Ebook: 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.

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Content:
Front Matter....Pages i-vii
Front Matter....Pages 1-1
Dual Quasitriangular Structures Related to the Temperley-Lieb Algebra....Pages 1-16
On The Quantization of Quadratic Poisson Brackets on a Polynomial Algebra of Four Variables....Pages 17-25
Two Types of Poisson Pencils and Related Quantum Objects....Pages 27-46
Wave Packet Transform in Symplectic Geometry and Asymptotic Quantization....Pages 47-69
On Quantum Methods in the Classical Theory of Representations....Pages 71-84
Front Matter....Pages 85-85
Multivalued Groups, n-Hopf Algebras and n-Ring Homomorphisms....Pages 85-107
Hypergroups and Differential Equations....Pages 109-115
Haar Measure on Locally Compact Hypergroups....Pages 117-131
Laguerre Hypergroup and Limit Theorem....Pages 133-145
The Representation of The Reproducing Kernel in Orthogonal Polynomials on Several Intervals....Pages 147-162
Front Matter....Pages 163-163
Homology Invariants of Homogeneous Complex Manifolds....Pages 163-179
Micromodules....Pages 181-198
A Spectral Sequence for the Tangent Sheaf Cohomology of a Supermanifold....Pages 199-215
On a Duality of Varieties of Representations of Ternary Lie and Super Lie Systems....Pages 217-230
Various Aspects and Generalizations of the Godbillon-Vey Invariant....Pages 231-247
Volume of Bounded Symmetric Domains and Compactification of Jordan Triple Systems....Pages 249-259
Front Matter....Pages 261-261
Asymptotic Behavior of the Poisson Transform on a Hyperboloid of One Sheet....Pages 261-284
Maximal Degenerate Representations, Berezin Kernels and Canonical Representations....Pages 285-298
Asymptotic Representation of Discrete Groups....Pages 299-312
Maximal Degenerate Series Representations of the Universal Covering of the Group SU(n,n)....Pages 313-336
Front Matter....Pages 261-261
Almost Representations and Quasi-Symmetry....Pages 337-358
Front Matter....Pages 359-359
Orbital Isomorphism Between Two Classical Integrable Systems....Pages 359-382
Noncommutative Deformation of The Kadomtsev-Petviashvili Hierarchy and The Universal Grassmann Manifold....Pages 383-391
Symmetries Of Completely Integrable Distributions....Pages 393-405
Algebras With Flat Connections and Symmetries of Differential Equations....Pages 407-424
On The Geometry of Current Groups and a Model of The Landau-Lifschitz Equation....Pages 425-433
Change of Variable Formulas for Gaussian Integrals over Spaces of Paths in Compact Riemannian Manifolds....Pages 435-442

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Content:
Front Matter....Pages i-vii
Front Matter....Pages 1-1
Dual Quasitriangular Structures Related to the Temperley-Lieb Algebra....Pages 1-16
On The Quantization of Quadratic Poisson Brackets on a Polynomial Algebra of Four Variables....Pages 17-25
Two Types of Poisson Pencils and Related Quantum Objects....Pages 27-46
Wave Packet Transform in Symplectic Geometry and Asymptotic Quantization....Pages 47-69
On Quantum Methods in the Classical Theory of Representations....Pages 71-84
Front Matter....Pages 85-85
Multivalued Groups, n-Hopf Algebras and n-Ring Homomorphisms....Pages 85-107
Hypergroups and Differential Equations....Pages 109-115
Haar Measure on Locally Compact Hypergroups....Pages 117-131
Laguerre Hypergroup and Limit Theorem....Pages 133-145
The Representation of The Reproducing Kernel in Orthogonal Polynomials on Several Intervals....Pages 147-162
Front Matter....Pages 163-163
Homology Invariants of Homogeneous Complex Manifolds....Pages 163-179
Micromodules....Pages 181-198
A Spectral Sequence for the Tangent Sheaf Cohomology of a Supermanifold....Pages 199-215
On a Duality of Varieties of Representations of Ternary Lie and Super Lie Systems....Pages 217-230
Various Aspects and Generalizations of the Godbillon-Vey Invariant....Pages 231-247
Volume of Bounded Symmetric Domains and Compactification of Jordan Triple Systems....Pages 249-259
Front Matter....Pages 261-261
Asymptotic Behavior of the Poisson Transform on a Hyperboloid of One Sheet....Pages 261-284
Maximal Degenerate Representations, Berezin Kernels and Canonical Representations....Pages 285-298
Asymptotic Representation of Discrete Groups....Pages 299-312
Maximal Degenerate Series Representations of the Universal Covering of the Group SU(n,n)....Pages 313-336
Front Matter....Pages 261-261
Almost Representations and Quasi-Symmetry....Pages 337-358
Front Matter....Pages 359-359
Orbital Isomorphism Between Two Classical Integrable Systems....Pages 359-382
Noncommutative Deformation of The Kadomtsev-Petviashvili Hierarchy and The Universal Grassmann Manifold....Pages 383-391
Symmetries Of Completely Integrable Distributions....Pages 393-405
Algebras With Flat Connections and Symmetries of Differential Equations....Pages 407-424
On The Geometry of Current Groups and a Model of The Landau-Lifschitz Equation....Pages 425-433
Change of Variable Formulas for Gaussian Integrals over Spaces of Paths in Compact Riemannian Manifolds....Pages 435-442
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