Perturbations of Lie algebra structures. (English) Zbl 0715.17009

Deformation theory of algebras and structures and applications, Nato Adv. Study Inst., Castelvecchio-Pascoli/Italy 1986, Nato ASI Ser., Ser. C 247, 265-355 (1988).
[For the entire collection see Zbl 0654.00006.]
This long article written in the style of lecture notes with lengthy calculations and many examples is meant to promote nonstandard methods for studying perturbations of Lie algebra structures like contractions and deformations.
Ch. 1 is meant to prove the superiority of nonstandard analysis by looking at the zeros of two polynomials of the same degree the coefficients of which are infinitesimally close to each other. I do not understand why the problem is solved by the statement “The roots of the standard polynomial are the shadows of the roots of the other.” Therefore, I lack the necessary enthusiasm to believe that the article is full of new results which cannot be obtained otherwise.
(On p. 337 the author states that the perturbation of the Lie algebra structure leading to a contraction is a powerful tool which does not appear in the “classical” works. This is not true since I have used it myself.)
The English the article is written in does not help either. One example (p. 339): “A Lie algebra whose if every non vanishing linear form...”
Reviewer: E.Weimar-Woods


17B05 Structure theory for Lie algebras and superalgebras
03H99 Nonstandard models
58H15 Deformations of general structures on manifolds