The Penrose Transform: Its Interaction with Representation TheoryCourier Dover Publications, 28. lis 2016. - Broj stranica: 256 "Brings to the reader a huge amount of information, well organized and condensed into less than two hundred pages." — Mathematical Reviews In recent decades twistor theory has become an important focus for students of mathematical physics. Central to twistor theory is the geometrical transform known as the Penrose transform, named for its groundbreaking developer. Geared toward students of physics and mathematics, this advanced text explores the Penrose transform and presupposes no background in twistor theory and a minimal familiarity with representation theory. An introductory chapter sketches the development of the Penrose transform, followed by reviews of Lie algebras and flag manifolds, representation theory and homogeneous vector bundles, and the Weyl group and the Bott-Borel-Weil theorem. Succeeding chapters explore the Penrose transform in terms of the Bernstein-Gelfand-Gelfand resolution, followed by worked examples, constructions of unitary representations, and module structures on cohomology. The treatment concludes with a review of constructions and suggests further avenues for research. |
Sadržaj
Minkowski space | 1 |
The Penrose transform on flag varieties | 9 |
Homogeneous Vector Bundles on | 23 |
The Weyl Group its Actions and Hasse Diagrams | 37 |
The BottBorelWeil Theorem | 47 |
Realizations of | 58 |
The Penrose Transform in Principle | 74 |
The Penrose Transform in Practice | 94 |
2 | 161 |
4 | 168 |
Conclusions and Outlook | 206 |
| 216 | |
| 218 | |
| 222 | |
| 227 | |
Constructing Unitary Representations | 155 |
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action affine Bernstein–Gelfand–Gelfand resolution BGG resolution chapter cohomology group complex composition series compute conformal conjugate consider construction defined denote dimensions direct images discrete series double fibration dual Dynkin diagram element equations example fibres finite dimensional flag manifolds flag varieties follows g-module geometric Hasse diagram Hermitian symmetric highest weight homogeneous bundles homogeneous sheaves homogeneous vector bundles homomorphism homomorphisms of Verma hypercohomology spectral sequence identified induced integral invariant differential operator isomorphism K-finite K-finite vectors K-types Lemma Lie algebra Lie group line bundle massless fields metric Minkowski space nodes non-standard homomorphisms non-trivial non-zero obtain orbit parabolic Penrose transform projective Remark rest mass fields scalar product sheaf spinors structure subalgebra subgroup subset subvariety surjective tangent bundle theorem twistor space twistor theory twistor transform vanishes vector bundles vector fields Verma modules Ward correspondence Weyl group zero rest mass