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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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##### References:
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