Universal enveloping algebra

Definition

Abstract category-theoretic definition

Note that for any commutative unital ring $R$, there is a forgetful functor between categories:

Associative unital $R$-algebras $\to$ $R$-Lie algebras

called the associated Lie ring functor: sends any associative unital algebra to the Lie algebra with the same $R$-module structure and where the Lie bracket is defined as the additive commutator, i.e., $[x,y]:=xy-yx$.

The universal enveloping algebra functor is the left adjoint functor for this forgetful functor.

Explicit category-theoretic definition

Suppose $R$ is a commutative unital ring and $L$ is a Lie algebra over $R$. The universal enveloping algebra of $L$ is the following data:

• An associative unital algebra over $R$, which we denote $U(L)$
• A set map $i:L\to U(L)$ that is a homomorphism of Lie algebras when $U(L)$ is viewed as a Lie algebra under the Lie bracket $[x,y]:=xy-yx$

such that the following holds: For any associative unital $R$-algebra $M$, and any homomorphism of $R$-Lie algebras $f:L\to M$ where $M$ is given a Lie bracket by $[x,y]:=xy-yx$, there is a unique associative unital $R$-algebra homomorphism $g:U(L)\to M$ such that $f=g\circ i$.

Note that with the abuse of notation, we typically simply define $U(L)$ to be the universal enveloping algebra and treat the map from $L$ to $U(L)$ as implicitly specified.

Explicit definition in terms of tensor algebra

Suppose $R$ is a commutative unital ring and $L$ is a Lie algebra over $R$. The universal enveloping algebra of $L$, denoted $U(L)$, is constructed as follows:

• Denote by $T(L)$ the tensor algebra on $L$ as a $R$-module.
• Take the quotient of $T(L)$ by all relations of the form $xy-yx-[x,y]$ for $x,y$ arising from the degree one component of $T(L)$, which is identified with $L$.

=Particular cases

• The abelian Lie algebra case: Universal enveloping algebra for abelian Lie algebra equals symmetric algebra on underlying module. This is because, in the explicit definition above, the relations by which we are quotienting out the tensor algebra are precisely the defining relations for the symmetric algebra.

Notes

Poincare-Birkhoff-Witt theorem

The Poincare-Birkhoff-Witt theorem says that the canonical mapping from a Lie algebra to its universal enveloping algebra is always injective. Thus, we often think of the Lie algebras as a subalgebra in its universal enveloping algebra via identification with its image.

Universal enveloping algebras are quite large

Contrary to what somebody not familiar with adjoint functors might expect, the universal enveloping algebra is not the smallest associative algebra containing the Lie algebra. In fact, it is likely to be much larger. Universal enveloping algebras over nonzero finite-dimensional Lie algebras are infinite-dimensional, even though many of these finite-dimensional Lie algebras embed inside finite-dimensional matrix algebras, which can be viewed as associative algebras.

The correct intuition is that the universal enveloping algebra is the 'freest associative unital algebra generated by the Lie algebra, so we consider elements as distinct as we can given the Lie algebra relations already between them.