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On certain current algebras related to finite-zone integration. (English) Zbl 1196.17022
Buchstaber, V. M. (ed.) et al., Geometry, topology, and mathematical physics. S. P. Novikov’s seminar: 2006–2007. Selected papers of the seminar, Moscow, Russia, 2006–2007. Dedicated to S. P. Novikov on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4674-2/hbk). Translations. Series 2. American Mathematical Society 224. Advances in the Mathematical Sciences 61, 271-284 (2008).
Lax operator algebras are infinite-dimensional Lie algebras of current type which are associated to higher genus Riemann surfaces with marked points. They were introduced by Krichever and Sheinman and are examples of almost-graded Lie algebras. By M. Schlichenmaier and O. K. Sheinman in [Russ. Math. Surv. 63, No. 4, 727–766 (2008); translation from Usp. Mat. Nauk 63, No. 4, 131–172 (2008; Zbl 1204.17016)] almost-graded central extensions of these algebras were studied and classified. The article under review gives the definition of the Lax operator algebras associated to the finite-dimensional Lie algebras for $$g$$ one of the algebras $$\text{gl}(n), \text{sl}(n), \text{so}(n), \text{sp}(2n)$$. The classification results of the above mentioned article on almost-graded central extensions are reviewed. As in the classical current algebra case there is essentially only one nontrivial central extension if the finite-dimensional algebra is simple. Lax operator algebras are related to Lax operators on Riemann surfaces. In the article also $$g$$-valued Lax equations and their corresponding phase spaces are discussed. Certain properties are shown. In particular their consistency is checked.
For the entire collection see [Zbl 1143.00004].

##### MSC:
 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 30F30 Differentials on Riemann surfaces 14H10 Families, moduli of curves (algebraic) 14H15 Families, moduli of curves (analytic) 14H55 Riemann surfaces; Weierstrass points; gap sequences 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 17B66 Lie algebras of vector fields and related (super) algebras