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 ) 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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