Universal enveloping algebra

The idea of the universal enveloping algebra is to embed a Lie algebra $\mathfrak{g}$ into an associative algebra $\mathcal{A}$ with identity in such a way that the abstract bracket operation in $\mathfrak{g}$ corresponds to the commutator x y − y x in and the algebra is generated by the elements of . There may be many ways to make such an embedding, but there is a unique "largest" such , called the universal enveloping algebra of .