quiver variety in nLab

Contents

Contents

Idea

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.

References

• Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), no. 2, 365–416 MR1302318

• Hiraku Nakajima, Quiver varieties and Kac-Moody algebras. Duke Math. J. 91 (1998), no. 3, 515–560 MR1604167

• 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

• H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238 MR1808477

• H. Nakajima, Quiver varieties and tt-analogs of qq-characters of quantum affine algebras, Ann. of Math. (2) 160 (2004), no. 3, 1057–1097 MR2144973

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:

• Davesh Maulik, Andrei Okounkov, Quantum groups and quantum cohomology, Asterisque 408(2019) (arXiv:1211.1287, ISBN: 978-2-85629-900-5)

Review in:

• Andrey Smirnov, Stable envelopes for A nA_n, A^ n\widehat A_n-quiver varieties, 2019 (pdf)