# Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

@article{Nakajima1994InstantonsOA, title={Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras}, author={Hiraku Nakajima}, journal={Duke Mathematical Journal}, year={1994}, volume={76}, pages={365-416} }

To Professor Shoshichi Kobayashi on his 60th birthday 1. Introduction. In this paper we shall introduce a new family of varieties, which we call quiver varieties, and study their geometric structures. They have close relation to the singularity theory and the representation theory of the Kac-Moody algebras. Our original motivation was to study solutions of the anti-self-dual Yang-Mills equations on a particular class of 4-dimensional noncompact complete manifolds, the so-called ALE spaces (or…

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## References

SHOWING 1-10 OF 79 REFERENCES

### Homology of moduli spaces of instantons on ALE spaces. I

- Mathematics
- 1994

In [13] P. B. Kronheimer and the author introduced a new class of hyper-Kahler manifolds which arise as moduli spaces of anti-self-dual connections on a certain class of 4-dimensional noncompact…

### Instantons and affine algebras I: The Hilbert scheme and vertex operators

- Mathematics
- 1995

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises…

### Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

- Mathematics
- 1988

The fundamental group is one of the most basic topological invariants of a space. The aim of this paper is to present a method of constructing representations of fundamental groups in complex…

### On crystal bases of the $Q$-analogue of universal enveloping algebras

- Mathematics
- 1991

0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the…

### Quivers, perverse sheaves, and quantized enveloping algebras

- Mathematics
- 1991

1. Preliminaries 2. A class of perverse sheaves on Ev 3. Multiplication 4. Restriction 5. Fourier-Deligne transform 6. Analysis of a sink 7. Multiplicative generators 8. Compatibility of…

### Langlands Reciprocity for Algebraic Surfaces

- Mathematics
- 1995

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface.…

### Closures of conjugacy classes of matrices are normal

- Mathematics
- 1979

Hanspeter Kraft 1 and Claudio Procesi 2 Sonderforschungsbereich Theoretische Mathematik, Universit~it Bonn D-5300 Bonn, Federal Republic of Germany 2 Istituto di Matematica, Universitfi di Roma,…

### THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE

- Mathematics
- 1987

In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean…

### Stratified symplectic spaces and reduction

- Mathematics
- 1991

Let (M, w) be a Hamiltonian G-space with proper momentum map J: M -> g*. It is well-known that if zero is a regular value of J and G acts freely on the level set J '(0), then the reduced space MO =…

### Instability in invariant theory

- Mathematics
- 1978

Let V be a representation of a reductive group G. A fundamental theorem in geometric invariant theory states that there are enough polynomial functions on V, which are invariant under G, to…