Universal Property
Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → UL, (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 → AL there exists a unique unital algebra homomorphism g: U → A such that: f(-) = gL (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 AL.
Read more about this topic: Universal Enveloping Algebra
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