Milnor–Moore theorem

In algebra, the Milnor–Moore theorem, introduced in (Milnor–Moore 1965), states: given a connected graded cocommutative Hopf algebra A over a field of characteristic zero with \dim A_n < \infty, the natural Hopf algebra homomorphism

U(P(A)) \to A

from the universal enveloping algebra of the "graded" Lie algebra P(A) of primitive elements of A to A is an isomorphism. (The universal enveloping algebra of a graded Lie algebra L is the quotient of the tensor algebra of L by the two-sided ideal generated by elements xy-yx - (-1)|x||y|[x, y].)

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