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• Conversely, suppose that the category A-mod is endowed with a braiding. Consider the
element
R := τ A,A (c A,A (1 A ⊗ 1 A )) ∈ A ⊗ A .
We have to show that R contains all information on the braiding on the category. To
this end, let V be an A-module; for any vector v ∈ V , consider the A-linear map which
realizes the isomorphism V ∼ = Hom A (A, V ):
v : A → V
a ↦→ av
Now consider two A-modules V, W and two vectors v ∈ V and w ∈ W . The naturality of
the braiding c applied to the morphism v ⊗ w implies
and thus
c V,W ◦ (v ⊗ w) = (w ⊗ v) ◦ c A,A
c V,W (v ⊗ w) = c V,W (v ⊗ w(1 A ⊗ 1 A ))
= (w ⊗ v)c A,A (1 A ⊗ 1 A ) [naturality, see above]
= τ V,W (v ⊗ w(R))
= τ V,W R.(v ⊗ w)
This shows that all the information on a braiding is contained in the element R ∈ A ⊗ A.
We have to derive the three relations on an R-matrix from the properties of a braiding.
We have for the action of any x ∈ A on c A,A (1 ⊗ 1) ∈ A ⊗ A:
x.c A,A (1 ⊗ 1) = x.τ A,A (R) = Δ(x) ∙ τ A,A (R)
On the other hand, the braiding c A,A is A-linear. Thus this expression equals
c A,A (x.1 ⊗ 1) = τ A,A R ∙ (Δ(x) ∙ 1 ⊗ 1) = τ A,A R ∙ Δ(x) .
Thus Δ(x) ∙ τ A,A (R) = τ A,A R ∙ Δ(x) which amounts to
RΔ(x) = τ A,A Δ(x)τ A,A (R) = Δ opp (x)R .
One can finally derive the two hexagon properties (10) and (11) an R-matrix from the
hexagon axioms for the braiding.
Let A be a quasi-triangular Hopf algebra. We conclude from proposition 4.2.1 that for any
A-module V , the automorphism
c R V,V : V ⊗ V → V ⊗ V
is a solution of the Yang-Baxter equation. This explains the name universal R-matrix. We note
some properties of this R-matrix.
Proposition 4.2.5.
Let (H, R) be a quasi-triangular bialgebra.
✷
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