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1. The invertible element
Q := R 21 ◦ R 12 ∈ H ⊗ H
is called the monodromy element. We write Q = Q (1) ⊗ Q (2) and note that
for all h ∈ H.
2. The linear map
is called the Drinfeld map.
Δ(h) ∙ Q = Q ∙ Δ(h)
F R : H ∗ → H
φ ↦→ (id H ⊗ φ)(R 21 ∙ R 12 ) = (id H ⊗ φ)Q
3. A quasi-triangular Hopf algebra is called factorizable, if the Drinfeld map is an isomorphism
of vector spaces.
Remark 4.4.7.
The word factorizable is justified as follows: let (b i ) i∈I be a basis of H and (b i ) i∈I the dual basis
of H ∗ . If H is factorizable, then the vectors c i := F R (b i ) form another basis of H. We write
Q = ∑ i,j
λ i,j b i ⊗ c j
with λ i,j ∈ K. We then have
c k = F R (b k ) = ∑ i,j
λ i,j b k (b i ) ⊗ c k = ∑ j∈I
λ k,j c k
and thus for the monodromy matrix
Q = ∑ i∈I
b i ⊗ c i ,
which explains the word factorizable.
We consider the following subspace of H ∗ :
C(H) := {f ∈ H ∗ | f(xy) = f(yS 2 (x)) for all x, y ∈ H}
We call this subspace the space of central forms or the space of class functions or the
character algebra. We relate it to the center of H.
Lemma 4.4.8.
Let H be a finite-dimensional unimodular Hopf algebra with non-zero left cointegral λ ∈ H ∗ .
Then by theorem 3.1.13 the map
H → H ∗
a ↦→ λ(a ∙ −) = (λ ↼ a)
is a bijection. It restricts to a bijection Z(H) ∼ = C(H). In particular dim K Z(H) = dim K C(H).
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