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H.
v
α (1) .x 1
v
B α (v,p) :
x n
x 1
x 2 p
↦→ α (2) .x 2 p
α (n) .x n
(x 1 ) (2)
v
= 〈α, S(x n ) (1) . . . (x 1 ) (1) )〉 p
(x 2 ) (2)
(x n ) (2)
We need the following
Lemma 5.5.15.
Let X be a representation of H, and Y a representation of H ∗ . For h ∈ H, α ∈ H ∗ , define the
endomorphisms p h , q α ∈ End(H ⊗ X ⊗ Y ⊗ H) by
p h (u ⊗ x ⊗ y ⊗ v) = h (1) u ⊗ h (2) x ⊗ y ⊗ vS(h (3) )
q α (u ⊗ x ⊗ y ⊗ v) = α (3) .u ⊗ x ⊗ α (2) .y ⊗ α (1) .v
Then these endomorphisms satisfy the straightening formula of D(H). Then the map
is a morphism of algebras.
D(H) → End(H ⊗ X ⊗ Y ⊗ H)
h ⊗ α ↦→ p h q α
Proof.
It is obvious that we have actions a ↦→ p a and α ↦→ p α of H and H ∗ . It remains to show that
these endomorphisms satisfy the straightening formula of D(H). This is done in a direct, but
tedious calculation in [BMCA, Lemma 1, Theorem 1].
✷
This allows us to show:
Theorem 5.5.16.
1. If v, w are distinct vertices of Δ, then the operators A a (v,p) , Ab (w,p ′ )
a, b ∈ H.
commute for any pair
2. Similarly, if p, q are distinct plaquettes, then the operators B(v,p) α , Bβ (v ′ ,q)
pair α, β ∈ H ∗ .
commute for any
3. If the sites are different, then the operators A h (v,p) and Bα (v ′ ,p ′ ) commute.
4. For a given site s = (v, p), the operators A h (v,p) and and Bα (v,p)
relations of the Drinfeld double Z(H): the map
satisfy the commutation
ρ s : D(R) → End(V (Σ, Δ)) (16)
a ⊗ α ↦→ A a (v,p)B α (v,p) (17)
is an algebra morphism.
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