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Central extensions of Lax operator algebras. (English. Russian original) Zbl 1204.17016
Russ. Math. Surv. 63, No. 4, 727-766 (2008); translation from Usp. Mat. Nauk 63, No. 4, 131-172 (2008).
The main result of this paper provides the classification of almost-graded central extensions of two-point Lax current algebras. Let $$\Sigma$$ be a compact Riemann surface of genus $$g$$, $$P_{\pm}$$ two points in $$\Sigma$$, $$n \in \mathbb N$$; fix a collection $$T = (\gamma_s, \alpha_s)_{1\leq s \leq ng}$$, where the $$\gamma_s\in \Sigma - \{P_+, P_-\}$$ are distinct and $$\alpha_s\in \mathbb C^n$$. Let also $$\mathfrak g$$ be either $${\mathfrak{gl}}(n, \mathbb C)$$, $${\mathfrak{sl}}(n, \mathbb C)$$, $${\mathfrak{so}}(n, \mathbb C)$$, $${\mathfrak{sp}}(n, \mathbb C)$$ or $${\mathfrak{s}}(n, \mathbb C)$$ (the algebra of scalar matrices inside $${\mathfrak{gl}}(n, \mathbb C)$$). The Lax current algebra $$\overline{\mathfrak g}$$ associated to this datum $$T$$ consists of the functions $$L: \Sigma \to \mathfrak g$$ that are holomorphic outside $$\{\gamma_1, \dots, \gamma_s, P_+, P_-\}$$ and have poles of at most some prescribed order in $$\{\gamma_1, \dots, \gamma_s\}$$ (that depends on $$\mathfrak g$$) and satisfy some other technical conditions.
These algebras, and their multi-point analogs, were introduced in [I. M. Krichever and O. K. Sheinman, Funct. Anal. Appl. 41, No. 4, 284–294 (2007); translation from Funkts. Anal. Prilozh. 41, No. 4, 46–59 (2007; Zbl 1160.17017)], generalizing previous work in [I. Krichever, Commun. Math. Phys. 229, No. 2, 229–269 (2002; Zbl 1073.14048)].
The algebras $$\overline{\mathfrak g}$$ are almost-graded, meaning that they bear a graded decomposition of the underlying vector space: $$\overline{\mathfrak g} = \bigoplus_{i\in \mathbb Z}\overline{\mathfrak g}_i$$ where $$\dim \overline{\mathfrak g}_i < \infty$$ and $$[\overline{\mathfrak g}_i, \overline{\mathfrak g}_j] \subset \bigoplus_{i+j - k_0 \leq k \leq i+j + k_0 }\overline{\mathfrak g}_k$$, where $$k_0$$ does not depend on $$i$$, $$j$$. A central extension corresponding to a cocycle $$\gamma$$ is almost-graded if there exists $$K\in\mathbb Z$$ such that $$\gamma(\overline{\mathfrak g}_i, \overline{\mathfrak g}_j) = 0$$ whenever $$| i+j| > K$$. Concretely, the authors show that there is essentially one non-trivial almost-graded central extension of $$\overline{\mathfrak g}$$ when $$\mathfrak g$$ is simple, and get similar results in the other cases.

##### MSC:
 17B65 Infinite-dimensional Lie (super)algebras 17B80 Applications of Lie algebras and superalgebras to integrable systems 14H70 Relationships between algebraic curves and integrable systems 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
##### Keywords:
Lax operator algebras; central extensions
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