The moment map for the variety of Leibniz algebras

TL;DR

This paper investigates the moment map for Leibniz algebras, characterizes extrema of a related functional, and classifies critical points for low dimensions, advancing understanding of algebraic structures via geometric methods.

Contribution

It describes the extrema of the moment map functional on Leibniz algebras, characterizes critical points using rationality, and classifies these points for dimensions two and three.

Findings

Maxima and minima of the functional are attained at symmetric Leibniz algebras.

Critical points of the functional are characterized by nonnegative rationality.

Complete classification of critical points for dimensions 2 and 3.

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.