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  "body": "\n## Summary\n\nThe universal enveloping algebra $U(\\mathfrak g)$ of a Lie algebra $\\mathfrak g$ is the most general associative algebra containing $\\mathfrak g$.\nIn it the Lie bracket becomes the commutator $xy - yx$.\nRepresentations of $\\mathfrak g$ are the same as modules over $U(\\mathfrak g)$, so Lie theory can use the tools of ring theory.\nBy the [[Poincaré–Birkhoff–Witt theorem|HFEA4FKGW1]], $U(\\mathfrak g)$ has a basis of ordered monomials in a basis of $\\mathfrak g$.\nIt is also a [[Hopf algebra|7XFYP0N84H]], and its deformations give the quantum groups.\n\n## Explanations\n\n### L1 human\n\nA Lie algebra has a bracket $[x, y]$ but no product $xy$.\nThe universal enveloping algebra adds a product, subject only to the rule $xy - yx = [x, y]$.\nFor an abelian Lie algebra, the rule says that the elements commute, and $U(\\mathfrak g)$ is a polynomial ring.\nFor a Lie algebra with basis $x, y$ and $[x, y] = y$, it is the algebra of noncommutative polynomials in $x, y$ with $xy - yx = y$.\n",
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