots, n]$ through an ideal generated by $n^3$ polynomials. We prove that ${\mathcal A}(\mathfrak{h})$ admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on $\mathfrak{h}$. The new object ${\mathcal A} (\mathfrak{h})$ is the key tool in answering two open problems in Lie algebra theory. First, we prove that the automorphism group ${\rm Aut}_{Lbz} (\mathfrak{h})$ of $\mathfrak{h}$ is isomorphic to the group $U \bigl( G({\mathcal A} (\mathfrak{h})^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual ${\mathcal A} (\mathfrak{h})^{\rm o}$. Secondly, for an abelian group $G$, we show that there exists a bijection between the set of all $G$-gradings on $\mathfrak{h}$ and the set of all bialgebra homomorphisms ${\mathcal A} (\mathfrak{h}) \to k[G]$. Based on this, all $G$-gradings on $\mathfrak{h}$ are explicitly classified and parameterized. ${\mathcal A} (\mathfrak{h})$ is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra $\mathfrak{h}$.