Universal Enveloping Algebra
The universal enveloping algebra of a free Lie algebra on a set X is the free associative algebra generated by X. By the Poincaré-Birkhoff-Witt theorem it is the "same size" as the symmetric algebra of the free Lie algebra (meaning that if both sides are graded by giving elements of X degree 1 then they are isomorphic as graded vector spaces). This can be used to describe the dimension of the piece of the free Lie algebra of any given degree.
Witt showed that the number of basic commutators of degree k in the free Lie algebra on an m-element set is given by the necklace polynomial:
where is the Möbius function.
The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra.
Read more about this topic: Free Lie Algebra
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