The moment map for the variety of associative 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 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.