I have been recently studying different methods to construct Hopf algebras.

In Theorem IX.2.3 of "*Quantum groups*" by Kassel the **bicrossed product** of a pair of matched bialgebras (or Hopf algebras) $X$ and $A$ is defined as the vector space $X \otimes A$ with unit $1 \otimes 1$ and the following structure:
$$ (x \otimes a) (y \otimes b) = \sum_{(a) (y)} x \cdot \alpha(a', y') \otimes \beta(a^{''},y^{''}) \cdot b $$
$$ \Delta(x \otimes a) = \sum_{(a) (y)} (x' \otimes a') \otimes (x'' \otimes a'') $$
$$ \epsilon(x \otimes a) = \epsilon(x) \epsilon(a) $$
where $\alpha: A \otimes X \to X$ and $\beta: A \otimes X \to A$ are the corresponding actions. The bicrossed product is denoted by $X \bowtie A$.

In Theorem 6.2.2 of "*Foundations of quantum group theory*" by Majid the **bicrossproduct** is defined in an alternative way. Here, $X \bowtie A$ is $X \rtimes A$ as an algebra and $X \ltimes A$ as a coalgebra, where these are defined in Proposition 1.6.6. The product of $X \rtimes A$ and therefore the product of $X \bowtie A$ is defined as
$$ (x \otimes a) (y \otimes b) = \sum_{(a)} x \cdot \alpha(a',y) \otimes a'' b $$
The coproduct is defined dually and depending on $\beta$.

I thought these two constructions were the same but were just given slightly different names. However, if $X = k[G]$ and $A = k[H]$ are group algebras with non-trivial $\beta$ then the first construction yields the group algebra $k[G \bowtie H]$, but the second does not as it is not cocommutative in general.

Both books refer back to "*Matched pairs of groups and bismash products of Hopf algebras*" by Takeuchi. Here, **bismash products** are defined similarly to the bicrossedproduct by Majid (pp. 845-847).

**Question:** Are the *bismash product by Takeuchi* and the *bicrosspuct by Majid* the same construction, but different to the *bicrossed product by Kassel*?