# Bicrossed and bismash product of Hopf algebras

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?

The two construction are similar but different. In the former, we deform the algebra structure with the two actions but preserve the coalgebra structure. For instance, if $$G$$ and $$H$$ are groups we have that $$\mathbb{C}[G \bowtie H] \cong \mathbb{C}[G] \bowtie \mathbb{C}[H]$$. In the second construction, we deform both, the algebra and the coalgebra structure and the previous isomorphism is no longer true. I suspect that Kassel defines $$X \bowtie A$$ as $$X^* \bowtie A$$ using Majid's and Takeuchi's products.