The moment map for the variety of associative algebras

Abstract

We consider the moment map m : PV n → iu(n) for the action of GL(n) on V n = ⊗ 2 (C n ) * ⊗ C n , and study the critical points of the functional F n = m 2 : PV n → R. Firstly, we prove that [µ] ∈ PV n is a critical point if and only if M µ = c µ I + D µ for some c µ ∈ R and D µ ∈ Der(µ), where m([µ]) = Mµ µ 2 . Then we show that any algebra µ admits a Nikolayevsky derivation φ µ which is unique up to automorphism, and if moreover, [µ] is a critical point of F n , then φ µ = − 1 cµ D µ . Secondly, we characterize the maxima and minima of the functional F n : A n → R, where A n denotes the projectivization of the algebraic varieties of all ndimensional associative algebras. Furthermore, for an arbitrary critical point [µ] of F n : A n → R, we also obtain a description of the algebraic structure of [µ]. Finally, we classify the critical points ofwhere (•, •) is an Ad(U(n))-invariant real inner product on iu(n), and ρ µ : GL(n) → R is defined by ρ µ (g) = g.µ, g.µ . The function m is the moment map from symplectic geometry, corresponding to the Hamiltonian action U(n) of V n on the symplectic manifold PV n (see [4,12]). In this paper, we study the critical points of the functional F n = m 2 : PV n → R, with an emphasis on the critical points that lie in the projectivization of the algebraic variety of all n-dimensional associative algebras A n .