Preprojective algebras impose a moment-map relation on a doubled quiver. They sit between ordinary quiver representation theory and symplectic geometry: modules are representations of arrows and reverse arrows whose signed compositions cancel at every vertex.
This chapter continues Quiver Representations, Nakajima Quiver Varieties, Symplectic Reflection Algebras and Cohomological Hall Algebras.
Double the quiver
For each arrow a:i→j in Q, add a reverse arrow a*:j→i. The doubled quiver \bar Q contains both orientations.
The path algebra C\bar Q contains compositions aa* and a*a that return to the starting vertices.
Preprojective relation
Choose an orientation of the underlying graph. The preprojective algebra is
Π(Q)=C\bar Q /(Σ_{a∈Q_1}(aa*−a*a))
with the relation interpreted vertex by vertex using idempotents. Equivalent sign conventions are common.
The remarkable fact is that Π(Q) is independent, up to natural isomorphism, of the chosen orientation of the underlying graph.
Moment-map interpretation
Fix vector spaces V_i. The representation space of the doubled quiver is naturally a cotangent bundle of the representation space of Q after choosing an orientation.
The base-change group G_V=∏_iGL(V_i) acts Hamiltonianly. Its moment map has components equal to the signed sums of B_aB_{a*}. The equation μ=0 is exactly the preprojective relation.
Thus Π(Q)-modules are points of a zero-level moment-map variety before quotienting by change of basis.
A2 example
Take Q:1→2 with arrow a and reverse a*:2→1. The relations at the two vertices force aa*=0 and a*a=0 in the undeformed preprojective algebra.
A representation therefore consists of maps A:V_1→V_2 and B:V_2→V_1 satisfying AB=0 and BA=0.
Already in this smallest nontrivial case, the doubled representation remembers extensions in both directions while the relation constrains their compositions.
Dynkin versus non-Dynkin
If the underlying graph is Dynkin, the preprojective algebra is finite-dimensional. For affine and indefinite graphs it is typically infinite-dimensional.
This mirrors root-system growth: finite root systems lead to bounded representation geometry, while affine and indefinite systems produce infinite families.
Roots and dimension vectors
Dimension vectors of indecomposable quiver representations are controlled by positive roots. For preprojective algebras, root systems continue to organize nonemptiness and dimensions of representation varieties, though the classification is richer than Gabriel’s finite-type theorem.
Crawley-Boevey’s results relate simple representations of deformed preprojective algebras to positive roots satisfying parameter and decomposition conditions.
Deformed preprojective algebras
Introduce parameters λ_i at the vertices and replace the zero moment-map equation by
Σ(aa*−a*a)=Σ_i λ_i e_i.
The resulting algebra Π^λ(Q) is a deformed preprojective algebra. A representation of dimension vector v can exist only if λ·v=0, obtained by taking traces of the relation.
Trace check
Trace of a commutator vanishes. Summing traces of the left-hand moment-map relation gives zero. The right side gives Σ_iλ_i dim V_i=λ·v.
Therefore λ·v=0 is an immediate necessary consistency condition before any deeper representation analysis.
McKay correspondence
For affine ADE quivers arising from finite subgroups Γ⊂SL_2(C), deformed preprojective algebras connect to deformations of Kleinian singularities C²/Γ.
The affine Dynkin graph records irreducible Γ-representations, while the deformation parameters correspond to directions in the singularity’s Poisson deformation space.
Nakajima quiver varieties
Nakajima adds framing maps I and J and imposes the framed moment-map equation before taking a stability quotient. The unframed preprojective algebra is therefore the algebraic core of the quiver-variety construction.
Framing and stability turn the raw zero-level representation variety into smooth or resolved symplectic moduli spaces under suitable conditions.
Lusztig nilpotent varieties
Nilpotent representations of preprojective algebras form Lusztig’s nilpotent varieties. Their irreducible components realize the semicanonical basis of U(n^−).
This is the preprojective counterpart to the perverse-sheaf construction of canonical bases in the first Batch 16 chapter.
2-Calabi–Yau behaviour
Stable categories and derived categories associated with preprojective algebras often exhibit 2-Calabi–Yau structures, especially in Dynkin/appropriate completed settings.
This connects preprojective representation theory to cluster categories and mutation.
Verification workflow
- Fix the orientation only as a presentation device.
- Write the doubled quiver explicitly.
- Check signs vertex by vertex.
- Distinguish ordinary and deformed preprojective algebras.
- Use λ·v=0 as a trace consistency check.
- Separate unframed moment-map varieties from Nakajima framed quotients.
- Do not identify canonical and semicanonical bases.
- State Dynkin/affine hypotheses before finite-dimensionality claims.
Practice
1. What is the doubled quiver? 2. State the preprojective relation. 3. Why is it a moment-map equation? 4. What condition follows by taking traces in the deformed case? 5. When is Π(Q) finite-dimensional? 6. What basis comes from nilpotent preprojective varieties?
Answers. Add a reverse arrow for every arrow; the signed sum of arrow/reverse-arrow compositions vanishes; it is the Hamiltonian moment map for base change on the doubled representation space; λ·v=0; for Dynkin underlying graphs; the semicanonical basis.
Representation Mathematics — Batch 16
Read Canonical Bases and Cohomological Hall Algebras, then continue to q-Schur Algebras. Return to the BTT Mathematics Learning Hub.