Synopses & Reviews
The orbit method influenced the development of several areas of mathematics in the second half of the 20th century and continues to be an important tool today. Among the distinguished names associated with the orbit method is that of A.A. Kirillov, whose pioneering paper on nilpotent orbits in 1962, places him as the founder of orbit theory. The origins of the orbit method lie in the search for a relationship between classical and quantum mechanics. Over the years, the orbit method has been used to link harmonic analysis (theory of unitary representations of Lie groups) with differential geometry (symplectic geometry of homogeneous spaces), and it has stimulated and invigorated many classical domains of mathematics, i.e., representation theory, integrable systems, complex algebraic geometry, to name several. It continues to be a useful and powerful tool in all of these areas of mathematics and mathematical physics. This volume, dedicated to A. A. Kirillov, covers a very broad range of key topics such as: * The orbit method in the theory of unitary representations of Lie groups * Infinite-dimensional Lie groups: their orbits and representations * Quantization and the orbit method; geometric quantization (old and new) * The Virasoro algebra; string and conformal field theories * Lie superalgebras and their representations * Combinatorial aspects of representation theory. The prominent contributors to this volume present original and expository invited papers in the areas of Lie theory, geometry, algebra, and mathematical physics. The work will be an invaluable reference for researchers in the above mentioned fields, as well as a useful text for graduate seminars and courses. Contributors include: A. Alekseev, J. Alev, R. Brylinski, J. Dixmier, D.R. Farkas, V. Ginzburg, V. Gorbounov, P. Grozman, E. Gutkin, A. Joseph, D. Kazhdan, A.A. Kirillov, B. Kostant, D. Leites, F. Malikov, A. Melnikov, Y.A. Neretin, A. Okounkov, G. Olshanski, F. Petrov, A. Polishchuk, W. Rossmann, A. Sergeev, V. Schechtman, I. Shchepochkina.
Review
"...the volume might be useful to a large number of potential readers interested in various fields, like representation theory of Lie groups, symplectic geometry, differential equations, combinatorics, etc. It is noteworthy that the history of mathematics can also be added to this list of topics, due to the nice article authored by J. Dixmier." --Romanian Journal of Pure and Appl. Math.
Synopsis
The orbit method influenced the development of several areas of mathematics in the second half of the 20th century and continues to be an important tool today. Among the distinguished names associated with the orbit method is that of A.A. Kirillov, whose pioneering paper on nilpotent orbits in 1962, places him as the founder of orbit theory. The origins of the orbit method lie in the search for a relationship between classical and quantum mechanics. Over the years, the orbit method has been used to link harmonic analysis (theory of unitary representations of Lie groups) with differential geometry (symplectic geometry of homogeneous spaces), and it has stimulated and invigorated many classical domains of mathematics, i.e., representation theory, integrable systems, complex algebraic geometry, to name several. It continues to be a useful and powerful tool in all of these areas of mathematics and mathematical physics. This volume, dedicated to A. A. Kirillov, covers a very broad range of key topics such as: * The orbit method in the theory of unitary representations of Lie groups * Infinite-dimensional Lie groups: their orbits and representations * Quantization and the orbit method; geometric quantization (old and new) * The Virasoro algebra; string and conformal field theories * Lie superalgebras and their representations * Combinatorial aspects of representation theory. The prominent contributors to this volume present original and expository invited papers in the areas of Lie theory, geometry, algebra, and mathematical physics. The work will be an invaluable reference for researchers in the above mentioned fields, as well as a useful text for graduate seminars and courses. Contributors include: A. Alekseev, J. Alev, R. Brylinski, J. Dixmier, D.R. Farkas, V. Ginzburg, V. Gorbounov, P. Grozman, E. Gutkin, A. Joseph, D. Kazhdan, A.A. Kirillov, B. Kostant, D. Leites, F. Malikov, A. Melnikov, Y.A. Neretin, A. Okounkov, G. Olshanski, F. Petrov, A. Polishchuk, W. Rossmann, A. Sergeev, V. Schechtman, I. Shchepochkina.
Table of Contents
A Principle of Variations in Representation Theory * Finite Group Actions on Poisson Algebras * Representations of Quantum Tori and G-bundles on Elliptic Curves * Dixmier Algebras for Classical Complex Nilpotent Orbits via Kraft-Procesi Models I * Gerbes of Chiral Differential Operators. III * Defining Relations for the Exceptional Superalgebras of Vector Fields * Schur-Weyl Duality and Representations of Permutation Groups * Quantiziation of Hypersurface Orbitla Varieties in sln * Generalization of a Theorem of Waldspurger to Nice Representations * Two More Variations on the Triangular Theme * The Generalized Cayley Map from an Algebraic Group to its Lie Algebra * Geometry of GLn(C) at Infinity: Hinges, Complete Collineations, Projective Compactifications, and Universal Boundary * Point Processes Related to the Infinite Symmetric Group * Some Toric Manifolds and a Path Integral * Projective Schur Functions as Bispherical Functions on Certain Homogeneous Superspaces * Maximal Subalgebras of the Classical Linear Lie Superalgebras