Lie Groups and Lie Algebras Their Representations, Generalisations and Applications

This collection contains papers conceptually related to the classical ideas of Sophus Lie (i.e., to Lie groups and Lie algebras). Obviously, it is impos­ sible to embrace all such topics in a book of reasonable size. The contents of this one reflect the scientific interests of those authors whose activities, to some extent at least, are associated with the International Sophus Lie Center. We have divided the book into five parts in accordance with the basic topics of the papers (although it can be easily seen that some of them may be attributed to several parts simultaneously). The first part (quantum mathematics) combines the papers related to the methods generated by the concepts of quantization and quantum group. The second part is devoted to the theory of hypergroups and Lie hypergroups, which is one of the most important generalizations of the classical concept of locally compact group and of Lie group. A natural harmonic analysis arises on hypergroups, while any abstract transformation of Fourier type is gen­ erated by some hypergroup (commutative or not). Part III contains papers on the geometry of homogeneous spaces, Lie algebras and Lie superalgebras. Classical problems of the representation theory for Lie groups, as well as for topological groups and semigroups, are discussed in the papers of Part IV. Finally, the last part of the collection relates to applications of the ideas of Sophus Lie to differential equations.

I: Quantum Mathematics.- Dual Quasitriangular Structures Related to the Temperley-Lieb Algebra.- On the Quantization of Quadratic Poisson Brackets on a Polynomial Algebra of Four Variables.- Two Types of Poisson Pencils and Related Quantum Objects.- Wave Packet Transform in Symplectic Geometry and Asymptotic Quantization.- On Quantum Methods in the Classical Theory of Representations.- II: Hypergroups.- Multivalued Groups, n-Hopf Algebras and n-ring Homomorphisms.- Hypergroups and Differential Equations.- The Haar Measure On Locally Compact Hypergroups.- Laguerre Hypergroup and Limit Theorem.- The Representation of the Reproducing Kernel in Orthogonal Polynomials on Several Intervals.- III: Homogenious Spaces And Lie Algebras and Superalgebras.- Homology Invariants of Homogeneous Complex Manifolds.- Micromodules.- A Spectral Sequence for the Tangent Sheaf Cohomology of a Supermanifold.- On a Duality of Varieties of Representations of Ternary Lie and Super Lie Systems.- Various Aspectsand Generalizations of the Godbillon-Vey Invariant.- Volume of Bounded Symmetric Domains and Compactification of Jordan Triple Systems.- IV: Representations.- Asymptotic Behavior of the Poisson Transform on a hyperboloid of one sheet.- Maximal Degenerate Representations, Berezin Kernels and Canonical Representations.- Asymptotic Representation of Discrete Groups.- Maximal Degenerate Series Representations of the Universal Covering of the Group SU(n, n).- Almost Representations and Quasi-Symmetry.- V: Differential Equations.- Orbital Isomorphism between Two Classical Integrable Systems.- Noncommutative Deformation of the KP Hierarchy and the Universal Grassmann Manifold.- Symmetries of Completely Integrable Distributions.- Algebras with Flat Connections and Symmetries of Differential Equations.- On the Geometry of Current Groups and a Model of the Landau-Lifschitz Equation.- Change Variable Formulas for Gaussian Integrals over Spaces of Paths in Compact Riemannian Manifolds.