Krichever–Novikov Type Algebras: Theory and Applications

Description

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. Preface 1 Some background on Lie algebras 1.1 Basic definitions on Lie algebras 1.2 Subalgebras and ideals 1.3 Lie homomorphism 1.4 Representations and modules 1.5 Simple Lie algebras 1.6 Direct sum and semidirect sum 1.7 Universal enveloping algebras 2 The higher genus algebras 2.1 Riemann surfaces 2.2 Meromorphic forms 2.3 Associative structure 2.4 Lie and Poisson algebra structure 2.5 The vector field algebra and the Lie derivative 2.6 The algebra of differential operators 2.7 Differential operators of all degrees 2.8 Lie superalgebras of half forms 2.8.1 Lie superalgebras 2.8.2 Jordan superalgebras 2.9 Higher genus current algebras 2.10 The generalized Krichever–Novikov situation 2.10.1 The global holomorphic situation 2.10.2 The one-point case 2.10.3 The generalized Krichever–Novikov algebras 2.11 The classical situation 2.11.1 The vector field algebra – the Witt algebra 2.11.2 The function algebra 2.11.3 The differential operator algebra 2.11.4 The Lie superalgebra 2.11.5 Current algebras 3 The almost-grading 3.1 Definition of an almost-graded structure 3.2 Separating cycle and Krichever–Novikov pairing 3.3 The homogeneous subspaces 3.4 The almost-graded structure for the introduced algebras 3.5 Triangular decomposition and filtrations 3.6 Equivalence of filtrations and almost-gradings 3.7 Inverted grading 3.8 The one-point situation 3.9 Level lines 3.10 Delta-distribution 4 Fixing the basis elements 4.1 The Riemann–Roch theorem 4.1.1 The language of divisors 4.1.2 Divisors and line bundles 4.1.3 The theorem 4.2 Choice of a basis for the generic case 4.2.1 Axiomatic characterisation 4.2.2 Realizing all splittings 4.3 The remaining cases 4.3.1 Genus greater or equal to two 4.3.2 Genus one 5 Explicit expressions for a system of generators 5.1 The construction via rational functions in the g = 0 case 5.2 The construction via theta functions and prime forms in the case g ≥ 1 (general case) 5.3 The construction via theta functions and prime forms in the case g ≥ 1 (exceptional cases) 5.4 Half-integer weights 5.5 The construction via the Weierstraß σ-function in the g = 1 case 6 Central extensions of Krichever–Novikov type algebras 6.1 Lie algebra cohomology 6.2 Central extensions and 2-cocycles 6.3 Projective actions and central extensions 6.4 Projective and affine connections 6.4.1 The definitions 6.4.2 Proof of existence of an affine connection 6.5 Geometric cocycles 6.5.1 Geometric cocycles for function algebra 6.5.2 Geometric cocycles for vector field algebra 6.5.3 Geometric cocycles for the differential operator algebra 6.5.4 Special integration curves 6.5.5 Geometric cocycles for the current algebra g 6.6 Uniqueness and classification of central extensions 6.7 The classical situation 6.8 Proofs for the classification results 6.8.1 The function algebra 6.8.2 Vector field algebra 6.8.3 Mixing cocycle for the differential operator algebra 6.9 Central extensions – the supercase 6.9.1 Proof of Theorem 6.91 6.9.2 The case of an odd central element 6.9.3 Examples 6.10 General cohomology of Krichever–Novikov algebras 6.10.1 Universal central extension 6.10.2 The full H2(L, C) 6.10.3 Some remarks on the continuous cohomology H·cont(L, C) 7 Semi-infinite wedge forms and fermionic Fock space representations 7.1 The infinite matrix algebra ḡl̅(∞) 7.1.1 The algebra and its central extension 7.1.2 Semi-infinite wedge representation for gl̅(∞) 7.2 Semi-infinite wedge forms of Krichever–Novikov type elements 7.2.1 Action of differential operators of all degrees 7.2.2 Fine structure of the representation space 7.3 Highest weight representations and Verma modules 7.3.1 Highest weight representations 7.3.2 Verma modules 7.4 Some remarks on the Heisenberg algebra representations 7.5 Left semi-infinite forms 8 b - c systems 8.1 The Clifford algebra like structure 8.2 Operator valued fields in conformal field theory 8.3 b - c fields 8.4 Energy-momentum tensor 8.5 Representation of the Heisenberg algebra via b - c systems 8.6 b - c systems and the algebra ḡl̅(∞) 9 Affine algebras 9.1 Higher genus current algebras 9.2 Central extensions 9.3 Local cocycles 9.4 L-invariant cocycles 9.5 Current algebras of reductive Lie algebras 9.6 Classification results 9.6.1 Cocycles for the simple case 9.6.2 Cocycles for the semisimple case 9.6.3 Cocycles for the abelian case 9.7 Algebras of g̅-valued differential operators 9.7.1 g-valued differential operators 9.7.2 Cocycles 9.7.3 The classification result for reductive Lie algebras 9.7.4 The proof 9.8 Examples: sl(n) and gl(n) 9.8.1 sl(n) 9.8.2 gl(n) 9.9 Verma modules 9.10 Fermionic representations 10 The Sugawara construction 10.1 The classical Sugawara construction 10.2 General Sugawara construction 10.2.1 The reductive case 10.2.2 Almost-graded structure 10.3 Verma module representations 10.4 The proofs 10.4.1 Proof of Proposition 10.24 10.4.2 Proof of Proposition 10.10 10.4.3 The case K > 1 11 Wess–Zumino–Novikov–Witten models and Knizhnik–Zamolodchikov connection 11.1 Moduli space of curves with marked points 11.2 Tangent spaces of the moduli spaces and the Krichever–Novikov vector field algebra 11.3 Sheaf versions of the Krichever–Novikov type algebras 11.4 The Knizhnik–Zamolodchikov connection 11.4.1 Variation of the complex structure 11.4.2 Defining the connection 11.4.3 Knizhnik–Zamolodchikov equations 11.4.4 Example g = 0 11.4.5 Example g = 1 12 Degenerations and deformations 12.1 Deformations of Lie algebras 12.2 Definition of a general deformation of a Lie algebra 12.3 The geometric families in the case of the torus 12.3.1 Complex tori 12.3.2 The family of elliptic curves 12.4 Basis for the meromorphic forms 12.5 Families of algebras 12.5.1 Function algebras 12.5.2 Vector field algebras 12.5.3 The current algebra 12.6 The geometric background of the degenerated cases 12.7 Algebras appearing in the degenerate cases 12.7.1 Witt algebra case 12.7.2 The genus zero and three-point situation 12.7.3 Subalgebras of the classical algebras 13 Lax operator algebras 13.1 Lax operator algebras 13.2 The geometric meaning of the Tyurin parameters 13.3 Module structure of Lax operator algebras 13.3.1 Structure over A 13.3.2 Structure over L 13.3.3 Structure over D1 and the algebra D1g 13.4 Almost-graded central extensions of Lax operator algebras 14 Some related developments 14.1 Vertex algebras 14.2 Other geometric algebras 14.3 Discretized and ??-deformed Krichever–Novikov type algebras 14.4 Genus zero multi-point algebras – integrable systems 14.5 Related works in theoretical physics Bibliography Index