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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