universal enveloping algebra

A universal enveloping algebra of a Lie algebraMathworldPlanetmath 𝔤 over a field k is an associative http://planetmath.org/node/AlgebraPlanetmathPlanetmathPlanetmathalgebra U (with unity) over k, together with a Lie algebra homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ι:𝔤U (where the Lie algebra structure on U is given by the commutatorPlanetmathPlanetmath), such that if A is a another associative algebra over k and ϕ:𝔤A is another Lie algebra homomorphism, then there exists a unique homomorphism ψ:UA of associative algebras such that the diagram