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The study of W algebras began in 1985 in the context of two-dimensional conf- mal field theories, the aim being to explore higher-spin extensions of the Virasoro algebra. Given the simultaneous growth in the understanding of two-dimensional metric gravity inspired by analyses of string models, it was inevitable that these algebras would be applied to give analogues of putative higher-spin gravity t- ories. This book is an exposition of the past few years of our work on such an application for the algebra: in particular, the BRST quantization of the n- critical 4D string. We calculate the physical spectrum as a problem in BRST cohomology. The corresponding operator cohomology forms a BV algebra, for which we provide a geometrical model. The algebra has one further generator, of spin three, in addition to the (spin two) energy-momentum tensor which generates the Virasoro algebra. C- trary to the Virasoro algebra, it is an algebra defined by nonlinear relations. In deriving our understanding of the resulting gravity theories we have had to - velop a number of results on the representation theory of W algebras, to replace the standard techniques that were so successful in treating linear algebras.
This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra g g over C C. The setting is the module category O O introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for g g such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory. Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of g g. Basic techniques in category O O such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan–Lusztig Conjecture (due to Beilinson–Bernstein and Brylinski–Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: D D-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category O O, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson–Ginzburg–Soergel.
These proceedings are from the Tenth International Conference on Representations of Algebras and Related Topics (ICRA X) held at The Fields Institute. In addition to the traditional ``instructional'' workshop preceding the conference, there were also workshops on ``Commutative Algebra, Algebraic Geometry and Representation Theory'', ``Finite Dimensional Algebras, Algebraic Groups and Lie Theory'', and ``Quantum Groups and Hall Algebras''. These workshops reflect the latest developments and the increasing interest in areas that are closely related to the representation theory of finite dimensional associative algebras. Although these workshops were organized separately, their topics are strongly interrelated. The workshop on Commutative Algebra, Algebraic Geometry and Representation Theory surveyed various recently established connections, such as those pertaining to the classification of vector bundles or Cohen-Macaulay modules over Noetherian rings, coherent sheaves on curves, or ideals in Weyl algebras. In addition, methods from algebraic geometry or commutative algebra relating to quiver representations and varieties of modules were presented. The workshop on Finite Dimensional Algebras, Algebraic Groups and Lie Theory surveyed developments in finite dimensional algebras and infinite dimensional Lie theory, especially as the two areas interact and may have future interactions. The workshop on Quantum Groups and Hall Algebras dealt with the different approaches of using the representation theory of quivers (and species) in order to construct quantum groups, working either over finite fields or over the complex numbers. In particular, these proceedings contain a quite detailed outline of the use of perverse sheaves in order to obtain canonical bases. The book is recommended for graduate students and researchers in algebra and geometry.
In this paper, the structure of generalized Verma modules is studied, as well as the structure of their projective covers in a suitable version of the Bernstein-Gelfand-Gelfand category [script capital]O.
Our work is aimed toward an understanding of higher-order extension groups in the category of ([bold italic]g, [bold italic]K)-modules. Although our final results concern rank one Lie groups, the ideas and approach are applicable to any semisimple matrix group.
The category of highest weight representations is of special interest withing the full set of representations of a real semisimple Lie group. This memoir describes the structure of the generalized Verma modules as well as the Kazhdan-Lusztig data for the simple modules in this category for the classical groups. In particular, explicit formulas for composition factors of generalized Verma modules and Kazhdan-Lusztig polynomials are given.
This paper investigates the question of linkage and block theory for Lie algebras of Cartan type. The second part of the paper deals mainly with block structure and projective modules of Lies algebras of types W and K.
This volume presents modern trends in the area of symmetries and their applications based on contributions from the workshop "Lie Theory and Its Applications in Physics", held near Varna, Bulgaria, in June 2015. Traditionally, Lie theory is a tool to build mathematical models for physical systems.Recently, the trend has been towards geometrization of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry, which is very helpful in understanding its structure. Geometrization and symmetries are employed in their widest sense, embracing representation theory, algebraic geometry, number theory, infinite-dimensional Lie algebras and groups, superalgebras and supergroups, groups and quantum groups, noncommutative geometry, symmetries of linear and nonlinear partial differential operators (PDO), special functions, and others. Furthermore, the necessary tools from functional analysis are included.“div>This is a large interdisciplinary and interrelated field, and the present volume is suitable for a broad audience of mathematicians, mathematical physicists, and theoretical physicists, including researchers and graduate students interested in Lie Theory.
We study conformal symmetry breaking differential operators which map dif-ferential forms on Rn to differential forms on a codimension one subspace Rn−1. These operators are equivariant with respect to the conformal Lie algebra of the subspace Rn−1. They correspond to homomorphisms of generalized Verma mod-ules for so(n, 1) into generalized Verma modules for so(n+1, 1) both being induced from fundamental form representations of a parabolic subalgebra. We apply the F -method to derive explicit formulas for such homomorphisms. In particular, we find explicit formulas for the generators of the intertwining operators of the re-lated branching problems restricting generalized Verma modules for so(n +1, 1) to so(n, 1). As consequences, we derive closed formulas for all conformal symmetry breaking differential operators in terms of the first-order operators d, δ, d¯ and δ¯ and certain hypergeometric polynomials. A dominant role in these studies is played by two infinite sequences of symmetry breaking differential operators which depend on a complex parameter λ. Their values at special values of λ appear as factors in two systems of factorization identities which involve the Branson-Gover opera- tors of the Euclidean metrics on Rn and Rn−1 and the operators d, δ, d¯ and δ¯ as factors, respectively. Moreover, they naturally recover the gauge companion and Q-curvature operators of the Euclidean metric on the subspace Rn−1, respectively.