Lie Groups Beyond an Introduction pp 164-180 | Cite as

# 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.

## Keywords

Associative Algebra Algebra Homomorphism Symmetric Algebra Unital Left Complex Vector Space Versus## Preview

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