We consider the moment map m: ℙVn → iu(n) for the action of GL(n) on Vn = ⊗2(ℂn)* ⊗ ℂn, and study the critical points of the functional Fn = ∥m∥2: ℙVn → ℝ. Firstly, we prove that [μ] ∈ ℙVn is a critical point if and only if Mμ = cμI + Dμ for some cμ ∈ ℝ and Dμ ∈ Der(μ), where \(m([\mu])={M_{\mu}\over{\Vert\mu\Vert^{2}}}\) . Then we show that any algebra μ admits a Nikolayevsky derivation ϕμ which is unique up to automorphism, and if moreover, [μ] is a critical point of Fn, then \(\phi_{\mu}=-{1\over c_{\mu}}D_{\mu}\) . Secondly, we characterize the maxima and minima of the functional \(F_{n}:\cal{A}_{n}\rightarrow\mathbb{R}\) , where \(\cal{A}_{n}\) denotes the projectivization of the algebraic varieties of all the n-dimensional associative algebras. Furthermore, for an arbitrary critical point [μ] of \(F_{n}:\cal{A}_{n}\rightarrow\mathbb{R}\) , we obtain a description of the algebraic structure of μ. Finally, we classify the critical points of \(F_{n}:\cal{A}_{n}\rightarrow\mathbb{R}\) for n = 2 and n = 3, respectively.
