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Proof.
Writing in a short form
R = ∑ i
(1 ⊗ e i ) ⊗ (e i ⊗ 1) = ∑ i
e i ⊗ e i
we find for the monodromy matrix of D(H)
Q = R 21 ∙ R 12 = ∑ i,j
(e i e j ) ⊗ (e i e j ) .
The family e i e j = e i ⊗ e j is a basis of D(H) = H ∗ ⊗ H. Moreover,
S(e i e j ) = S(e j ) ∙ S(e i ) = S(e j ) ⊗ S(e i )
and since S is invertible, also e i e j is a basis of D(H). Thus by remark 4.4.7 D(H) is factorizable.
✷
We can treat a left action of H ∗ for a right coaction of H by
Δ V :
V id V ⊗b
−→
H
V ⊗ H ∗ ⊗ H τ V,H ∗ ⊗id H
−→
H ∗ ⊗ V ⊗ H ρ⊗id H
−→ V ⊗ H
and conversely,
ρ V :
H ∗ ⊗ V id H ∗ ⊗Δ V
−→
H ∗ ⊗ V ⊗ H id H ∗ ⊗τ V,H
−→
H ∗ ⊗ H ⊗ V d H⊗id
−→
V
V
Put differently, we have
f.v = 〈f, v (H) 〉v (V ) .
The definition of the Drinfeld double D(H) has been made in such a way that the following
assertion holds:
Proposition 4.5.12.
Let H be a finite-dimensional Hopf algebra.
1. By treating the right H-coaction for a left H ∗ -action as above, any left D(H)-module
becomes a Yetter-Drinfeld module in H YD H .
2. Conversely, any Yetter-Drinfeld module in H YD H has a natural structure of a left module
over the Drinfeld double D(H).
(∗)
Proof.
We note that the structure of a left D(H)-module on a vector space V consists of the structure
of an H-module and of an H ∗ -module such that for all f ∈ H ∗ , h ∈ H and v ∈ V the following
consequence of the straightening formula holds:
a.(f.v) = f(S −1 (a (3) )?a (1) ). ( a (2) .v ) .
To show the second claim, we have to derive this relation from the Yetter-Drinfeld condition:
f(S −1 (a (3) ?a (1) ). ( a (2) .v ) = 〈f, S −1 (a (3) )(a (2) v) (H) a (1) 〉(a (2) v) (V ) [equation (∗)]
= 〈f, S −1 (a (3) )a (2) v (H) 〉a (1) v (V ) [YD condition]
= ɛ(a (2) )〈f, v (H) 〉a (1) v (V ) [lemma 2.5.7]
= 〈f, v (H) 〉av (V ) = a.(f.v) [equation (∗)]
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