The moment map for the variety of associative algebras

TL;DR

This paper studies the moment map for associative algebras, characterizing critical points, maxima, minima, and classifying these points for low-dimensional cases, revealing algebraic structures associated with these critical points.

Contribution

It provides a complete characterization of critical points of the moment map functional on associative algebras and classifies them for dimensions two and three, introducing new structural insights.

Findings

Critical points characterized by specific algebraic conditions.

Existence and uniqueness of Nikolayevsky derivation for each algebra.

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 critical points of the functional $F_n=\|m\|^{2}: \mathbb{P} V_n \rightarrow \mathbb{R}$. Firstly, we prove that $[\mu]\in \mathbb{P}V_n$ is a critical point if and only if $\text{M}_{\mu}=c_{\mu} I+D_{\mu}$ for some $c_{\mu} \in \mathbb{R}$ and $D_{\mu} \in \text{Der}(\mu),$ where $m([\mu])=\frac{\text{M}_{\mu}}{\|\mu\|^{2}}$. Then we show that any algebra $\mu$ admits a Nikolayevsky derivation $\phi_\mu$ which is unique up to automorphism, and if moreover, $[\mu]$ is a critical point of $F_n$, then $\phi_\mu=-\frac{1}{c_\mu}D_\mu.$ Secondly, we characterize the maxima and minima of the functional $F_n: \mathcal{A}_n \rightarrow \mathbb{R}$, where $\mathcal{A}_n$ denotes the projectivization of the algebraic varieties of all $n$-dimensional associative algebras. Furthermore, for an arbitrary critical point $[\mu]$ of $F_n: \mathcal{A}_n \rightarrow \mathbb{R}$, we also obtain a description of the algebraic structure of $[\mu]$. Finally, we classify the critical points of $F_n: \mathcal{A}_n \rightarrow \mathbb{R}$ for $n=2$, $3$, respectively.