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Nilpotent Lie Algebras (Hardcover, 1996 ed.)
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Nilpotent Lie Algebras (Hardcover, 1996 ed.)
Series: Mathematics and Its Applications, 361
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Nilpotent Ue algebras have played an Important role over the last
ye!US : either In the domain at Algebra when one considers Its role
In the classlftcation problems of Ue algebras, or In the domain of
geometry since one knows the place of nilmanlfolds In the
Illustration, the description and representation of specific
situations. The first fondamental results In the study of nilpotent
Ue algebras are obvlsouly, due to Umlauf. In his thesis (leipZig,
1991), he presented the first non trlvlal classifications. The
systematic study of real and complex nilpotent Ue algebras was
Independently begun by D1xmler and Morozov. Complete
classifications In dimension less than or equal to six were given
and the problems regarding superior dimensions brought to light,
such as problems related to the existence from seven up, of an
infinity of non Isomorphic complex nilpotent Ue algebras. One can
also find these losts (for complex and real algebras) In the books
about differential geometry by Vranceanu. A more formal approach
within the frame of algebraiC geometry was developed by Michele
Vergne. The variety of Ue algebraiC laws Is an affine algebraic
subset In this view the role variety and the nilpotent laws
constitute a Zarlski's closed of Irreduclbl~ components appears
naturally as well the determination or estimate of their numbers.
Theoritical physiCiSts, Interested In the links between diverse
mechanics have developed the Idea of contractions of Ue algebras
(Segal, Inonu, Wlgner). That Idea was In fact very convenient In
the determination of components.
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