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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $Q = (Q_0, Q_1, s, t)$ be a [[quiver|RE9YFFS3TS]].\nA *representation* $V$ of $Q$ over $k$ consists of:\n\n1. a [[vector space|000AYPA9AJ]] $V_i$ over $k$ for each vertex $i \\in Q_0$;\n2. a [[linear map|0NRYHJPQG2]] $V_a \\colon V_{s(a)} \\to V_{t(a)}$ for each arrow $a \\in Q_1$.\n\nA *morphism* $\\varphi \\colon V \\to W$ of representations is a family of linear maps $\\varphi_i \\colon V_i \\to W_i$, $i \\in Q_0$, such that $\\varphi_{t(a)} \\circ V_a = W_a \\circ \\varphi_{s(a)}$ for every arrow $a$.\nIf $Q_0$ is finite and each $V_i$ is finite-dimensional, $V$ is *finite-dimensional*, and its *dimension vector* is $(\\dim V_i)_{i \\in Q_0}$.\n\n## Remarks\n\nExample: for the quiver $1 \\to 2$ with arrow $a$, the data $V_1 = V_2 = k$ and $V_a = \\id_k$ form a representation with dimension vector $(1, 1)$.\nNon-example: for the same quiver, a vector space at each vertex with a linear map $V_2 \\to V_1$ is not a representation.\nClause 2 asks for a map from the source of $a$ to its target.\nDirect sums are taken vertexwise, and a nonzero representation is indecomposable if it is not a direct sum of two nonzero representations.\nFor a finite quiver, representations of $Q$ correspond to left modules over the [[path algebra|2PYANBGXFD]] $kQ$, and this is an equivalence of categories.\n",
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