The term quiver variety refers to a number of algebraic varieties constructed as moduli spaces of quiver representations.
The most important are the Nakajima quiver varieties (Nakajima 98), which give the moduli of stable representations of preprojective algebras?. These are notable for allowing a geometrical construction of the universal enveloping algebra of Kac-Moody algebras acting on the cohomology of quiver varieties and related spaces.
Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), no. 2, 365–416
Hiraku Nakajima, Quiver varieties and Kac-Moody algebras. Duke Math. J. 91 (1998), no. 3, 515–560.
Bernard Leclerc, Quantum loop algebras, quiver varieties, and cluster algebraṣ, 117-152 in in A. Skowronski, K. Yamagata K. (eds.), Representations of algebras and related topics, Eur. Math. Soc. 2011
On equivariant elliptic cohomology of quiver varieties in relation to the AGT correspondence:
Mina Aganagic, Andrei Okounkov, Elliptic stable envelopes (arXiv:1604.00423)
Andrei Okounkov, Inductive construction of stable envelopes and applications, I. Actions of tori. Elliptic cohomology and K-theory (arXiv:2007.09094)
following the analogous non-elliptic discussion in:
Review in:
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