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If the braided category is not strict, this amounts to a commuting diagram with 12 corners, a
dodecagon. The reader should draw the graphical representation of this identity.
In particular c V,V ∈ Aut (V ⊗V ) is a solution of the Yang-Baxter equation. Thus any object
V of a braided tensor category provides a group homomorphism B n → Aut(V ⊗n ).
Proof.
The equality is a direct consequence of the hexagon axiom from definition 4.1.1 and the functoriality
of the braiding:
(c V,W ⊗ id U ) ◦ (id V ⊗ c U,W ) ◦ (c U,V ⊗ id W )
= (c V,W ⊗ id U ) ◦ c U,V ⊗W [Hexagon axiom]
= c U,W ⊗V ◦ (id U ⊗ c V,W ) [Functoriality, applied to c U,V ⊗W ]
= (id W ⊗ c U,V ) ◦ (c U,W ⊗ id V ) ◦ (id U ⊗ c V,W ) [Hexagon axiom]
It is an obvious question to ask what kind of structure on a Hopf algebra induces the
structure of a braiding on its representation category.
Definition 4.2.2
1. Let H be a bialgebra. The structure of a quasi-cocommutative bialgebra is the choice of
an invertible element R in the algebra H ⊗ H such that for all x ∈ H
Δ opp (x)R = RΔ(x) . (9)
R is called a universal R-matrix. A quasi-triangular Hopf algebra is a Hopf algebra together
with the choice of a universal R-matrix.
2. A quasi-cocommutative bialgebra H is called quasi-triangular, if its universal R-matrix
obeys the relations in H ⊗3 (Δ ⊗ id H )(R) = R 13 ∙ R 23 (10)
✷
(id H ⊗ Δ)(R) = R 13 ∙ R 12 (11)
with
R 12 := R ⊗ 1, R 23 := 1 ⊗ R and R 13 := (τ H,H ⊗ id H )(1 ⊗ R) .
3. A morphism f : (H, R) → (H ′ , R ′ ) of quasi-triangular Hopf algebras is a morphism
f : H → H ′ of Hopf algebras such that R ′ = (f ⊗ f)(R).
Remarks 4.2.3.
1. In Sweedler notation R = R (1) ⊗ R (2) , the relations read
x (2) R (1) ⊗ x (1) R (2) = R (1) x (1) ⊗ R (2) x (2)
(R (1) ) (1) ⊗ (R (1) ) (2) ⊗ R (2) = R (1) ⊗ R ′ (1) ⊗ R (2)R ′ (2)
R (1) ⊗ (R (2) ) (1) ⊗ (R (2) ) (2) = R (1) R ′ (1) ⊗ R′ (2) ⊗ R (2)
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