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Almost-graded central extensions of Lax operator algebras

Tom 93 / 2011

Martin Schlichenmaier Banach Center Publications 93 (2011), 129-144 MSC: Primary 17B65; Secondary 17B67, 17B80, 14D20, 14H55, 14H60, 14H70, 30F30, 81R10, 81T40. DOI: 10.4064/bc93-0-11

Streszczenie

Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for $\def\gl{\mathfrak{gl}}\gl(n)$, with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and extended to more general groups. These algebras are almost-graded. In this article their definition is recalled and classification and uniqueness results for almost-graded central extensions for this new class of algebras are presented. The explicit forms of the defining cocycles are given. If the finite-dimensional Lie algebra on which the Lax operator algebra is based is simple then, up to equivalence and rescaling of the central element, there is a unique non-trivial almost-graded central extension. Some results are joint work with Oleg Sheinman.

Autorzy

  • Martin SchlichenmaierMathematics Research Unit, FSTC,
    University of Luxembourg
    6, rue Coudenhove-Kalergi
    L-1359 Luxembourg-Kirchberg
    e-mail

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