Chapter III. Universal Enveloping Algebra

Abstract

For a complex Lie algebra $\mathfrak g$, the universal enveloping algebra $U(\mathfrak g)$ is an explicit complex associative algebra with identity having the property that any Lie algebra homomorphism of $\mathfrak g$ into an associative algebra $A$ with identity “extends” to an associative algebra homomorphism of $U(\mathfrak g)$ into $A$ and carrying 1 to 1. The algebra $U(\mathfrak g)$ is a quotient of the tensor algebra $T(\mathfrak g)$ and is a filtered algebra as a consequence of this property. The Poincaré-Birkhoff-Witt Theorem gives a vector-space basis of $U(\mathfrak g)$ in terms of an ordered basis of $\mathfrak g$.

One consequence of this theorem is to identify the associated graded algebra for $U(\mathfrak g)$ as canonically isomorphic to the symmetric algebra $S(\mathfrak g)$. This identification allows the construction of a vector-space isomorphism called “symmetrization” from $S(\mathfrak g)$ onto $U(\mathfrak g)$. When $\mathfrak g$ is a direct sum of subspaces, the symmetrization mapping exhibits $U(\mathfrak g)$ canonically as a tensor product.

Another consequence of the Poincaré-Birkhoff-Witt Theorem is the existence of a free Lie algebra on any set $X$. This is a Lie algebra $\mathfrak F$ with the property that any function from $X$ into a Lie algebra extends uniquely to a Lie algebra homomorphism of $\mathfrak F$ into the Lie algebra.