Lax operator algebras and gradings on semisimple Lie algebras.

*(English. Russian original)*Zbl 1362.17046
Dokl. Math. 91, No. 2, 160-162 (2015); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 461, No. 2, 143-145 (2015).

Author’s introduction: Lax operator algebras (LOAs) were introduced in [I. M. Krichever and the author, Funct. Anal. Appl. 41, No. 4, 284–294 (2007); translation from Funkts. Anal. Prilozh. 41, No. 4, 46–59 (2007; Zbl 1160.17017)]. The most complete existing exposition of the theory and applications of LOAs for the case of classical root systems, as well as bibliography, is contained in the author’s monograph [Current algebras on Riemann surfaces. New results and applications. Berlin: de Gruyter (2012; Zbl 1258.81002)]. In the author’s paper [Dokl. Math. 89, No. 2, 151–153 (2014); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 455, No. 1, 23–25 (2014; Zbl 1360.17029)] (an extended version appeared in [Transl. Ser. 2. Am. Math. Soc. 234. Adv. Math. Sci. 67, 373–392 (2014; Zbl 1360.17030)]), LOAs for singular root systems of type \(G_2\) were constructed. Lax operator algebras are classified with almost graded current algebras on pointed Riemann surfaces. As such, they generalize loop algebras and the Krichever-Novikov affine algebras. Lax operator algebras are closely related to the theory of integrable systems.

In this paper, we propose a general construction of LOA for all finite irreducible reduced root systems. Thereby, we solve the problem posed by the author in [Dokl. etc. (loc. cit.)] and in earlier works. The new construction makes it possible to give a unified proof of the presence of an almost graded structure and central extensions for each LOA and, moreover, construct and classify the latter. It should be mentioned that, up to now, these problems were solved for each type of LOAs separately; moreover, for the root systems \(F_4, E_6, E_7\), and \(E_8\), even an LOA construction did not exist. It seems important that the previous approach to the study of LOAs was based on the Tyurin parameters of holomorphic bundles on Riemann surfaces, while the approach presented here is based on the structure theory of semisimple Lie algebras. This suggests the existence of a fundamental relationship between these theories.

In this paper, we propose a general construction of LOA for all finite irreducible reduced root systems. Thereby, we solve the problem posed by the author in [Dokl. etc. (loc. cit.)] and in earlier works. The new construction makes it possible to give a unified proof of the presence of an almost graded structure and central extensions for each LOA and, moreover, construct and classify the latter. It should be mentioned that, up to now, these problems were solved for each type of LOAs separately; moreover, for the root systems \(F_4, E_6, E_7\), and \(E_8\), even an LOA construction did not exist. It seems important that the previous approach to the study of LOAs was based on the Tyurin parameters of holomorphic bundles on Riemann surfaces, while the approach presented here is based on the structure theory of semisimple Lie algebras. This suggests the existence of a fundamental relationship between these theories.

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\textit{O. K. Sheinman}, Dokl. Math. 91, No. 2, 160--162 (2015; Zbl 1362.17046); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 461, No. 2, 143--145 (2015)

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##### References:

[1] | Krichever, I M, No article title, Commun. Math. Phys., 229, 229-269, (2002) · Zbl 1073.14048 |

[2] | Krichever, I M; Novikov, S P, No article title, Funct. Anal. Appl., 21, 126-142, (1987) · Zbl 0634.17010 |

[3] | Krichever, I M; Sheinman, O K, No article title, Funct. Anal. Appl., 41, 284-294, (2007) · Zbl 1160.17017 |

[4] | Schlichenmaier, M, No article title, Lett. Math. Phys., 19, 151-165, (1990) · Zbl 0691.30037 |

[5] | Schlichenmaier, M, No article title, Lett. Math. Phys., 19, 327-336, (1990) · Zbl 0703.30038 |

[6] | Schlichenmaier, M, No article title, Sb. Math., 205, 722-762, (2014) · Zbl 1361.17021 |

[7] | Schlichenmaier, M; Sheinman, O K, No article title, Russ. Math. Surveys, 63, 727-766, (2008) · Zbl 1204.17016 |

[8] | O. K. Sheinman, Current Algebras on Riemann Surfaces (Walter de Gruyter, Berlin, 2012). · Zbl 1258.81002 |

[9] | Sheinman, O K, No article title, Dokl. Math., 89, 151-153, (2014) · Zbl 1360.17029 |

[10] | O. K. Sheinman, ArXiv:1406.5017. |

[11] | Vinberg, E B; Gorbatsevich, V V; Onishchik, A L, No article title, Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat. Fundam. Napravleniya, 41, 5-258, (1990) |

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