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 $k[X_{ij}\mathrm{\mid }i,j=1,\cdots ,n]$ 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$\mathfrak{h}$isisomorphictothegroup$U \bigl( G({\mathcal A} (\mathfrak{h})^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual $\mathcal{A}(\mathfrak{h}{)}^{\mathrm{o}}$. 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 $\mathfrak{h}$.