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Affine Krichever-Novikov algebras, their representations and applications. (English) Zbl 1081.17014
Buchstaber, V. M. (ed.) et al., Geometry, topology, and mathematical physics. Selected papers from S. P. Novikov’s seminar held in Moscow, Russia, 2002–2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3613-7/hbk). Translations. Series 2. American Mathematical Society. 212. Advances in the Mathematical Sciences 55, 297-316 (2004).
Krichever-Novikov (KN) type algebras are algebras of meromorphic objects on compact Riemann surfaces of arbitrary genus. The meromorphic objects are allowed to have possibly poles at a fixed set of points. By the disjoint splitting of this set into two subsets an almost-grading is introduced. See the review of [O. K. Sheinman, Funct. Anal. Appl. 27, No. 4, 266–272 (1993; Zbl 0820.17036)] for more background information. The most important examples are vector field algebras, differential operator algebras, current algebras, and their corresponding central extensions. For the current algebras the central extensions are the affine KN algebras.
The author gives a review of the current state of the theory of the affine KN algebras, on their invariants and representations, and on some applications of them. After reviewing the basic definitions, results on local central extensions of these algebras (mainly due to the reviewer) are recalled. The representations of the affine KN algebras are discussed in detail. Starting from a certain KN basis of the global meromorphic sections of a holomorphic vector bundle, the author of the article under review constructed representations in some earlier work. This construction is recalled. In this context an important role is played by the higher genus Sugawara construction. It allows to give a relation between tangent directions at the moduli space of compact Riemann surfaces with marked points (represented by certain meromorphic vector fields on the Riemann surface) and operators (the Sugawara operators) on the representation space of the affine KN algebra. This relation is necessary to construct geometric connections.
In the frame of a global operator approach to WZNW model it will be the higher genus Knizhnik-Zamolodchikov connection, see [M. Schlichenmaier and O. K. Sheinman, “Knizhnik-Zamolodchikov equations for positive genus and Krichever-Novikov algebras”, Russ. Math. Surv. 59, 737–770 (2004; Zbl 1066.17014)]. The author presents results on coadjoint orbits of affine KN algebras and discusses relations to the Riemann-Hilbert problem, Tyurin invariants, Casimirs, semi-Casimirs, Hitchin integrals, and the global geometric Langlands correspondence.
For the entire collection see [Zbl 1051.00009].

##### MSC:
 17B65 Infinite-dimensional Lie (super)algebras 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B68 Virasoro and related algebras 22E67 Loop groups and related constructions, group-theoretic treatment 14H60 Vector bundles on curves and their moduli 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 35Q15 Riemann-Hilbert problems in context of PDEs 14H55 Riemann surfaces; Weierstrass points; gap sequences