Lax算子代数的几乎分次中心扩张
Almost-graded central extensions of Lax operator algebra
摘要：
Lax算子代数构成一类新的几何起源的无限维李代数。更确切地说，它们是矩阵代数，其元素是紧致黎曼曲面上的亚纯函数。它们推广了经典流代数和Krichever-Novikov型流代数。由Krichever引入的$\gl(n)$的Lax算子，其谱参数在Riemann曲面上.在联合工程的克里切弗和Sheinman他们的代数结构被发现，并扩大到更一般的群体。这些代数是几乎分次的。在这篇文章中，他们的定义回顾和分类和唯一性结果几乎分次中心扩张这一类新的代数。给出了定义上圈的显式形式。如果Lax算子代数所基于的有限维李代数是简单的，那么，直到等价和重标度的中心元素，有一个唯一的非平凡的几乎分次中心扩张。这些结果是与Oleg Sheinman的联合工作。这是在第五届波罗的海-北欧AGMP研讨会上发表的演讲的扩展写作：Bedlewo，2009年10月12日至16日
英文摘要：
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 $\gl(n)$, 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. These results are joint work with Oleg Sheinman. This is an extended write-up of a talk presented at the 5 th Baltic-Nordic AGMP Workshop: Bedlewo, 12-16 October, 2009
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