On the rigidity of solvable Lie algebras



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Ancochea Bermúdez , José María (1988) On the rigidity of solvable Lie algebras. In Deformation theory of algebras and structures and applications. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences (247). Kluwer Academic , Dordrecht, pp. 403-445. ISBN 90-277-2804-6


The author gives a method for the construction of solvable rigid Lie algebras in the varieties defined by Jacobi's identities. His method does not use tables of nilpotent Lie algebras and the rigidity is not obtained by usual cohomological criteria. He uses the fact that such an algebra g 0 is algebraic and has a Chevalley splitting T⊕n , where n is the maximal nilpotent ideal and T is a subalgebra constituted by ad g 0 -semisimple elements (by the adjoint representation ad g 0 we can also identify T to a maximal torus of derivations over n) . The author takes a regular semisimple element X (X∈T) and studies all possible systems of relations (S) checked for the eigenvalues of the operator ad g 0 X . Certain systems (S) are not suitable and are directly eliminated. For the remaining cases, the rigidity is established by the method of perturbations in nonstandard analysis.
As applications of his method the author gives a classification of solvable rigid Lie algebras of dimension ≤8 and a list of 48 rigid algebras in dimension 9.

Item Type:Book Section
Additional Information:

Papers from the NATO Advanced Study Institute held in Il Ciocco, June 1–14, 1986

Uncontrolled Keywords:dimension eight; solvable Lie algebras; open orbit; inner derivations; rigidity
Subjects:Sciences > Mathematics > Algebra
ID Code:21163
Deposited On:29 Apr 2013 17:55
Last Modified:12 Dec 2018 15:13

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