Nakajima quiver varieties turn a graph into a symplectic moduli space whose geometry carries representations of Kac–Moody algebras. The construction doubles the quiver, adds framing, imposes a moment-map equation and then performs a stability quotient. Correspondences between the resulting varieties act as Chevalley generators on homology or K-theory.
The architecture unifies the main ideas of Batch 11. Quiver moduli supplies the representation variety and stability quotient. Geometric invariant theory controls the quotient. Symplectic reduction supplies the moment-map perspective. The output connects directly to Kac–Moody representation theory.
Doubled quiver · Framing · Moment map · Quiver variety · Worked A1 example · Hecke correspondences · Kac–Moody action · Practice
Why double the quiver?
Start with a quiver Q. For every arrow h:i→j, introduce a reverse arrow \bar h:j→i. The doubled quiver \overline Q therefore contains both orientations.
If the original representation space is built from Hom(V_i,V_j), the doubled space contains a matching Hom(V_j,V_i). These paired directions naturally form cotangent coordinates.
Indeed, after choosing one orientation, the representation space of the doubled quiver can be identified with the cotangent bundle of the original representation space.
That cotangent structure is why a natural symplectic form appears.
The doubled representation space
Choose vector spaces V_i of dimensions v_i at the vertices. For the doubled arrows define
M_Q(V)=⊕_{h∈\overline Q_1} Hom(V_{s(h)},V_{t(h)}).
For each original arrow h and reverse arrow \bar h, the trace pairing
Tr(B_{\bar h} dB_h)
provides the canonical cotangent pairing. Summing across arrows gives the symplectic form.
The choice of orientation affects signs in formulas but not the underlying doubled geometry.
Framing adds external data
Choose framing spaces W_i of dimensions w_i. Add maps
I_i:W_i→V_i, J_i:V_i→W_i.
The full framed representation space is
M(V,W)=M_Q(V)⊕⊕_i Hom(W_i,V_i)⊕⊕_i Hom(V_i,W_i).
Framing breaks unwanted scalar symmetries and fixes a highest-weight parameter. In the eventual Kac–Moody representation, w determines the highest weight.
The gauge group
The group
G_V=∏_i GL(V_i)
acts by changing bases in the internal spaces V_i. It conjugates arrow maps and acts naturally on I_i and J_i.
The framing spaces W_i are not quotiented by G_V. They serve as external boundary data.
The complex moment map
The G_V-action is Hamiltonian. For each vertex i, the complex moment map has the form
μ_i(B,I,J)=Σ_{h:t(h)=i} ε(h) B_h B_{\bar h}+I_iJ_i,
where ε(h)=±1 is determined by the chosen orientation.
The moment-map equation
μ(B,I,J)=0
is a matrix relation at every vertex.
Without framing, these are closely related to preprojective-algebra relations.
Preprojective algebra
The preprojective algebra of a quiver is obtained from the doubled path algebra by imposing the relation
Σ_h ε(h) h\bar h=0
at the vertices.
A representation of the preprojective algebra is therefore a doubled-quiver representation satisfying the unframed moment-map relation.
This algebraic reformulation connects symplectic geometry back to ordinary module theory.
The Nakajima quiver variety
Choose a stability parameter θ. The Nakajima quiver variety is, schematically,
M_θ(v,w)=μ^{-1}(0)^{θ-stable}/G_V
or the corresponding GIT quotient including semistable points, depending on convention.
Thus the construction has four stages:
- double the quiver;
- add framing;
- impose the moment-map equation;
- take a stability quotient.
Each stage has a distinct job. Removing any one changes the object substantially.
Dimension formula
For simply-laced quivers and stable smooth points, a standard complex-dimension formula is
dim M(v,w)=2(w·v)−(v,Cv),
where C is the symmetric Cartan matrix associated with the underlying graph and the exact pairing convention determines the factor notation.
Equivalently, many references write 2d(v,w), with
d(v,w)=w·v−(1/2)(v,Cv).
The even complex dimension reflects the holomorphic symplectic structure.
Worked A1 example
Take a quiver with one vertex and no arrows. Let dim V=v and dim W=w.
The framed data is just
I:W→V, J:V→W.
The moment-map equation is
IJ=0
as an endomorphism of V.
With the usual stability forcing I to be surjective in one chamber, the quotient is naturally isomorphic to the cotangent bundle of the Grassmannian Gr(v,W).
Thus
M(v,w)≈T*Gr(v,w)
for the standard A1 Nakajima setup.
Dimension check for the A1 model
Gr(v,w) has complex dimension v(w−v). Its cotangent bundle therefore has dimension
2v(w−v).
For A1 the Cartan matrix is [2]. The general formula gives
2wv−2v²=2v(w−v),
exactly matching the cotangent-bundle calculation.
This is a strong internal consistency check on the construction.
Why cotangent bundles keep appearing
Doubling a representation space produces cotangent directions. Moment-map reduction of a cotangent bundle is a standard source of symplectic quotients.
Therefore many quiver varieties are symplectic resolutions of singular affine quotients.
This geometric architecture parallels the Springer resolution T*B→N studied in Springer Theory.
Affine quotient and resolution
Besides the smooth stable quiver variety M_θ(v,w), one can form the affine quotient
M_0(v,w)=μ^{-1}(0)//G_V.
There is a projective morphism
π:M_θ(v,w)→M_0(v,w).
For generic stability and favorable dimension vectors, this is a symplectic resolution.
The fibers of π carry important representation-theoretic information, much as Springer fibers do.
Lagrangian core
A central subvariety is the fiber over the origin
L(v,w)=π^{-1}(0).
Under standard conditions it is Lagrangian inside the quiver variety.
Top Borel–Moore homology of these Lagrangian fibers is where highest-weight representations are often realized.
Hecke correspondences
To change v by one simple root direction, compare quiver varieties M(v,w) and M(v+e_i,w).
A Hecke correspondence parametrizes pairs of stable framed representations related by an inclusion or quotient at vertex i.
As a correspondence between two varieties, it induces maps on homology by pull-push operations.
These maps become the geometric raising and lowering operators e_i and f_i.
Why correspondences act like operators
Suppose Z⊂X×Y is a correspondence. A cycle or cohomology class on X can be pulled back to Z and then pushed forward to Y.
Schematically,
H_*(X) → H_*(Z) → H_*(Y).
Composition of correspondences becomes composition of operators.
Geometric relations among the correspondences then prove algebraic relations among the operators.
Kac–Moody representations from geometry
Let the underlying graph define a symmetric generalized Cartan matrix and hence a Kac–Moody algebra g.
Nakajima constructs an action of g on a direct sum of homology groups over all dimension vectors v:
⊕_v H_{top}(L(v,w)).
The framing vector w determines the dominant highest weight
Λ_w=Σ_i w_i Λ_i.
The component v=0 supplies the highest-weight vector.
Hecke correspondences supply e_i and f_i, and geometric intersection arguments verify the Chevalley–Serre relations.
Weight of a component
The quiver variety indexed by v contributes to weight
Λ_w−Σ_i v_i α_i.
Increasing v_i by one moves the weight down by the simple root α_i.
This makes the dimension vector a geometric record of how far the representation has moved from the highest weight.
A1 as an sl2 representation
For the one-vertex graph A1 with framing dimension w, the quiver varieties T*Gr(v,w) exist for v=0,…,w.
Their Lagrangian cores include the zero sections Gr(v,w). The direct sum of their top homologies has one basis line for each v.
There are w+1 such lines, matching the dimension of the irreducible sl2 representation of highest weight w.
The Hecke correspondences between adjacent Grassmannians realize the familiar sl2 raising and lowering chain.
This is the geometric analogue of the crystal B(w) chain in Crystal Bases.
Irreducible components and crystal bases
Irreducible components of Nakajima’s Lagrangian quiver varieties carry a crystal structure.
The geometric crystal operators modify the representation at one vertex and match Kashiwara operators on the crystal basis of the highest-weight representation.
Thus the q→0 combinatorial crystal can be recovered from actual irreducible geometric components.
This is a deep return path across the estate:
quiver geometry → irreducible components → crystal graph → highest-weight representation.
K-theory and quantum groups
Ordinary homology realizes enveloping-algebra actions. Equivariant K-theory introduces deformation parameters and supports actions of quantum loop algebras or related quantum groups.
This geometrizes the transition from Kac–Moody algebras to quantum affine algebras.
The exact quantum algebra depends on the quiver, equivariance and chosen K-theoretic framework, so one should state the acting algebra precisely.
ADHM and instantons
Certain quiver varieties reproduce moduli spaces of framed instantons through ADHM data.
For the Jordan quiver, matrices B_1,B_2 together with framing maps I,J satisfy the ADHM equation
[B_1,B_2]+IJ=0.
After stability and quotienting, the resulting moduli spaces connect gauge theory, Hilbert schemes and representation theory.
This is one reason quiver varieties appear far beyond abstract graph representations.
Hilbert schemes
The Hilbert scheme of n points on C² can be realized through a rank-one framed Jordan-quiver construction.
Across all n, their cohomology carries Heisenberg-algebra actions constructed by correspondences.
Thus the same correspondence philosophy builds not only Kac–Moody representations but also oscillator-type representations.
Relation to Springer theory
Springer theory uses a symplectic resolution T*B→N and extracts Weyl-group representations from fiber cohomology.
Nakajima theory uses families of symplectic quiver varieties and correspondences between different dimension vectors to construct Kac–Moody actions.
Both turn resolution geometry into representation theory, but the groups/algebras and operators differ.
Springer should therefore be read as a close geometric relative, not as the same construction.
Relation to geometric Satake
Geometric Satake builds the Langlands dual group from a tensor category of sheaves on an affine Grassmannian.
Nakajima varieties instead construct specific highest-weight representations through homology/K-theory of symplectic moduli spaces.
Both use correspondences and geometric categories, but Satake is Tannakian reconstruction while Nakajima is a geometric representation construction.
A verification workflow
- 1. Fix the underlying graph, orientation and vectors v,w.
- 2. Double every arrow.
- 3. Add framing maps I and J.
- 4. Write the G_V action.
- 5. Fix the orientation signs in the moment map.
- 6. Solve μ=0 before quotienting.
- 7. State the stability chamber.
- 8. Check the dimension formula.
- 9. Identify the affine quotient and resolution map.
- 10. For representation theory, track how changing v_i shifts the weight by α_i.
Common mistakes
- Using the original quiver without doubling when writing the symplectic moment map.
- Forgetting framing spaces.
- Quotienting before imposing μ=0.
- Ignoring stability parameters.
- Mixing moment-map sign conventions across orientations.
- Assuming every quiver variety is smooth for every parameter.
- Confusing homology Kac–Moody actions with K-theoretic quantum actions.
- Calling Springer and Nakajima constructions identical.
Practice questions
1. What is the doubled quiver? 2. Why does doubling produce cotangent-like coordinates? 3. What are the framing maps? 4. What group is quotiented?
5. State the moment-map equation schematically. 6. For A1, what is the resulting quiver variety? 7. Verify its dimension. 8. What is the affine quotient M_0?
9. What do Hecke correspondences do? 10. How does v determine representation weight? 11. What does w determine? 12. How are crystal operators seen geometrically?
Worked answers
1. The quiver obtained by adding a reverse arrow for every original arrow.
2. Every arrow coordinate is paired with a reverse-arrow coordinate, giving the canonical form of a cotangent representation space.
3. I_i:W_i→V_i and J_i:V_i→W_i.
4. G_V=∏_i GL(V_i).
5. Σ ε(h)B_hB_{\bar h}+IJ=0 at each vertex.
6. In the standard chamber, T*Gr(v,w).
7. 2v(w−v), matching 2wv−2v² from the A1 Cartan matrix.
8. μ^{-1}(0)//G_V, the singular affine quotient underlying the projective resolution.
9. They relate varieties whose dimension vectors differ by a simple root and induce raising/lowering operators on homology.
10. The v-component has weight Λ_w−Σ_i v_iα_i.
11. The dominant highest weight Λ_w=Σ_iw_iΛ_i.
12. By modifying irreducible components through correspondences at one vertex, producing the Kashiwara crystal moves.
Sources and further study
Hiraku Nakajima’s foundational papers on quiver varieties and Kac–Moody algebras establish the geometric construction of highest-weight representations. His lectures on quiver varieties provide the standard entry point. Lusztig’s quiver-variety work and preprojective-algebra constructions form a closely related route to canonical bases and quantum groups.
Representation Mathematics — Batch 11
Begin with Geometric Invariant Theory, apply stability to Quiver Moduli, and move from modules to complexes in Derived Categories and Exceptional Collections. Return to the BTT Mathematics Learning Hub.