A new invariant for finite dimensional Leibniz/Lie algebras

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. Furthermore, ${\mathcal A}(\mathfrak{h})$ admits a unique bialgebra structure which makes it an initial object among all bialgebras coacting on $\mathfrak{h}$ through a Leibniz/Lie algebra homomorphism. The bialgebra ${\mathcal A}(\mathfrak{h})$ is the key object in approaching three classical and open problems in Lie algebra theory. First, we prove that the automorphisms 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 ${\mathcal A} (\mathfrak{h}) \to k[G]$. Finally, for a finite group $G$, we prove that the set of all actions as automorphisms of $G$ on $\mathfrak{h}$ is parameterized by the set of all bialgebra homomorphisms ${\mathcal A} (\mathfrak{h}) \to k[G]^*$. ${\mathcal A} (\mathfrak{h})$ is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra $\mathfrak{h}$.