For the moment map m : PVn -> iu(n) for the action of GL(n) on V-n = Lambda(3)(C-n)(*) circle times C-n , we will study the critical points of F-n = IImII(2) : P V-n -> R in this paper. We first prove that [mu] is an element of PVn is a critical point if and only if m([mu]) = c(mu)I+D-mu for some c(mu) is an element of R and D-mu is an element of Der(mu), and that there exists a constant c > 0 such that the eigenvalues of cD(mu) are integers prime to each other and some eigenvalues can be strictly negative. Then we give a description of the maxima and minima of F-n : L-n -> R, where L-n is the projectivization of all n-dimensional 3-Lie algebras. Furthermore, the structure of the critical points of F-n is discussed. Finally, as an application, we show that every three-dimensional 3-Lie algebra is isomorphic to a critical point of F-3; and there exists a curve of nonisomorphic four-dimensional 3-Lie algebras which are not isomorphic to any critical point of F-4. (c) 2022 Elsevier Inc. All rights reserved.
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