[[Lie algebras MOC]]
# Universal enveloping algebra

Let $\mathfrak{g}$ be a [[Lie algebra]] over $\mathbb{K}$.
The **universal enveloping algebra** $U(\mathfrak{g})$ is the most general [[K-monoid]] with the Lie bracket of $\mathfrak{g}$ as its [[commutator]], as formalized by the [[#Universal property]] and the [[Poincaré-Birkhoff-Witt theorem]].
In particular, this means any [[Lie algebra representation]] of $\mathfrak{g}$ uniquely corresponds to a $U(\mathfrak{g})$-[[Module over a unital associative algebra|module]], motivating the abuse of terminology [[module over a Lie algebra]].

## Universal property

Let $\mathfrak{g}$ be a [[Lie algebra]] over $\mathbb{K}$.
The **universal enveloping algebra** is a pair consisting of a [[K-monoid]] $U(\mathfrak{g})$ and a [[Lie algebra homomorphism]] $\iota : \mathfrak{g} \to U(\mathfrak{g})$[^comm]
such that given any [[K-monoid]] $A$ and Lie algebra homomorphism $f : \mathfrak{g} \to A$,
there exists a unique [[Algebra homomorphism|unital algebra homomorphism]] $\bar{f}: U(\mathfrak{g} ) \to A$ such that the following diagram commutes: #m/def/lie 

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$U : \cat{Lie}_{\mathbb{K}} \to \cat{AsAlg}_{\mathbb{K}}$ has a unique extension to a [[functor]] such that $\iota : 1 \Rightarrow U : \cat{Lie}_{\mathbb{K}} \to \cat{Lie}_{\mathbb{K}}$ becomes a [[natural transformation]].

It is not immediately clear from the universal property that $\iota$ should be an injection,
but this is guaranteed by the [[Poincaré-Birkhoff-Witt theorem]],
so indeed $U(\mathfrak{g})$ contains $\mathfrak{g}$ as a [[Lie subalgebra]],
whence every Lie algebra is a Lie subalgebra of some unital associative algebra.

## Construction

Let $T^\bullet \mathfrak{g}$ be the [[tensor algebra]] of $\mathfrak{g}$ with inclusion $j : \mathfrak{g} \hookrightarrow T^\bullet\mathfrak{g}$ and let $I$ be the (two-sided) [[Algebra ideal|ideal]] generated by any terms of the form
$$
\begin{align*}
x \otimes y - y \otimes x - [x,y]
\end{align*}
$$
for $x,y \in \mathfrak{g}$.
We construct the universal enveloping algebra as the quotient module
$$
\begin{align*}
U(\mathfrak{g}) = T^\bullet \mathfrak{g} / I
\end{align*}
$$
with its natural projection $\pi : T^\bullet V \twoheadrightarrow U(\mathfrak{g})$.
The map $\iota = \pi \circ j$.

> [!missing]- Proof
> #missing/proof

  [^comm]: As usual we regard an associative algebra as a Lie algebra under its [[commutator]].

## Graded structure

Let $\mathfrak{g}$ be a $\mathfrak{A}$-[[graded Lie algebra]].
Then $U(\mathfrak{g})$ is a [[graded algebra]] such that $\mathfrak{g}_{\alpha} \mathfrak{g}_{\beta} \leq \mathfrak{g}_{\alpha + \beta}$.
This is the same as the gradation given by the [[quotient graded algebra]] in the construction above.

## Filtered structure

#to/complete 

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