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, Band 53, 375 Seiten

Reihe: De Gruyter Studies in MathematicsISSN

Autoren/Hrsg.

Schlichenmaier, Martin

Fachgebiete

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

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.