The moment map for the variety of Leibniz algebras

Abstract

We consider the moment map $m:\mathbb{P}{V}_n\to \text{i}\mathfrak{u}(n)$ for the action of $\text{GL}(n)$ on $V_n={\otimes }^2({\mathbb{C}}^n{)}^{\ast }\otimes {\mathbb{C}}^n$, and study the functional $F_n=\Vert m{\Vert }^2$ 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 $F_n:L_n\to \mathbb{R}$, proving that they are actually attained at the symmetric Leibniz algebras. Then, for an arbitrary critical point $[\mu ]$ of $F_n:S_n\to \mathbb{R}$, we characterize the structure of $[\mu ]$ by virtue of the nonnegative rationality. Finally, we classify the critical points of $F_n:S_n\to \mathbb{R}$ for $n=2$, $3$, respectively.

Some typos are corrected