1.3. ENVELOPING ALGEBRAS 5

Proposition 1.10. Let g − Mod be the category of representations of a Lie

algebra g, and let U(g) − Mod be the category of representations of the enveloping

algebra U(g). We define functors F, G as follows.

F : g − Mod −→ U(g) − Mod

(ρ, V ) → (φ, V )

f ∈ Homg(V1,V2) → f ∈ HomU(g)(V1,V2)

G : U(g) − Mod −→ g − Mod

(φ, V ) → (φ ◦ ι, V )

f ∈ HomU(g)(V1,V2) → f ∈ Homg(V1,V2)

Then F, G give isomorphisms of categories.

By virtue of the proposition, we have

• V is a submodule of W as a g-module if and only if V is a submodule of W

as a U(g)-module.

• V is a quotient module of W as a g-module if and only if V is a quotient

module of W as a U(g)-module.

• 0 → U → V → W → 0 is an exact sequence of g-modules if and only if it is

an exact sequence of U(g)-modules.

We have another important operation in g − Mod.

Definition 1.11. Let (ρ1,V1), (ρ2,V2) be representations of g. Then we can

make V1 ⊗ V2 into a representation (ρ1 ⊗ ρ2,V1 ⊗ V2) of g via

(ρ1 ⊗ ρ2)(X) = ρ1(X) ⊗ 1 + 1 ⊗ ρ2(X).

This representation is called the tensor product representation of V1 and V2.

If we consider the extension of the tensor product representation to a rep-

resentation of U(g), it is the tensor product representation of U(g) in the usual

sense. That is, if we denote by φi (i = 1, 2) the representations of U(g) which are

extensions of ρi respectively, then the extension of ρ1 ⊗ ρ2 is φ1 ⊗ φ2.

To explain this, we start with the warning that not all categories A − mod have

tensor product representations. To have this operation, the algebra A needs to be

equipped with an algebra homomorphism Δ : A → A ⊗ A. In these cases, we may

consider the tensor product M1 ⊗ M2 of two A-modules M1,M2 as an A-module

via a · (m1

⊗ m2) = Δ(a)(m1 ⊗ m2).

In the case of the enveloping algebra, the map Δ defined by the following

proposition induces the tensor product representations.

Proposition 1.12. There exists a unique algebra homomorphism

Δ : U(g) → U(g) ⊗ U(g)

satisfying

Δ ◦ ι(X) = ι(X) ⊗ 1 + 1 ⊗ ι(X) (X ∈ g),

such that for any tensor product representation ρ1 ⊗ 1+1 ⊗ ρ2 of g, its extension to

U(g) is given by (φ1 ⊗ φ2) ◦ Δ. In other words, we have the following commutative

diagram.