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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]], with vertex set $Q_0$, arrow set $Q_1$, and source and target maps $s, t \\colon Q_1 \\to Q_0$.\nA *path* of length $m \\ge 1$ is a sequence $p = \\alpha_1 \\alpha_2 \\cdots \\alpha_m$ of arrows with $t(\\alpha_i) = s(\\alpha_{i+1})$ for $1 \\le i \u003c m$; it has source $s(p) = s(\\alpha_1)$ and target $t(p) = t(\\alpha_m)$.\nFor each vertex $v$ there is also a path $e_v$ of length $0$ with $s(e_v) = t(e_v) = v$.\nThe *path algebra* $kQ$ is the [[vector space|000AYPA9AJ]] over $k$ with [[basis|0KCQ9Y3QKA]] the set of all paths.\nThe product of two paths $p$ and $q$ is the concatenation $pq$ if $t(p) = s(q)$ and $0$ otherwise.\nHere $e_v$ acts as an identity on paths through $v$, and the product on $kQ$ is the bilinear extension.\n\n## Remarks\n\nThe product is associative, so $kQ$ is an [[associative algebra|3CTW1P4HBP]]; it has an identity $\\sum_{v \\in Q_0} e_v$ when $Q_0$ is finite.\nThe $e_v$ are orthogonal idempotents: $e_v e_w = e_v$ if $v = w$ and $0$ otherwise.\nExample: let $Q$ have vertices $1, 2$ and one arrow $\\alpha \\colon 1 \\to 2$.\nThe paths are $e_1, e_2, \\alpha$, and the nonzero products of basis elements are $e_1 e_1 = e_1$, $e_2 e_2 = e_2$ and $e_1 \\alpha = \\alpha = \\alpha e_2$.\nSo $kQ$ has dimension $3$, and it is isomorphic to the algebra of upper triangular $2 \\times 2$ matrices over $k$, via $e_1 \\mapsto E_{11}$, $e_2 \\mapsto E_{22}$, $\\alpha \\mapsto E_{12}$.\nNon-example: for the quiver with one vertex and two loops $a, b$, the paths $ab$ and $ba$ are different basis elements.\nSo $kQ$ is the [[free algebra|2P3Y2FRCSZ]] $k\\langle a, b \\rangle$ and not the commutative ring $k[a, b]$.\n$kQ$ is finite-dimensional if and only if $Q$ is finite and has no oriented cycles.\nSome books compose paths in the opposite order, which replaces $kQ$ by its opposite algebra.\n",
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