Schlichenmaier Krichever–Novikov Type Algebras

Theory and Applications
1. Auflage 2014
ISBN: 978-3-11-027964-1
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 375 Seiten

Reihe: ISSN

Krichever and Novikov introduced certain classes of infinite dimensional Lie algebras to extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them to a more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are still manageable.

This book gives an introduction for the newcomer to this exciting field of ongoing research in mathematics and will be a valuable source of reference for the experienced researcher. Beside the basic constructions and results also applications are presented.

Zielgruppe

Graduate and PhD students, researchers in Lie algebras, algebraic

Autoren/Hrsg.

Schlichenmaier, Martin

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

1;Preface;5
2;1 Some background on Lie algebras;17
2.1;1.1 Basic definitions on Lie algebras;17
2.2;1.2 Subalgebras and ideals;18
2.3;1.3 Lie homomorphism;19
2.4;1.4 Representations and modules;19
2.5;1.5 Simple Lie algebras;20
2.6;1.6 Direct sum and semidirect sum;21
2.7;1.7 Universal enveloping algebras;22
3;2 The higher genus algebras;24
3.1;2.1 Riemann surfaces;24
3.2;2.2 Meromorphic forms;25
3.3;2.3 Associative structure;28
3.4;2.4 Lie and Poisson algebra structure;28
3.5;2.5 The vector field algebra and the Lie derivative;30
3.6;2.6 The algebra of differential operators;31
3.7;2.7 Differential operators of all degrees;32
3.8;2.8 Lie superalgebras of half forms;33
3.8.1;2.8.1 Lie superalgebras;33
3.8.2;2.8.2 Jordan superalgebras;35
3.9;2.9 Higher genus current algebras;36
3.10;2.10 The generalized Krichever–Novikov situation;37
3.10.1;2.10.1 The global holomorphic situation;37
3.10.2;2.10.2 The one-point case;38
3.10.3;2.10.3 The generalized Krichever–Novikov algebras;38
3.11;2.11 The classical situation;38
3.11.1;2.11.1 The vector field algebra – the Witt algebra;41
3.11.2;2.11.2 The function algebra;42
3.11.3;2.11.3 The differential operator algebra;42
3.11.4;2.11.4 The Lie superalgebra;42
3.11.5;2.11.5 Current algebras;43
4;3 The almost-grading;44
4.1;3.1 Definition of an almost-graded structure;45
4.2;3.2 Separating cycle and Krichever–Novikov pairing;46
4.3;3.3 The homogeneous subspaces;47
4.4;3.4 The almost-graded structure for the introduced algebras;50
4.5;3.5 Triangular decomposition and filtrations;54
4.6;3.6 Equivalence of filtrations and almost-gradings;56
4.7;3.7 Inverted grading;57
4.8;3.8 The one-point situation;57
4.9;3.9 Level lines;58
4.10;3.10 Delta-distribution;62
5;4 Fixing the basis elements;65
5.1;4.1 The Riemann–Roch theorem;65
5.1.1;4.1.1 The language of divisors;65
5.1.2;4.1.2 Divisors and line bundles;66
5.1.3;4.1.3 The theorem;67
5.2;4.2 Choice of a basis for the generic case;73
5.2.1;4.2.1 Axiomatic characterisation;73
5.2.2;4.2.2 Realizing all splittings;79
5.3;4.3 The remaining cases;81
5.3.1;4.3.1 Genus greater or equal to two;82
5.3.2;4.3.2 Genus one;85
6;5 Explicit expressions for a system of generators;86
6.1;5.1 The construction via rational functions in the g = 0 case;88
6.2;5.2 The construction via theta functions and prime forms in the case g = 1 (general case);89
6.3;5.3 The construction via theta functions and prime forms in the case g = 1 (exceptional cases);96
6.4;5.4 Half-integer weights;98
6.5;5.5 The construction via the Weierstraß s-function in the g = 1 case;100
7;6 Central extensions of Krichever–Novikov type algebras;103
7.1;6.1 Lie algebra cohomology;103
7.2;6.2 Central extensions and 2-cocycles;105
7.3;6.3 Projective actions and central extensions;109
7.4;6.4 Projective and affine connections;111
7.4.1;6.4.1 The definitions;112
7.4.2;6.4.2 Proof of existence of an affine connection;113
7.5;6.5 Geometric cocycles;115
7.5.1;6.5.1 Geometric cocycles for function algebra;118
7.5.2;6.5.2 Geometric cocycles for vector field algebra;119
7.5.3;6.5.3 Geometric cocycles for the differential operator algebra;122
7.5.4;6.5.4 Special integration curves;125
7.5.5;6.5.5 Geometric cocycles for the current algebra g;126
7.6;6.6 Uniqueness and classification of central extensions;127
7.7;6.7 The classical situation;135
7.8;6.8 Proofs for the classification results;138
7.8.1;6.8.1 The function algebra;139
7.8.2;6.8.2 Vector field algebra;145
7.8.3;6.8.3 Mixing cocycle for the differential operator algebra;153
7.9;6.9 Central extensions – the supercase;158
7.9.1;6.9.1 Proof of Theorem 6.91;164
7.9.2;6.9.2 The case of an odd central element;165
7.9.3;6.9.3 Examples;166
7.10;6.10 General cohomology of Krichever–Novikov algebras;167
7.10.1;6.10.1 Universal central extension;168
7.10.2;6.10.2 The full H2(L, C);170
7.10.3;6.10.3 Some remarks on the continuous cohomology H·cont(L, C);171
8;7 Semi-infinite wedge forms and fermionic Fock space representations;173
8.1;7.1 The infinite matrix algebra g¯l¯(8);174
8.1.1;7.1.1 The algebra and its central extension;174
8.1.2;7.1.2 Semi-infinite wedge representation for gl¯(8);178
8.2;7.2 Semi-infinite wedge forms of Krichever–Novikov type elements;184
8.2.1;7.2.1 Action of differential operators of all degrees;190
8.2.2;7.2.2 Fine structure of the representation space;191
8.3;7.3 Highest weight representations and Verma modules;195
8.3.1;7.3.1 Highest weight representations;195
8.3.2;7.3.2 Verma modules;197
8.4;7.4 Some remarks on the Heisenberg algebra representations;200
8.5;7.5 Left semi-infinite forms;202
9;8 b - c systems;205
9.1;8.1 The Clifford algebra like structure;205
9.2;8.2 Operator valued fields in conformal field theory;210
9.3;8.3 b - c fields;215
9.4;8.4 Energy-momentum tensor;216
9.5;8.5 Representation of the Heisenberg algebra via b - c systems;223
9.6;8.6 b - c systems and the algebra g¯l¯(8);225
10;9 Affine algebras;228
10.1;9.1 Higher genus current algebras;228
10.2;9.2 Central extensions;229
10.3;9.3 Local cocycles;230
10.4;9.4 L-invariant cocycles;233
10.5;9.5 Current algebras of reductive Lie algebras;234
10.6;9.6 Classification results;237
10.6.1;9.6.1 Cocycles for the simple case;238
10.6.2;9.6.2 Cocycles for the semisimple case;239
10.6.3;9.6.3 Cocycles for the abelian case;240
10.7;9.7 Algebras of g¯-valued differential operators;242
10.7.1;9.7.1 g-valued differential operators;242
10.7.2;9.7.2 Cocycles;243
10.7.3;9.7.3 The classification result for reductive Lie algebras;245
10.7.4;9.7.4 The proof;246
10.8;9.8 Examples: sl(n) and gl(n);250
10.8.1;9.8.1 sl(n);250
10.8.2;9.8.2 gl(n);251
10.9;9.9 Verma modules;252
10.10;9.10 Fermionic representations;257
11;10 The Sugawara construction;263
11.1;10.1 The classical Sugawara construction;263
11.2;10.2 General Sugawara construction;265
11.2.1;10.2.1 The reductive case;272
11.2.2;10.2.2 Almost-graded structure;274
11.3;10.3 Verma module representations;275
11.4;10.4 The proofs;277
11.4.1;10.4.1 Proof of Proposition 10.24;280
11.4.2;10.4.2 Proof of Proposition 10.10;285
11.4.3;10.4.3 The case K > 1;290
12;11 Wess–Zumino–Novikov–Witten models and Knizhnik–Zamolodchikov connection;291
12.1;11.1 Moduli space of curves with marked points;292
12.2;11.2 Tangent spaces of the moduli spaces and the Krichever–Novikov vector field algebra;297
12.3;11.3 Sheaf versions of the Krichever–Novikov type algebras;300
12.4;11.4 The Knizhnik–Zamolodchikov connection;305
12.4.1;11.4.1 Variation of the complex structure;305
12.4.2;11.4.2 Defining the connection;310
12.4.3;11.4.3 Knizhnik–Zamolodchikov equations;313
12.4.4;11.4.4 Example g = 0;314
12.4.5;11.4.5 Example g = 1;316
13;12 Degenerations and deformations;319
13.1;12.1 Deformations of Lie algebras;320
13.2;12.2 Definition of a general deformation of a Lie algebra;324
13.3;12.3 The geometric families in the case of the torus;325
13.3.1;12.3.1 Complex tori;325
13.3.2;12.3.2 The family of elliptic curves;326
13.4;12.4 Basis for the meromorphic forms;329
13.5;12.5 Families of algebras;330
13.5.1;12.5.1 Function algebras;330
13.5.2;12.5.2 Vector field algebras;331
13.5.3;12.5.3 The current algebra;333
13.6;12.6 The geometric background of the degenerated cases;334
13.7;12.7 Algebras appearing in the degenerate cases;336
13.7.1;12.7.1 Witt algebra case;336
13.7.2;12.7.2 The genus zero and three-point situation;336
13.7.3;12.7.3 Subalgebras of the classical algebras;338
14;13 Lax operator algebras;340
14.1;13.1 Lax operator algebras;340
14.2;13.2 The geometric meaning of the Tyurin parameters;345
14.3;13.3 Module structure of Lax operator algebras;347
14.3.1;13.3.1 Structure over A;347
14.3.2;13.3.2 Structure over L;347
14.3.3;13.3.3 Structure over D1 and the algebra D1g;349
14.4;13.4 Almost-graded central extensions of Lax operator algebras;350
15;14 Some related developments;356
15.1;14.1 Vertex algebras;356
15.2;14.2 Other geometric algebras;357
15.3;14.3 Discretized and ??-deformed Krichever–Novikov type algebras;357
15.4;14.4 Genus zero multi-point algebras – integrable systems;358
15.5;14.5 Related works in theoretical physics;358
16;Bibliography;361
17;Index;373

Über Autor(innen)

Martin Schlichenmaier, University of Luxembourg, Luxembourg.