Krichever-Novikov Type Algebras (eBook)

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