Universal Enveloping Algebra

Abstract

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

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

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