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.

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.

Reviewer: NicolĂˇs Andruskiewitsch (Cordoba)

##### 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 |