# universal enveloping algebra

A universal enveloping algebra of a Lie algebra^{} $\U0001d524$ over
a field $k$ is an associative http://planetmath.org/node/Algebra^{}algebra $U$ (with unity) over $k$, together
with a Lie algebra homomorphism^{} $\iota :\U0001d524\to U$ (where
the Lie algebra structure on $U$ is given by the commutator^{}), such that
if $A$ is a another associative algebra over $k$ and $\varphi :\U0001d524\to A$ is another Lie algebra homomorphism, then there exists a
unique homomorphism $\psi :U\to A$ of associative algebras such
that the diagram