Differential operators on the moduli space of G-bundles on algebraic curve and Lie algebra cohomologies

  • B.ย L.ย Feigin
Conference paper


In this paper we discuss the following problem. Let ๐”Š be a semi-simple complex Lie algebra, ๐”Šฬ‚ โ€” the corresponding affine Kac-Moody algebra, K is the central element of ๐”Šฬ‚, U k (๐”Šฬ‚), the universal enveloping algebra of ๐”Šฬ‚ where we suppose, that K is equal to the number k โˆˆ ๐‚ ; U k (๐”Šฬ‚) contains infinite combination of the generators, which are acting in the representations of (๐”Šฬ‚) with highest weight. It was noticed some time ago that there is a remarkable value of k, which is equal to โ€”g, where g is the dual Coxeter number of ๐”Š. For such k the algebra U -g (๐”Šฬ‚) has a large center. Algebra U k (๐”Šฬ‚) is a quantization of the Lie algebra of currents on the circle ๐”Š S , k is the paramenter of quantization. In some sense the current algebra ๐”Š S is a sum of algebras โŠ•ฯ†๐”Š(ฯ†), where ๐”Š(ฯ†) is ๐”Š attached to a point ฯ† โˆˆ S. Each ๐”Š(ฯ†) has a family of Casimir operators, which are destroyed after the quantization. For example, the famous Sugawara construction provides the Virasoro algebra from the quadratic central elements of ๐”Š. But the center appears again for the special value of k. It was proved by Hayashi [5], Malikov [4], and Goodman and Wallach [9].


Modulus Spaceย Spectral Sequenceย Hamiltonian Structureย Projective Connectionย Dual Coxeter Numberย 
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Copyright information

ยฉย Springer-Verlag Tokyoย 1991

Authors and Affiliations

  • B.ย L.ย Feigin
    • 1
  1. 1.Institute for Solid State Phys.Shernogolovka Noginskyi DistrictMoscowUSSR

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