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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $\\mathfrak g$ be a Lie algebra over $k$ with bracket $[\\cdot, \\cdot]$.\nLet $T(\\mathfrak g)$ be the [[tensor algebra|P4J5JZDW0B]] of the vector space $\\mathfrak g$.\nLet $I$ be the [[two-sided ideal|0CD87SFM2N]] of $T(\\mathfrak g)$ generated by the elements $x \\otimes y - y \\otimes x - [x, y]$ for $x, y \\in \\mathfrak g$.\nThe *universal enveloping algebra* of $\\mathfrak g$ is $U(\\mathfrak g) = T(\\mathfrak g) / I$, together with the map $\\iota \\colon \\mathfrak g \\to U(\\mathfrak g)$ induced by the inclusion of $\\mathfrak g$ in $T(\\mathfrak g)$.\n\n## Remarks\n\nThe map $\\iota$ is linear and satisfies $\\iota([x, y]) = \\iota(x)\\iota(y) - \\iota(y)\\iota(x)$.\nUniversal property: for every [[associative algebra|3CTW1P4HBP]] $A$ over $k$ with identity and every linear map $\\varphi \\colon \\mathfrak g \\to A$ with $\\varphi([x, y]) = \\varphi(x)\\varphi(y) - \\varphi(y)\\varphi(x)$, there is a unique algebra homomorphism $\\tilde\\varphi \\colon U(\\mathfrak g) \\to A$ with $\\tilde\\varphi \\circ \\iota = \\varphi$.\nHence representations of $\\mathfrak g$ on a vector space $V$ are the same as left $U(\\mathfrak g)$-module structures on $V$.\nExample: if $\\mathfrak g$ is abelian, all brackets are $0$, so $I$ is generated by the elements $x \\otimes y - y \\otimes x$.\nThen $U(\\mathfrak g)$ is the [[symmetric algebra|PBGF9TXMB7]] $S(\\mathfrak g)$, a polynomial ring in a basis of $\\mathfrak g$.\nNon-example: for the Lie algebra $\\mathfrak g$ with basis $x, y$ and $[x, y] = y$, the tensor algebra $T(\\mathfrak g)$ itself does not satisfy $xy - yx = [x, y]$, since $x \\otimes y - y \\otimes x \\ne y$ there; one must pass to the quotient.\n",
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