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Krichever-Novikov type algebras : theory and applications
Martin Schlichenmaier 1952- author.
2014
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Title:
Krichever-Novikov type algebras : theory and applications
Author:
Martin Schlichenmaier 1952- author.
Description:
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 superalgebras2.8.2 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 subspaces3.4 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
6.4 Projective and affine connections6.4.1 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
Krichever and Novikov introduced certain classes of infinite dimensionalLie algebrasto extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them toa 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 origi
Publication Date:
2014
Publisher:
Berlin ; Boston : De Gruyter
Format:
1 online resource (378 p.).
Identifier:
ISBN 3-11-038147-8
Related Titles:
Series: De Gruyter studies in mathematics ; volume 53.
Subjects:
Infinite dimensional Lie algebras
;
Electronic books
Language:
English
Source:
01DAL UDM ALMA
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