Universal Enveloping Algebra

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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.