universal enveloping algebra

A universal enveloping algebra of a Lie algebra $𝔤$ over a field $k$ is an associative http://planetmath.org/node/Algebraalgebra $U$ (with unity) over $k$, together with a Lie algebra homomorphism $\iota :𝔤\to U$ (where the Lie algebra structure on $U$ is given by the commutator), such that if $A$ is a another associative algebra over $k$ and $\varphi :𝔤\to A$ is another Lie algebra homomorphism, then there exists a unique homomorphism $\psi :U\to A$ of associative algebras such that the diagram