In the theory of Lie algebras, we have the notion of representation of a Lie algebra \(\mathfrak{g}\), which consists of a vector space \(V\), and a morphism of Lie algebras \(\mathfrak{g}\rightarrow \mathfrak{gl}(V)\).
We are so used to thinking of \(\mathfrak{gl}(V)\) as a Lie algebra, that, we might not remember that the underlying set \(End(V)\) has a structure of an associative algebra, and that we made the underlying set into a Lie algebra by considering the binary operation \([f,g]=fg-gf\) for \(f,g\in End(V)\). This is where the notion of enveloping algebra comes into picture.
Given a Lie algebra \(\mathfrak{g}\), an enveloping algebra of \(\mathfrak{g}\) consists of
- an associative algebra \(A\)
- a morphism of Lie algebras \(\mathfrak{g}\rightarrow A\), where we see \(A\) as a Lie algebra with the Lie bracket \([a,b]=ab-ba\).
With that definition, any representation \((V,\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(V))\) of a Lie algebra \(\mathfrak{g}\) gives an enveloping algebra \(End(V)\) for \(\mathfrak{g}\).
Once we have a notion of a structure, we would ask for a "universal property"; the best among the possibilities. We have seen such "universal" ideas before, for example universal covering space of a topological space, abelianization of a group, tensor product of two \(R\)-modules.
In the case of enveloping algebras, one can ask a similar question.
Given a Lie algebra \(\mathfrak{g}\), can there be a "best" associative algebra \(A\), along with a morphism of Lie algebras \(\Phi:\mathfrak{g}\rightarrow A\) such that, for any other associative algebra \(B\) and a morphism of Lie algebras \(\Psi:\mathfrak{g}\rightarrow B\), there exists a unique morphism of associative algebras/Lie algebras \(\theta:A\rightarrow B\) such that \(\theta\circ \Phi=\Psi\), as in the diagram below,
Such enveloping algebra is called the universal enveloping algebra of \(\mathfrak{g}\).
In case of representations of Lie algebras, we have seen some notion of "best" but it is not unique; the notion of irreducible representation. Given a Lie algebra \(\mathfrak{g}\), there may be many irreducible representations of same/different dimensions.
Question :
- Does an irreducible representation of a Lie algebra \(\mathfrak{g}\) help as a starting point to understand universal enveloping algebra of \(\mathfrak{g}\)?
- Given a Lie algebra \(\mathfrak{g}\), is it possible for universal enveloping algebra of \(\mathfrak{g}\) to be of the form \(End(V)\) for some vector space \(V\)?
- Are there any other notions in representation theory of Lie algebras that are related to the notion of universal enveloping algebra?