The Lie Theory Workshop, founded by Joe Wolf (UC, Berkeley), has been running for over two decades. These workshops have been sponsored by the NSF, noting the talks have been seminal in describing new perspectives in the field covering broad areas of current research. At the beginning, the top universities in California and Utah hosted the meetings which continue to run on a quarterly basis. Experts in representation theory/Lie theory from various parts of the US, Europe, Asia (China, Japan, Singapore, Russia), Canada, and South and Central America were routinely invited to give talks at these meetings. Nowadays, the workshops are also hosted at universities in Louisiana, Virginia, and Oklahoma. The contributors to this volume have all participated in these Lie theory workshops and include in this volume expository articles which cover representation theory from the algebraic, geometric, analytic, and topological perspectives with also important connections to math physics. These survey articles, review and update the prominent seminal series of workshops in representation/Lie theory mentioned-above, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, number theory, and mathematical physics. Many of the contributors have had prominent roles in both the classical and modern developments of Lie theory and its applications.

Includes contributions from an international group of experts in representation and Lie theory Reflects the widespread influence of the Lie Theory Workshop on areas such as harmonic analysis, differential and algebraic geometry, and number theory Contains articles covering representation theory from algebraic, geometric, analytic, and topological perspectives

Klappentext
This volume reviews and updates a prominent series of workshops in representation/Lie theory, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, and mathematical physics. Many of the contributors have had leading roles in both the classical and modern developments of Lie theory and its applications. This Work, entitled Developments and Retrospectives in Lie Theory, and comprising 26 articles, is organized in two volumes: Algebraic Methods and Geometric and Analytic Methods. This is the Algebraic Methods volume.The Lie Theory Workshop series, founded by Joe Wolf and Ivan Penkov and joined shortly thereafter by Geoff Mason, has been running for over two decades. Travel to the workshops has usually been supported by the NSF, and local universities have provided hospitality. The workshop talks have been seminal in describing new perspectives in the field covering broad areas of current research. Most of the workshops have taken place at leading public and private universities in California, though on occasion workshops have taken place in Oregon, Louisiana and Utah. Experts in representation theory/Lie theory from various parts of the Americas, Europe and Asia have given talks at these meetings. The workshop series is robust, and the meetings continue on a quarterly basis. Contributors to the Algebraic Methods volume:Y. Bahturin, C. P. Bendel, B.D. Boe, J. Brundan, A. Chirvasitu, B. Cox, V. Dolgushev, C.M. Drupieski, M.G. Eastwood, V. Futorny, D. Grantcharov, A. van Groningen, M. Goze, J.-S. Huang, A.V. Isaev, I. Kashuba, R.A. Martins, G. Mason, D. Milii, D.K., Nakano, S.-H. Ng, B.J. Parshall, I. Penkov, C. Pillen, E. Remm, V. Serganova, M.P. Tuite, H.D. Van, J.F. Willenbring, T. Willwacher, C.B. Wright, G. Yamskulna, G. Zuckerman

Inhalt
Group gradings on Lie algebras with applications to geometry. I (Y. Bahturin, M. Goze, E. Remm).- Bounding the dimensions of rational cohomology groups (C.P. Bendel, B.D. Boe, C.M. Drupieski, D.K. Nakano, B.J. Parshall, C. Pillen, C.B. Wright).- Representations of the general linear Lie superalgebra in the BGG Category {$\mathcal O$} (J. Brundan).- Three results on representations of Mackey Lie algebras (A. Chirvasitu).- Free field realizations of the DateJimboKashiwaraMiwa algebra (B. Cox, V. Futorny, R.A. Martins).- The deformation complex is a homotopy invariant of a homotopy algebra (V. Dolgushev, T. Willwacher).- Invariants of Artinian Gorenstein algebras and isolated hypersurface singularities (M.G. Eastwood, A.V. Isaev).- Generalized loop modules for affine KacMoody algebras (V. Futorny, I. Kashuba).- Twisted localization of weight modules (D. Grantcharov).- Dirac cohomology and generalization of classical branching rules (J.-S. Huang).- Cleft extensions and quotients of twisted quantum doubles (G. Mason, S.-H. Ng).- On the structure of ${\Bbb N}$-graded vertex operator algebras (G. Mason, G. Yamskulna).- Variations on a CasselmanOsborne theme (D. Milii).- Tensor representations of Mackey Lie algebras and their dense subalgebras (I. Penkov, V. Serganova).- Algebraic methods in the theory of generalized HarishChandra modules (I. Penkov, G. Zuckerman).- On exceptional vertex operator (super) algebras (M.P. Tuite, H.D. Van).- The cubic, the quartic, and the exceptional group $G_2$ (A. van Groningen, J.F. Willenbring).