Preface ......................................................... v
1 Some background on Lie algebras .............................. 1
1.1 Basic definitions on Lie algebras ....................... 1
1.2 Subalgebras and ideals .................................. 2
1.3 Lie homomorphism ........................................ 3
1.4 Representations and modules ............................. 3
1.5 Simple Lie algebras ..................................... 4
1.6 Direct sum and semidirect sum ........................... 5
1.7 Universal enveloping algebras ........................... 6
2 The higher genus algebras .................................... 8
2.1 Riemann surfaces ........................................ 8
2.2 Meromorphic forms ....................................... 9
2.3 Associative structure .................................. 12
2.4 Lie and Poisson algebra structure ...................... 12
2.5 The vector field algebra and the Lie derivative ........ 14
2.6 The algebra of differential operators .................. 15
2.7 Differential operators of all degrees .................. 16
2.8 Lie superalgebras of half forms ........................ 17
2.8.1 Lie superalgebras ............................... 17
2.8.2 Jordan superalgebras ............................ 19
2.9 Higher genus current algebras .......................... 20
2.10 The generalized Krichever-Novikov situation ............ 21
2.10.1 The global holomorphic situation ................ 21
2.10.2 The one-point case .............................. 22
2.10.3 The generalized Krichever-Novikov algebras ...... 22
2.11 The classical situation ................................ 22
2.11.1 The vector field algebra - the Witt algebra ..... 25
2.11.2 The function algebra ............................ 26
2.11.3 The differential operator algebra ............... 26
2.11.4 The Lie superalgebra ............................ 26
2.11.5 Current algebras ................................ 27
3 The almost-grading .......................................... 28
3.1 Definition of an almost-graded structure ............... 29
3.2 Separating cycle and Krichever-Novikov pairing ......... 30
3.3 The homogeneous subspaces .............................. 31
3.4 The almost-graded structure for the introduced
algebras ............................................... 34
3.5 Triangular decomposition and filtrations ............... 38
3.6 Equivalence of filtrations and almost-gradings ......... 40
3.7 Inverted grading ....................................... 41
3.8 The one-point situation ................................ 41
3.9 Level lines ............................................ 42
3.10 Delta-distribution ..................................... 46
4 Fixing the basis elements ................................... 49
4.1 The Riemann-Roch theorem ............................... 49
4.1.1 The language of divisors ........................ 49
4.1.2 Divisors and line bundles ....................... 50
4.1.3 The theorem ..................................... 51
4.2 Choice of a basis for the generic case ................. 57
4.2.1 Axiomatic characterisation ...................... 57
4.2.2 Realizing all splittings ........................ 63
4.3 The remaining cases .................................... 65
4.3.1 Genus greater or equal to two ................... 66
4.3.2 Genus one ....................................... 69
5 Explicit expressions for a system of generators ............. 70
5.1 The construction via rational functions in the g =
0 case ................................................. 72
5.2 The construction via theta functions and prime forms
in the case g ≥ 1 (general case) ....................... 73
5.3 The construction via theta functions and prime forms
in the case g ≥ 1 (exceptional cases) .................. 80
5.4 Half-integer weights ................................... 82
5.5 The construction via the Weierstrafi σ-function in
the g = 1 case ......................................... 84
6 Central extensions of Krichever-Novikov type algebras ....... 87
6.1 Lie algebra cohomology ................................. 87
6.2 Central extensions and 2-cocycles ...................... 89
6.3 Projective actions and central extensions .............. 93
6.4 Projective and affine connections ...................... 95
6.4.1 The definitions ................................. 96
6.4.2 Proof of existence of an affine connection ...... 97
6.5 Geometric cocycles ..................................... 99
6.5.1 Geometric cocycles for function algebra ........ 102
6.5.2 Geometric cocycles for vector field algebra .... 103
6.5.3 Geometric cocycles for the differential
operator algebra ............................... 106
6.5.4 Special integration curves ..................... 109
6.5.5 Geometric cocycles for the current algebra  ... 110
6.6 Uniqueness and classification of central extensions ... 111
6.7 The classical situation ............................... 119
6.8 Proofs for the classification results ................. 122
6.8.1 The function algebra ........................... 123
6.8.2 Vector field algebra ........................... 129
6.8.3 Mixing cocycle for the differential operator
algebra ........................................ 137
6.9 Central extensions - the supercase .................... 142
6.9.1 Proof of Theorem 6.91 .......................... 148
6.9.2 The case of an odd central element ............. 149
6.9.3 Examples ....................................... 150
6.10 General cohomology of Krichever-Novikov algebras ...... 151
6.10.1 Universal central extension .................... 152
6.10.2 The full h2( , ) ............................... 154
6.10.3 Some remarks on the continuous cohomology
H*cont(,) .................................... 155
7 Semi-infinite wedge forms and fermionic Fock space
representations ............................................ 157
7.1 The infinite matrix algebra  (∞) ...................... 158
7.1.1 The algebra and its central extension .......... 158
7.1.2 Semi-infinite wedge representation for  (∞) .... 162
7.2 Semi-infinite wedge forms of Krichever-Novikov type
elements .............................................. 168
7.2.1 Action of differential operators of all
degrees ........................................ 174
7.2.2 Fine structure of the representation space ..... 175
7.3 Highest weight representations and Verma modules ...... 179
7.3.1 Highest weight representations ................. 179
7.3.2 Verma modules .................................. 181
7.4 Some remarks on the Heisenberg algebra
representations ....................................... 184
7.5 Left semi-infinite forms .............................. 186
8 b-c systems ................................................ 189
8.1 The Clifford algebra like structure ................... 189
8.2 Operator valued fields in conformal field theory ...... 194
8.3 b-c fields ............................................ 199
8.4 Energy-momentum tensor ................................ 200
8.5 Representation of the Heisenberg algebra via b-c
systems ............................................... 207
8.6 b-c systems and the algebra (∞) ..................... 209
9 Affine algebras ............................................ 212
9.1 Higher genus current algebras ......................... 212
9.2 Central extensions .................................... 213
9.3 Local cocycles ........................................ 214
9.4 L-invariant cocycles .................................. 217
9.5 Current algebras of reductive Lie algebras ............ 218
9.6 Classification results ................................ 221
9.6.1 Cocycles for the simple case ................... 222
9.6.2 Cocycles for the semisimple case ............... 223
9.6.3 Cocycles for the abelian case .................. 224
9.7 Algebras of g-valued differential operators ........... 226
9.7.1 g-valued differential operators ................ 226
9.7.2 Cocycles ....................................... 227
9.7.3 The classification result for reductive Lie
algebras ....................................... 229
9.7.4 The proof ...................................... 230
9.8 Examples: sl(n) and gl(n) ............................. 234
9.8.1 sl(n) .......................................... 234
9.8.2 gl(n) .......................................... 235
9.9 Verma modules ......................................... 236
9.10 Fermionic representations ............................. 241
10 The Sugawara construction .................................. 247
10.1 The classical Sugawara construction ................... 247
10.2 General Sugawara construction ......................... 249
10.2.1 The reductive case ............................. 256
10.2.2 Almost-graded structure ........................ 258
10.3 Verma module representations .......................... 259
10.4 The proofs ............................................ 261
10.4.1 Proof of Proposition 10.24 ..................... 264
10.4.2 Proof of Proposition 10.10 ..................... 269
10.4.3 The case K > l ................................. 274
11 Wess-Zumino-Novikov-Witten models and Knizhnik-
Zamolodchikov connection ................................... 275
11.1 IVIoduli space of curves with marked points ........... 276
11.2 Tangent spaces of the moduli spaces and the
Krichever-Novikov vector field algebra ................ 281
11.3 Sheaf versions of the Krichever-Novikov type
algebras .............................................. 284
11.4 The Knizhnik-Zamolodchikov connection ................. 289
11.4.1 Variation of the complex structure ............. 289
11.4.2 Defining the connection ........................ 294
11.4.3 Knizhnik-Zamolodchikov equations ............... 297
11.4.4 Example g = 0 .................................. 298
11.4.5 Example g = l .................................. 300
12 Degenerations and deformations ............................. 303
12.1 Deformations of Lie algebras .......................... 304
12.2 Definition of a general deformation of a Lie algebra .. 308
12.3 The geometric families in the case of the torus ....... 309
12.3.1 Complex tori ................................... 309
12.3.2 The family of elliptic curves .................. 310
12.4 Basis for the meromorphic forms ....................... 313
12.5 Families of algebras .................................. 314
12.5.1 Function algebras .............................. 314
12.5.2 Vector held algebras ........................... 315
12.5.3 The current algebra ............................ 317
12.6 The geometric background of the degenerated cases ..... 318
12.7 Algebras appearing in the degenerate cases ............ 320
12.7.1 Witt algebra case .............................. 320
12.7.2 The genus zero and three-point situation ....... 320
12.7.3 Subalgebras of the classical algebras .......... 322
13 Lax operator algebras ...................................... 324
13.1 Lax operator algebras ................................. 324
13.2 The geometric meaning of the Tyurin parameters ........ 329
13.3 Module structure of Lax operator algebras ............. 331
13.3.1 Structure over  ............................... 331
13.3.2 Structure over ............................... 331
13.3.3 Structure over  1 and the algebra 1g .......... 333
13.4 Almost-graded central extensions of Lax operator
algebras .............................................. 334
14 Some related developments .................................. 340
14.1 Vertex algebras ....................................... 340
14.2 Other geometric algebras .............................. 341
14.3 Discretized and q-deformed Krichever-Novikov type
algebras .............................................. 341
14.4 Genus zero multi-point algebras - integrable systems .. 342
14.5 Related works in theoretical physics .................. 342
Bibliography .................................................. 345
Index ......................................................... 357
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