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Let be a Lie algebra. The universal enveloping algebra is the quotient of the tensor algebra by the ideal J generated by x ⊗ y — y ⊗ x — [x, y], .

LEMMA 1.1.– J is a Hopf ideal, that is, Δ(J) ⊂ ℋ ⊗ J + J ⊗ ℋ and S(J) ⊂ J.

PROOF.– The ideal J is generated by primitive elements (according to Proposition 1.5 below), and any ideal generated by primitive elements is a Hopf ideal (very easy and left to the reader). □

The quotient of a Hopf algebra by a Hopf ideal is a Hopf algebra. Hence, the universal enveloping algebra is a cocommutative Hopf algebra.