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Lax operator algebras. (English. Russian original) Zbl 1160.17017
Funct. Anal. Appl. 41, No. 4, 284-294 (2007); translation from Funkts. Anal. Prilozh. 41, No. 4, 46-59 (2007).
Let \(\Gamma\) be an algebraic curve. The notion of Lax operator on \(\Gamma\) was introduced by I. Krichever [Commun. Math. Phys. 229, No. 2, 229–269 (2002; Zbl 1073.14048)]; this is a \(\mathfrak{gl}(n)\)-valued function \(L\) holomorphic outside outside a fixed finite set of points, where \(L\) has at worst simple poles, satisfying some other conditions. In the present paper, it is observed that the space of Lax operators is an associative algebra. Then orthogonal and symplectic analogs of Lax operators are introduced; namely, these are functions \(L\), with values in \(\mathfrak{so}(n)\), resp. \(\mathfrak{sp}(n)\), holomorphic outside outside a fixed finite set of points and satisfying some conditions. It is proved that the space of orthogonal, resp. symplectic, Lax operators is an almost graded Lie algebras, in the sense of I. M. Krichever and S. P. Novikov [Funct. Anal. Appl. 21, 126–142 (1987); translation from Funkts. Anal. Prilozh. 21, No. 2, 46–63 (1987; Zbl 0634.17010)]. Furthermore, the authors construct case-by-case local central extensions of these Lie algebras, by explicitly describing the corresponding cocycles.

MSC:
17B65 Infinite-dimensional Lie (super)algebras
14H70 Relationships between algebraic curves and integrable systems
14H60 Vector bundles on curves and their moduli
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