M. Goze - Böcker

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.

This volume presents the lectures given during the Second French-Uzbek Colloquium on Algebra and Operator Theory which took place in Tashkent in 1997, at the Mathematical Institute of the Uzbekistan Academy of Sciences. Among the algebraic topics discussed here are deformation of Lie algebras, cohomology theory, the algebraic variety of the laws of Lie algebras, Euler equations on Lie algebras, Leibniz algebras, and real K-theory. Some contributions have a geometrical aspect, such as supermanifolds. The papers on operator theory deal with the study of certain types of operator algebras. The volume also contains a detailed introduction to the theory of quantum groups. This book is intended for graduate students specializing in algebra, differential geometry, operator theory, and theoretical physics, and for researchers in mathematics and theoretical physics.

The geometrical view of mechanics is based on the study of certain exterior systems, the most classical of which are Pfaffian systems. This text presents the classification theorems (Frobenius, Darboux) and the local classification of Pfaffian systems of five variables, following Cartan. It also presents a new class of exterior systems, called k-symplectic systems, generalizing the notion of symplectic form. These systems permit the authors to write in the language of exterior forms the equations proposed by Nambu for a model of statistical mechanics. This work is aimed at graduate students and at research workers in the field of mathematics, differential geometry, statistical mechanics, mathematics of physics and lie algebras.

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.

The geometrical view of mechanics is based on the study of certain exterior systems, the most classical of which are Pfaffian systems. In this book, we present the classification theorems (Frobenius, Darboux) and the local classification of Pfaffian systems of five variables, following Cartan. We also present a new class of exterior systems, called k-symplectic systems, generalizing the notion of symplectic form. These systems permit us to write in the language of exterior forms the equations proposed by Nambu for a model of statistical mechanics. Audience: This book is aimed at graduate students and at research workers in the field of mathematics, differential geometry, statistical mechanics, mathematics of physics and Lie algebras.