---
tag: SY22C7XVCW
kind: definition
name: Universal enveloping algebra
msc: [17B35, 16S30]
defines: SCZ8PRVY00
uses: [04C5ZBG1WT, P4J5JZDW0B, 0CD87SFM2N, 3CTW1P4HBP, PBGF9TXMB7]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_enveloping_algebra
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $\mathfrak g$ be a Lie algebra over $k$ with bracket $[\cdot, \cdot]$.
Let $T(\mathfrak g)$ be the [[tensor algebra|P4J5JZDW0B]] of the vector space $\mathfrak g$.
Let $I$ be the [[two-sided ideal|0CD87SFM2N]] of $T(\mathfrak g)$ generated by the elements $x \otimes y - y \otimes x - [x, y]$ for $x, y \in \mathfrak g$.
The *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 $T(\mathfrak g)$.

## Remarks

The map $\iota$ is linear and satisfies $\iota([x, y]) = \iota(x)\iota(y) - \iota(y)\iota(x)$.
Universal 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 $\tilde\varphi \circ \iota = \varphi$.
Hence representations of $\mathfrak g$ on a vector space $V$ are the same as left $U(\mathfrak g)$-module structures on $V$.
Example: if $\mathfrak g$ is abelian, all brackets are $0$, so $I$ is generated by the elements $x \otimes y - y \otimes x$.
Then $U(\mathfrak g)$ is the [[symmetric algebra|PBGF9TXMB7]] $S(\mathfrak g)$, a polynomial ring in a basis of $\mathfrak g$.
Non-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.