universal enveloping algebra

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