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.

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