The moment map for the variety of Leibniz algebras

Abstract: We consider the moment map $m:\mathbb{P}V_n\rightarrow \text{i}\mathfrak{u}(n)$ for the action of $\text{GL}(n)$ on $V_n=\otimes^{2}(\mathbb{C}^{n})^{*}\otimes\mathbb{C}^{n}$, and study the functional $F_n=|m|^{2}$ restricted to the projectivizations of the algebraic varieties of all $n$-dimensional Leibniz algebras $L_n$ and all $n$-dimensional symmetric Leibniz algebras $S_n$, respectively. Firstly, we give a description of the maxima and minima of the functional $F_n: L_n \rightarrow \mathbb{R}$, proving that they are actually attained at the symmetric Leibniz algebras. Then, for an arbitrary critical point $[\mu]$ of $F_n: S_n \rightarrow \mathbb{R}$, we characterize the structure of $[\mu]$ by virtue of the nonnegative rationality. Finally, we classify the critical points of $F_n: S_n \rightarrow \mathbb{R}$ for $n=2$, $3$, respectively.