Universal Enveloping Algebra - Universal Property

Universal Property

Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U_L, (notation as above) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A_L there exists a unique unital algebra homomorphism g: U → A such that: f(-) = g_L (h(-)).

This is the universal property expressing that the functor sending X to its universal enveloping algebra is left adjoint to the functor sending a unital associative algebra A to its Lie algebra A_L.