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Dirac Operators in Representation Theory

By: Huang, Jing-Song [author.].
Contributor(s): Pandžić, Pavle [author.] | SpringerLink (Online service).
Material type: materialTypeLabelBookSeries: Mathematics: Theory & Applications: Publisher: Boston, MA : Birkhäuser Boston, 2006.Description: XII, 200 p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9780817644932.Subject(s): Mathematics | Group theory | Topological groups | Lie groups | Operator theory | Differential geometry | Physics | Mathematics | Topological Groups, Lie Groups | Group Theory and Generalizations | Differential Geometry | Operator Theory | Mathematical Methods in PhysicsDDC classification: 512.55 | 512.482 Online resources: Click here to access online
Contents:
Lie Groups, Lie Algebras and Representations -- Clifford Algebras and Spinors -- Dirac Operators in the Algebraic Setting -- A Generalized Bott-Borel-Weil Theorem -- Cohomological Induction -- Properties of Cohomologically Induced Modules -- Discrete Series -- Dimensions of Spaces of Automorphic Forms -- Dirac Operators and Nilpotent Lie Algebra Cohomology -- Dirac Cohomology for Lie Superalgebras.
In: Springer eBooksSummary: This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of discrete series representations). The related concept of Dirac cohomology, which is defined using Dirac operators, is a far-reaching generalization that connects index theory in differential geometry to representation theory. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective. Key topics covered include: * Proof of Vogan's conjecture on Dirac cohomology * Simple proofs of many classical theorems, such as the Bott–Borel–Weil theorem and the Atiyah–Schmid theorem * Dirac cohomology, defined by Kostant's cubic Dirac operator, along with other closely related kinds of cohomology, such as n-cohomology and (g,K)-cohomology * Cohomological parabolic induction and $A_q(\lambda)$ modules * Discrete series theory, characters, existence and exhaustion * Sharpening of the Langlands formula on multiplicity of automorphic forms, with applications * Dirac cohomology for Lie superalgebras An excellent contribution to the mathematical literature of representation theory, this self-contained exposition offers a systematic examination and panoramic view of the subject. The material will be of interest to researchers and graduate students in representation theory, differential geometry, and physics.
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Item type Current location Call number Status Date due Barcode Item holds
E books E books PK Kelkar Library, IIT Kanpur
Available EBK7819
Total holds: 0

Lie Groups, Lie Algebras and Representations -- Clifford Algebras and Spinors -- Dirac Operators in the Algebraic Setting -- A Generalized Bott-Borel-Weil Theorem -- Cohomological Induction -- Properties of Cohomologically Induced Modules -- Discrete Series -- Dimensions of Spaces of Automorphic Forms -- Dirac Operators and Nilpotent Lie Algebra Cohomology -- Dirac Cohomology for Lie Superalgebras.

This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of discrete series representations). The related concept of Dirac cohomology, which is defined using Dirac operators, is a far-reaching generalization that connects index theory in differential geometry to representation theory. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective. Key topics covered include: * Proof of Vogan's conjecture on Dirac cohomology * Simple proofs of many classical theorems, such as the Bott–Borel–Weil theorem and the Atiyah–Schmid theorem * Dirac cohomology, defined by Kostant's cubic Dirac operator, along with other closely related kinds of cohomology, such as n-cohomology and (g,K)-cohomology * Cohomological parabolic induction and $A_q(\lambda)$ modules * Discrete series theory, characters, existence and exhaustion * Sharpening of the Langlands formula on multiplicity of automorphic forms, with applications * Dirac cohomology for Lie superalgebras An excellent contribution to the mathematical literature of representation theory, this self-contained exposition offers a systematic examination and panoramic view of the subject. The material will be of interest to researchers and graduate students in representation theory, differential geometry, and physics.

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