Sheĭnman, O. K. Affine Lie algebras on Riemann surfaces. (English. Russian original) Zbl 0820.17036 Funct. Anal. Appl. 27, No. 4, 266-272 (1993); translation from Funkts. Anal. Prilozh. 27, No. 4, 54-62 (1993). The theory of highest weight representations for affine Kac-Moody algebras is a well developed subject. These algebras are constructed as follows: Starting from a finite-dimensional simple Lie algebra \(\mathfrak g\) (or from the abelian Lie algebra \(\mathfrak{gl} (1)\)) one takes the ring \(A= \mathbb C [t, t^{-1}]\), constructs the current (or loop) algebra \(\mathfrak g\otimes A\) and takes its nontrivial (one-dimensional) central extension \(\widehat {\mathfrak g}\). For \(\mathfrak{gl} (1)\) one obtains the Heisenberg algebra. If one interprets the algebra \(A\) as the algebra of meromorphic functions on \(\mathbb P_ 1 (\mathbb C)\) (the Riemann surface of genus 0) which are holomorphic outside \(\{0,\infty\}\) it is possible to generalize the situation to higher genus Riemann surfaces \(X\). \(A\) is now the algebra of meromorphic functions on \(X\) which are holomorphic outside two fixed points \(z_ +, z_ -\). In this way one obtains higher genus affine Kac-Moody algebras. This goes back to I. M. Krichever and S. P. Novikov [Funkts. Anal. Prilozh. 21, 46–63 (1987; Zbl 0634.17010); Funkts. Anal. Prilozh. 21, 47–61 (1987; Zbl 0659.17012); Funkts. Anal. Prilozh 23, 24–40 (1989; Zbl 0684.17012)]. For a multi-point generalization, see also the reviewer [Differential operator algebras on compact Riemann surfaces, in: Generalized symmetries in physics, Clausthal, 1993, World Scientific, 425–434 (1994)]. The author of the article under review started the examination of these algebras in the case that the Riemann surface is an elliptic curve (e.g. a one-dimensional complex torus) and 2 points for allowed poles in [Funkts. Anal. Prilozh. 24, 51–61 (1990; Zbl 0715.17023)]. In this paper, he generalizes his approach to higher genus Riemann surfaces. The current algebra and certain central extensions (not necessarily one-dimensional ones) are studied. The current group is introduced and the orbits of the adjoint and coadjoint action are studied. One main tool is the generalization of the approach of Frenkel and Segal relating the orbits of Kac-Moody algebras to monodromy groups of certain differential equations. The author defines the Weyl groups for these higher genus Kac-Moody algebras. He shows that the space of “principal orbits” (orbits corresponding to commutative monodromy groups) of the adjoint representation is in 1-1 correspondence to a quotient of the first homology space of \(X\) with certain points removed (depending on the cocycles considered) with values in the Cartan subalgebra of \({\mathfrak g}\) under this Weyl group. A similar theorem is shown for the coadjoint action. Reviewer: Martin Schlichenmaier (Mannheim) Cited in 7 ReviewsCited in 4 Documents MSC: 17B65 Infinite-dimensional Lie (super)algebras 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties Keywords:Krichever-Novikov algebras; Riemann-Hilbert problem; higher genus Riemann surfaces; current algebra; central extensions; monodromy groups; Weyl groups; higher genus Kac-Moody algebras Citations:Zbl 0634.17010; Zbl 0659.17012; Zbl 0684.17012; Zbl 0715.17023 × Cite Format Result Cite Review PDF References: [1] V. I. Arnold, ”Remarks on perturbation theory for problems of Mathieu type,” Usp. Mat. Nauk,38, No. 4, 189–203 (1983). [2] V. I. Arnold, ”Small denominators I. Mappings of the circumference onto itself,” Trans. Amer. Math. Soc.,46, 213–284 (1965). · Zbl 0152.41905 [3] O. G. Galkin, ”Phase-locking for Mathieu-type vector fields on the torus,” Funkts. Anal. 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