Recent Advances in Lie Theory, Research and Exposition in Mathematics, vol. 25

This volume contains updated versions of selected contributions at the First Colloquium on Lie Theory and Applications, held in Vigo, Spain in the year 2000. There are altogether 23 contributions on the theory of Lie groups and algebras. Three of them were related to series of lectures at the Colloquium. The first paper, written by D. V. Alexeevsky and A. F. Spiro, describes flag manifolds and homogeneous CR structures. The first part of the paper contains an introduction to the geometry of flag manifolds with a short description of the structure theory of semisimple Lie algebras and parabolic subalgebras. The second part describes the theory of compact, homogeneous Levi non-degenerate CR manifolds and their classification. The second paper by Yu. A. Brailov and A. T. Fomenko deals with Lie groups and Hamiltonian systems as. It consists again of two parts. The first part contains a review of integration methods for special classes of Hamiltonian systems on Lie groups, symmetric spaces and homogeneous Riemanniaan spaces. The second part contains new results in the singularity theory of integrable Hamiltonian systems. The third paper is written by M. Scheunert and contains an introduction to the cohomology of Lie superalgebras and some applications. The cohomology of colour Lie algebras is introduced, and some classical results on Lie algebra cohomology are generalized to this setting. There are also applications of the theory to the study of formal deformations of the enveloping algebra of a colour Lie algebra and toHochschild cohomology.

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