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Abstract: The space of all complex Lie algebras of a given dimension n can be naturally identified with the set L_n of all Lie brackets on C^n. Since the Jacobi identity is determined by polynomial conditions, L_n is an algebraic subset of the vector space V_n of bilinear forms from C^n\timesC^n to C^n. The isomorphism class of a Lie algebra \mu\inL_n is then given by the orbit \Gl(n).\mu under the `change of basis' action of \Gl(n) on V_n. This action is very unpleasent from the point of view of invariant theory since any \mu\in V_n is unstable (i.e. 0\in\overline{\Gl(n).\mu}), which makes very difficult the study of the quotient space L_n/\Gl(n) parameterizing Lie algebras up to isomorphism.
Nevertheless, F. Kirwan and L. Ness have shown that the momentum map for an action can be used to study the orbit space of the null-cone (set of unstable vectors). We consider the momentum map m:P V_n -> iu(n) for the action of \Gl(n) on V_n, where iu(n) denotes the space of hermitian matrices. The critical points of the functional F_n=||m||^2:P V_n\mapstoR are studied, in order to understand the stratification of L_n\subsetP XSV_n defined by the negative gradient flow of F_n, where L_n is the projectivization of L_n. We obtain a description of the critical points which lie in L_n in terms of those which are nilpotent, as well as the minima and maxima of F_n:L_n\mapstoR. A characterization of the critical points modulo isomorphism, as the categorical quotient of a suitable action is considered, and some applications to the study of L_n are given.
This talk will be accessible to graduate students.
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