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Quiver varieties and Kac-Moody algebras. (English) Zbl 0970.17017
The author intends to characterize the modified enveloping algebra of Kac-Moody algebras geometrically. At first he changes the definition of a quiver variety given by himself in an earlier paper. The new definition makes sense over algebraically closed fields of positive characteristics too, though in this paper the essential considerations need the field of complex numbers. Considering Lagrangian subvarieties of products of quiver varieties called Hecke correspondences, generators of Kac-Moody algebras are obtained. Following ideas of Ginzburg a convolution product of Lagrangian subvarieties of products of quiver varieties is defined and once more following Ginzburg, a relation between the convolution algebra and the modified enveloping algebra of Kac-Moody algebras is stated. Finally, natural modules of the convolution algebra are identified with integrable highest weight representations of Kac-Moody algebras and the intersection form of the quiver varieties is identified with the invariant inner product on the highest weight representations.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B35 Universal enveloping (super)algebras
53D12 Lagrangian submanifolds; Maslov index
14R20 Group actions on affine varieties
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