Sheinman Current Algebras on Riemann Surfaces

New Results and Applications
1. Auflage 2012
ISBN: 978-3-11-026452-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 163 Seiten

Reihe: ISSN

This monograph is an introduction into a new and fast developing field on the crossroads of infinite-dimensional Lie algebra theory and contemporary mathematical physics. It contains a self-consistent presentation of the theory of Krichever-Novikov algebras, Lax operator algebras, their interaction, representation theory, relations to moduli spaces of Riemann surfaces and holomorphic vector bundles on them, to Lax integrable systems, and conformal field theory.

For beginners, the book provides a short way to join in the investigations in these fields. For experts, it sums up the recent advances in the theory of almost graded infinite-dimensional Lie algebras and their applications.
The book may serve as a base for semester lecture courses on finite-dimensional integrable systems, conformal field theory, almost graded Lie algebras. Majority of results are presented for the first time in the form of monograph.

Zielgruppe

Researchers, Lecturers, PhD and Graduate Students in Mathematics and Mathematical Physics; Academic Libraries

Autoren/Hrsg.

Sheinman, Oleg K.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

1;1 Krichever-Novikov algebras: basic definitions and structure theory;15
1.1;1.1 Current, vector field, and other Krichever-Novikov algebras;15
1.2;1.2 Meromorphic .-forms and Krichever-Novikov duality;16
1.3;1.3 Krichever-Novikov bases;18
1.4;1.4 Almost-graded structure, triangle decompositions;20
1.5;1.5 Central extensions and 2-cohomology; Virasoro-type algebras;23
1.6;1.6 Affine Krichever-Novikov, in particular Kac-Moody, algebras;27
1.7;1.7 Central extensions of the Lie algebra D1g;29
1.8;1.8 Local cocycles for sl(n) and gl(n);30
2;2 Fermion representations and Sugawara construction;33
2.1;2.1 Admissible representations and holomorphic bundles;33
2.2;2.2 Holomorphic bundles in the Tyurin parametrization;35
2.3;2.3 Krichever-Novikov bases for holomorphic vector bundles;37
2.4;2.4 Fermion representations of affine algebras;40
2.5;2.5 Verma modules for affine algebras;43
2.6;2.6 Fermion representations of Virasoro-type algebras;45
2.7;2.7 Sugawara representation;48
2.8;2.8 Proof of the main theorems for the Sugawara construction;53
2.8.1;2.8.1 Main theorems in the form of relations with structure constants;54
2.8.2;2.8.2 End of the proof of the main theorems;57
3;3 Projective flat connections on the moduli space of punctured Riemann surfaces and the Knizhnik-Zamolodchikov equation;69
3.1;3.1 Virasoro-type algebras and moduli spaces of Riemann surfaces;70
3.2;3.2 Sheaf of conformal blocks and other sheaves on the moduli space M(1,0)g,N+1;76
3.3;3.3 Differentiation of the Krichever-Novikov objects in modular variables;77
3.4;3.4 Projective flat connection and generalized Knizhnik-Zamolodchikov equation;81
3.5;3.5 Explicit form of the Knizhnik-Zamolodchikov equations for genus 0 and genus 1;86
3.5.1;3.5.1 Explicit form of the equations for g = 0;86
3.5.2;3.5.2 Explicit form of the equations for g = 1;90
3.6;3.6 Appendix: the Krichever-Novikov base in the elliptic case;95
4;4 Lax operator algebras;98
4.1;4.1 Lax operators and their Lie bracket;99
4.1.1;4.1.1 Lax operator algebras for gl(n) and sl(n);99
4.1.2;4.1.2 Lax operator algebras for sv(n);100
4.1.3;4.1.3 Lax operator algebras for sp(2n);102
4.2;4.2 Almost-graded structure;104
4.3;4.3 Central extensions of Lax operator algebras: the construction;106
4.4;4.4 Uniqueness theorem;112
5;5 Lax equations on Riemann surfaces, and their hierarchies;115
5.1;5.1 M-operators;117
5.2;5.2 L-operators and Lax operator algebras from M-operators;120
5.3;5.3 g-valued Lax equations;121
5.4;5.4 Hierarchies of commuting flows;125
5.5;5.5 Symplectic structure;127
5.6;5.6 Hamiltonian theory;131
5.7;5.7 Examples: Calogero-Moser systems;138
6;6 Lax integrable systems and conformal field theory;143
6.1;6.1 Conformal field theory related to a Lax integrable system;143
6.2;6.2 From Lax operator algebra to commutative Krichever-Novikov algebra;145
6.3;6.3 The representation of AL;146
6.4;6.4 Sugawara representation;148
6.5;6.5 Conformal blocks and the Knizhnik-Zamolodchikov connection;149
6.6;6.6 The representation of the algebra of Hamiltonian vector fields and commuting Hamiltonians;149
6.7;6.7 Unitarity;150
6.8;6.8 Relation to geometric quantization and quantum integrable systems;152
6.9;6.9 Remark on the Seiberg-Witten theory;152
7;Bibliography;155
8;Notation;161
9;Index;163

Über Autor(innen)

Sheinman, Oleg K.
Oleg K. Sheinman, Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia; Independent University of Moscow, Russia.