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On quiver varieties. (English) Zbl 0915.17008
Let \(U\) denote the enveloping algebra of the affine Lie algebra with symmetric Cartan matrix corresponding to a finite subgroup \(\Gamma\) in \(SL(T)\). Here \(T\) is a 2-dimensional complex vector space. In this paper the author gives an explicit description of the quiver variety corresponding to this situation. This is a projective algebraic variety (introduced by H. Nakajima, [see Duke J. Math. 76, 365-416 (1994; Zbl 0826.17026) and Duke J. Math. 91, 515-560 (1998)] which plays an important role in the theory of highest weight modules for \(U\). The result is given in terms of a natural bijection between the quiver variety and the variety consisting of certain \(S^\bullet(T) \times\Gamma\)-modules.
The construction is also carried over to the case where \(U\) is replaced by the enveloping algebra of the corresponding finite dimensional simple Lie algebra. In this case it is shown that the quiver variety describes the finite dimensional (highest weight) modules.

MSC:
17B35 Universal enveloping (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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