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(a) The functor F is essentially surjective, i.e. for any W ∈ Obj(D) there is V ∈ Obj(C) such
that F (V ) ∼ = W in D.
(b) The functor F is fully faithful: for any pair V, V ′ of objects in C, the maps
on Hom-spaces are bijective.
F : Hom C (V, V ′ ) → Hom D (F (V ), F (V ′ ))
Proof: see [Kassel, p. 278]. The proof uses the axiom of choice.
As a further consequence of the universal property of the enveloping algebra U(g), we get
from maps of Lie algebras maps of unital associative algebras:
g → K gives ɛ : U(g) → K
x ↦→ 0
g → g × g ⊂ U(g × g) gives Δ : U(g) → U(g × g) ∼ = U(g) ⊗ U(g)
x ↦→ (x, x)
g → g opp ⊂ U(g opp ) gives S : U(g) → U(g opp ) ∼ = U(g) opp
x ↦→ −x
These maps are explicitly on the generators x ∈ g ⊂ U(g)
ɛ(x) = 0
Δ(x) = 1 ⊗ x + x ⊗ 1
S(x) = −x
These maps allow us to endow tensor products of representations of g, the dual of a vector
space underlying a representation of g and the ground field K with the structure of g-
representations.
Observation 2.1.23.
• Let V, W be representations of g. The U(g)-module structure on the tensor product V ⊗W
is then obtained from
U(g)
Δ
−→ U(g) ⊗ U(g) ρ V ⊗ρ W
−→ End(V ) ⊗ End(W ) → End(V ⊗ W ) .
This is uniquely determined by the condition
x.(v ⊗ w) = x.v ⊗ w + v ⊗ x.w for all v ∈ V, w ∈ W and x ∈ g .
• The U(g)-module structure on the ground field K is obtained from
U(g)
ɛ
−→ K ∼ = End K (K) .
This is uniquely determined by the condition x.v = 0 for all x ∈ g and v ∈ K.
• The U(g)-module structure on V ∗ is then obtained via the transpose from
U(g)
S
−→ U(g) opp → ρt
End(V ∗ ) .
• Again, in physics, the representation on K is used to produce invariant states, and the
representation on V ⊗ W corresponds to the “coupling of two systems” for symmetries
leading to additive quantum numbers.
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