nLab quiver variety

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.

Related concepts

moduli spaces

• moduli space of complex structures

• moduli space of conformal structures

• moduli space of curves

• moduli space of Riemannian metrics

• moduli stack of elliptic curves

• moduli space of Calabi-Yau spaces

• moduli stack of tori

• moduli stack of formal groups

• moduli space of connections, moduli space of flat connections

• Hitchin moduli space

• moduli space of monopoles

• moduli space of framed manifolds

• moduli space of bundles

• moduli space of (higher) line bundles

• quiver variety

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 $t$-analogs of $q$-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 $A_n$, $\widehat A_n$-quiver varieties, 2019 (pdf)