A universal enveloping algebra of a Lie algebra $\mathfrak{g}$ over a field $k$ is an associative algebra $U$ (with unity) over $k$, together with a Lie algebra homomorphism $\iota:\mathfrak{g}\rightarrow U$ (where the Lie algebra structure on $U$ is given by the commutator), such that if $A$ is a another associative algebra over $k$ and $\phi:\mathfrak{g}\rightarrow A$ is another Lie algebra homomorphism, then there exists a unique homomorphism $\psi:U\rightarrow A$ of associative algebras such that the diagram

commutes. Any $\mathfrak{g}$ has a universal enveloping algebra: let $T$ be the associative tensor algebra generated by the vector space $\mathfrak{g}$, and let $I$ be the two-sided ideal of $T$ generated by elements of the form

then $U=T/I$ is a universal enveloping algebra of $\mathfrak{g}$. Moreover, the universal property above ensures that all universal enveloping algebras of $\mathfrak{g}$ are canonically isomorphic; this justifies the standard notation $U(\mathfrak{g})$.

Some remarks:

1. 1.

   By the Poincaré-Birkhoff-Witt theorem, the map $\iota$ is injective; usually $\mathfrak{g}$ is identified with $\iota(\mathfrak{g})$. From the construction above it is clear that this space generates $U(\mathfrak{g})$ as an associative algebra with unity.

2. 2.

   By definition, the (left) representation theory of $U(\mathfrak{g})$ is identical to that of $\mathfrak{g}$. In particular, any irreducible $\mathfrak{g}$-module corresponds to a maximal left ideal of $U(\mathfrak{g})$.

Example: let $\mathfrak{g}$ be the Lie algebra generated by the elements $p,q,$ and $e$ with Lie bracket determined by $[p,q]=e$ and $[p,e]=[q,e]=0$. Then $U(g)/(e-1)$ (where $(e-1)$ denotes the two-sided ideal generated by $e-1$) is isomorphic to the skew polynomial algebra $k[x,\frac{\partial}{\partial x}]$, the isomorphism being determined by