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## On the space of all Lie algebras of a given dimension

### Jorge Lauret

Yale University (and University of Cordoba)

###
Thursday, November 15, 2001

102 Bradley Hall, 4 pm

Tea 3:30 pm, Math Lounge

**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\times**C**^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\mapsto**R** are studied, in order to
understand the stratification of *L_n\subset***P** 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\mapsto***R**. 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.