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  "body": "\n## Statement\n\nLet $k$ be an algebraically closed field and let $Q$ be a finite [[quiver|RE9YFFS3TS]] without loops.\nLet $B_{ij}$ be the number of edges between $i$ and $j$ in the underlying graph of $Q$, and let $C = 2I - B$.\nLet $\\Delta_+$ be the set of positive roots of the Kac–Moody algebra with generalized Cartan matrix $C$.\n\n1. The dimension vectors of the finite-dimensional [[indecomposable|E2P29GCD9T]] [[representations|TXVWFWP6VQ]] of $Q$ over $k$ are exactly the elements of $\\Delta_+$.\nIn particular, this set does not depend on the orientation of $Q$.\n2. If $\\alpha \\in \\Delta_+$ is a real root, there is exactly one indecomposable representation of dimension vector $\\alpha$, up to isomorphism.\n3. If $\\alpha \\in \\Delta_+$ is an imaginary root, there are infinitely many pairwise non-isomorphic indecomposable representations of dimension vector $\\alpha$.\n\n## Remarks\n\nVictor Kac proved the theorem by counting representations over finite fields and passing to algebraically closed fields.\nOver a finite field with $q$ elements, count the isomorphism classes of absolutely indecomposable representations of dimension vector $\\alpha$.\nThis number is a polynomial in $q$ that does not depend on the orientation.\nExample: for the Kronecker quiver, the root system is of affine type $\\tilde{A}_1$, and $(1, 1)$ is an imaginary root.\nThe representations $k \\rightrightarrows k$ given by the pairs of scalars $(1, \\lambda)$ are indecomposable and pairwise non-isomorphic, as part 3 predicts.\nFor Dynkin quivers all roots are real and finite in number, and the theorem reduces to [[Gabriel's theorem|FC78F2E87E]].\n\n## Explanations\n\n### L1 human\n\nEvery quiver, not only the Dynkin ones, has a root system attached to it, built from its graph.\nKac's theorem says that the possible sizes of the indecomposable representations are exactly the positive roots.\nReal roots give a single representation each, and imaginary roots give infinite families.\nSo the root system controls the shape of the classification, even when a full list is out of reach.\n\n### L2 human\n\nThe theorem generalizes [[Gabriel's theorem|FC78F2E87E]] from Dynkin quivers to all quivers without loops.\nReal roots are the images of simple roots under the Weyl group, and they satisfy $q(\\alpha) = 1$ for the Tits form $q$.\nThe counting polynomials in the remarks are the Kac polynomials, studied in later work on quiver varieties.\n",
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