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while for the right hand side, we find by an analogous computation
(id ⊗ Δ F )Δ F (a) = (id ⊗ Δ F )(F Δ(a)F −1 )
= (id ⊗ Δ F )(F ) ∙ F 23 ∙ (id ⊗ Δ)Δ(a) ∙ F −1
23 ∙ (id ⊗ Δ F )(F −1 )
A sufficient condition for coassociativity to hold is the identity
(Δ F ⊗ id)(F ) ∙ F 12 = (id ⊗ Δ F )(F ) ∙ F 23
which, by taking inverses in the algebra H ⊗3 , implies
and which is equivalent to
F −1
12 (Δ ⊗ id)(F −1 ) = F −1
23 (id ⊗ Δ F )(F −1 )
F 12 (Δ ⊗ id H )(F ) = F 23 (id H ⊗ Δ)(F ) .
2. We leave the rest of the proofs to the reader.
✷
Definition 4.5.17
Let H be a Hopf algebra with invertible antipode. An invertible element F ∈ H ⊗ H satisfying
F 12 (Δ ⊗ id)(F ) = F 23 (id ⊗ Δ)(F )
(id ⊗ ɛ)(F ) = (ɛ ⊗ id)(F ) = 1
is called a 2-cocycle for H or a gauge transformation. We denote the twisted Hopf algebra with
coproduct Δ F , counit ɛ and antipode S F by H F .
Examples 4.5.18.
1. Let H be a finite-dimensional Hopf algebra with basis {e i } and dual basis {e i }. Consider
˜H := H ∗ ⊗ H opp
and in ˜H ⊗ ˜H the basis-independent element
˜F =
dim
∑H
i=1
(1 H ∗ ⊗ e i ) ⊗ (e i ⊗ 1 H )
A direct calculation shows that this element is a 2-cocycle for ˜H.
2. Let (H, R) be a finite-dimensional quasi-triangular Hopf algebra. Then
F R := 1 ⊗ R 2 ⊗ R 1 ⊗ 1
is a 2-cocycle for H ⊗ H with the tensor product Hopf algebra structure. For a proof, we
refer to [S, Theorem 4.3].
The proof of the following theorem can be found in [S, Theorem 4.3]:
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