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Lax operator algebras. (English. Russian original) Zbl 1160.17017
Funct. Anal. Appl. 41, No. 4, 284-294 (2007); translation from Funkts. Anal. Prilozh. 41, No. 4, 46-59 (2007).
Let \(\Gamma\) be an algebraic curve. The notion of Lax operator on \(\Gamma\) was introduced by I. Krichever [Commun. Math. Phys. 229, No. 2, 229–269 (2002; Zbl 1073.14048)]; this is a \(\mathfrak{gl}(n)\)-valued function \(L\) holomorphic outside outside a fixed finite set of points, where \(L\) has at worst simple poles, satisfying some other conditions. In the present paper, it is observed that the space of Lax operators is an associative algebra. Then orthogonal and symplectic analogs of Lax operators are introduced; namely, these are functions \(L\), with values in \(\mathfrak{so}(n)\), resp. \(\mathfrak{sp}(n)\), holomorphic outside outside a fixed finite set of points and satisfying some conditions. It is proved that the space of orthogonal, resp. symplectic, Lax operators is an almost graded Lie algebras, in the sense of I. M. Krichever and S. P. Novikov [Funct. Anal. Appl. 21, 126–142 (1987); translation from Funkts. Anal. Prilozh. 21, No. 2, 46–63 (1987; Zbl 0634.17010)]. Furthermore, the authors construct case-by-case local central extensions of these Lie algebras, by explicitly describing the corresponding cocycles.

17B65 Infinite-dimensional Lie (super)algebras
14H70 Relationships between algebraic curves and integrable systems
14H60 Vector bundles on curves and their moduli
Full Text: DOI
[1] I. M. Krichever, ”Vector bundles and Lax equations on algebraic curves,” Comm. Math. Phys., 229:2 (2002), 229–269; http://arxiv.org/abs/hep-th/0108110 . · Zbl 1073.14048 · doi:10.1007/s002200200659
[2] I. M. Krichever, ”Isomonodromy equations on algebraic curves, canonical transformations and Witham equations,” Mosc. Math. J., 2:4 (2002), 717–752, 806; http://arxiv.org/abs/hep-th/0112096 . · Zbl 1044.70010
[3] I. M. Krichever and S. P. Novikov, ”Algebras ofVirasoro type, Riemann surfaces and structures of the theory of solitons,” Funkts. Anal. Prilozhen., 21:2 (1987), 46–63. · Zbl 0634.17010
[4] I. M. Krichever and S. P. Novikov, ”Virasoro type algebras, Riemann surfaces and strings in Minkowski space,” Funkts. Anal. Prilozhen., 21:4 (1987), 47–61. · Zbl 0659.17012
[5] I. M. Krichever and S. P. Novikov, ”Algebras of Virasoro type, energy-momentum tensors and decompositions ofoperators on Riemann surfaces,” Funkts. Anal. Prilozhen., 23:1 (1989), 46–63. · Zbl 0695.35171 · doi:10.1007/BF01078572
[6] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles and commuting difference operators. Two-point constructions,” Uspekhi Mat. Nauk, 55:4 (2000), 181–182. · Zbl 0978.35066
[7] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles on Riemann surfaces and Kadomtsev-Petviashvili equation (KP). I,” Funkts. Anal. Prilozhen., 12:4 (1978), 41–52. · Zbl 0393.35061
[8] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles on algebraic curves and nonlinear equations,” Uspekhi Math. Nauk, 35:6 (1980), 47–68. · Zbl 0501.35071
[9] M. Schlichenmaier, ”Local cocycles and central extensions for multi-point algebras of Krichever-Novikov type,” J. Reine Angew. Math., 559 (2003), 53–94. · Zbl 1124.17305 · doi:10.1515/crll.2003.052
[10] M. Schlichenmaier, ”Higher genus affine algebras of Krichever-Novikov type,” Moscow Math. J., 3:4 (2003), 1395–1427; http://arxiv.org/abs/math/0210360 . · Zbl 1115.17010
[11] O. K. Sheinman, ”Affine Krichever-Novikov algebras, their representations and applications,” in: Geometry, Topology and Mathematical Physics. S. P. Novikov’s Seminar 2002–2003, Amer. Soc. Transl. (2), vol. 212 (eds. V. M. Buchstaber, I. M. Krichever), Amer. Math. Soc., Providence, R.I., 2004, 297–316; http://arxiv.org/abs/Math.RT/0304020 . · Zbl 1081.17014
[12] A. N. Tyurin, ”The classification of vector bundles over an algebraic curve of arbitrary genus,” Izv. Akad. Nauk SSSR Ser. Mat., 29 (1965), 657–688. · Zbl 0207.51603
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