Math 2.0

Associative rings and algebras Representations of algebras Theorem

Kac's theorem

Builds on QuiverRepresentation of a quiverIndecomposable moduleGabriel's theorem

Statement

Let kk be an algebraically closed field and let QQ be a finite quiver without loops. Let BijB_{ij} be the number of edges between ii and jj in the underlying graph of QQ, and let C=2I−BC = 2I - B. Let Δ+\Delta_+ be the set of positive roots of the Kac–Moody algebra with generalized Cartan matrix CC.

  1. The dimension vectors of the finite-dimensional indecomposable representations of QQ over kk are exactly the elements of Δ+\Delta_+. In particular, this set does not depend on the orientation of QQ.
  2. If α∈Δ+\alpha \in \Delta_+ is a real root, there is exactly one indecomposable representation of dimension vector α\alpha, up to isomorphism.
  3. If α∈Δ+\alpha \in \Delta_+ is an imaginary root, there are infinitely many pairwise non-isomorphic indecomposable representations of dimension vector α\alpha.

Remarks

Victor Kac proved the theorem by counting representations over finite fields and passing to algebraically closed fields. Over a finite field with qq elements, count the isomorphism classes of absolutely indecomposable representations of dimension vector α\alpha. This number is a polynomial in qq that does not depend on the orientation. Example: for the Kronecker quiver, the root system is of affine type A~1\tilde{A}_1, and (1,1)(1, 1) is an imaginary root. The representations k⇉kk \rightrightarrows k given by the pairs of scalars (1,λ)(1, \lambda) are indecomposable and pairwise non-isomorphic, as part 3 predicts. For Dynkin quivers all roots are real and finite in number, and the theorem reduces to Gabriel's theorem.

Going deeper

For a student

The theorem generalizes Gabriel's theorem from Dynkin quivers to all quivers without loops. Real roots are the images of simple roots under the Weyl group, and they satisfy q(α)=1q(\alpha) = 1 for the Tits form qq. The counting polynomials in the remarks are the Kac polynomials, studied in later work on quiver varieties.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Dynkin quiver, concept, no evidence levelDynkin quiverconceptIndecomposable module, concept, no evidence levelIndecomposable moduleconceptQuiver, concept, no evidence levelQuiverconceptRepresentation of a quiver, concept, no evidence levelRepresentation of a q…conceptGabriel's theorem, statement, no evidence levelGabriel's theoremstatementKac's theorem, statement, no evidence levelKac's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Dynkin quiver concept
  2. Indecomposable module concept
  3. Quiver concept
  4. Representation of a quiver concept
  5. Gabriel's theorem statement
  6. Kac's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Quiver_(mathematics)
About this page
Tag59QJC3ZXA1
URIm2://59QJC3ZXA1
Kindstatement, theorem
WritingW0 stub
MSC16G20, 17B67
StatusThis copy of the site has no build data. Run m2 status 59QJC3ZXA1.
AboutRepresentation of a quiver
UsesQuiver, Representation of a quiver, Indecomposable module, Gabriel's theorem
Changed2 November 2026

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