----
> [!definition] Definition. ([[universal enveloping algebra]])
> Let $\mathsf{Lie}$ denote the [[category]] of [[Lie algebra|Lie algebras]] and [[Lie algebra homomorphism|LAHs]]. Let $\mathsf{Alg}$ denote the [[category]] of unital associative [[algebra|algebra]] and [[algebra homomorphism|algebra morphisms]]. There is a [[covariant functor|functor]] $\text{Lie}(-):\mathsf{Alg} \to \mathsf{Lie}$ sending an [[algebra]] $A$ to $\text{Lie}(A)=(A, [-,-])$. It has a [[adjoint functor|left-adjoint]] $\mathcal{U}:\mathsf{Lie} \to \mathsf{Alg}$. If $\mathfrak{g}$ is a [[Lie algebra]], then $\mathcal{U}(\mathfrak{g})$, together with a natural map $\iota:\mathfrak{g} \to \mathcal{U}(\mathfrak{g})$, is called the **universal enveloping algebra** corresponding to $\mathfrak{g}$.
>
> This information is best encoded is the following [[universal property]]. Let $A$ be a unital associative [[algebra]], $f:\mathfrak{g} \to (A, [-,-])$ a [[Lie algebra homomorphism]]. Then there exists a unique [[algebra homomorphism]] $\mathcal{U}(f):\mathcal{U}(\mathfrak{g}) \to A$ satisfying $\mathcal{U}(f)\circ \iota=f$, i.e., there's a commutative diagram
>
>
> ```tikz
> \usepackage{tikz-cd}
> \usepackage{amsmath}
> \usepackage{amsfonts}
> \begin{document}
> % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYAFUAvgApOPfgDMATnQDWwAOZiAlCDGl0mXPkIoAjOSq1GLNgEFtukBmx4CRMsfP1mrRB268+CspqYtrmMFCq8ESgChBcSABM1DgQSGQWXmyc+Dh0djHycYnJqYimGVY+siDUDHQARjAMAAr6zkYg8liqfDj5ILHxiOkpSOWelb4wAB5YcDhwAAQAhIvS-kIMopKyWrVYYN4gUHRwfOGhYkA
> \begin{tikzcd}
> \mathcal{U}(\mathfrak{g}) \arrow[r, "\exists ! \mathcal{U}(f)", dashed] & A \\
> \mathfrak{g} \arrow[u, "\iota"] \arrow[ru, "f"'] &
> \end{tikzcd}
> \end{document}
> ```
> or more precisely a commutative diagram
>
> ```tikz
> \usepackage{tikz-cd}
> \usepackage{amsmath}
> \usepackage{amsfonts}
> \begin{document}
> % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYAFUAvgApOPfgDMATnQDWwAOZiAlCDGl0mXPkIoAjOSq1GLNgEFtukBmx4CRMsfP1mrRB268+CspqYnZ6ToZEpu7UnlY+nDgwAB44wAAyWDCS1lpi5jBQqvBEoAoQXEgATNQ4EEhkFl5snPg4dKEgZRWIDbVIpo1xvslYcDhwAAQAhBPS-kIMopKyWtQMWGDeIFB0cHwFHV1VNXWIAMwxlluyIGt0AEYwDAAK+s5GIPJYqnw4h-LlJAXEB9RADWJbBLJVJYKAhO6PF5vCI+L4-P55MRAA
> \begin{tikzcd}
> \mathcal{U}(\mathfrak{g}) \arrow[r, "\exists ! \mathcal{U}(f)", dashed] & A \\
> \mathfrak{g} \arrow[u, "\iota"] \arrow[r, "f"'] & \text{Lie}(A) \arrow[u, "\text{id}"']
> \end{tikzcd}
> \end{document}
> ```
> and thus a natural [[bijection]] $\text{Hom}_{\mathsf{Lie}}(\mathfrak{g}, \text{Lie}(A)) \cong \text{Hom}_{\mathsf{Alg}}(\mathcal{U}(\mathfrak{g}), A).$
>
>
> ```tikz
> \usepackage{tikz-cd}
> \usepackage{amsmath}
> \begin{document}
> % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACzgAzYABksMAL4gJpdJlz5CKAIzkqtRizace-IcACCDAOZSJ6mFGPwioQQCcIXJGRA4ISVSABGMMFCQAZld6ZlZEDm5ePgBjRmAAVTNZEAcnT2p3F2pffyCQzXDInBgADxxRcQkACgBaAEppCgkgA
> \begin{tikzcd}
> \mathsf{Lie} \arrow[r, "\mathcal{U}", bend left] & \mathsf{Alg} \arrow[l, "\text{Lie}(-)", bend left]
> \end{tikzcd}
> \end{document}
> ```
>
>
>
> This defines $\mathcal{U}(\mathfrak{g})$ up to [[isomorphism]], if it exists. Indeed, exist it does: it is instantiated by taking the [[tensor algebra]] $\mathbb{T}^{\bullet}(\mathfrak{g})$ and forcing its [[Lie algebra|Lie bracket]] (the [[commutator]]) to agree with $\mathfrak{g}