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Bicrossed products with the Taft algebra. (English) Zbl 1447.16025

Let \(A\) and \(B\) be two Hopf algebras over a field \(K\). A bicrossed product of \(A\) and \(B\) is any Hopf algebra structure on \(A\otimes B\), equal to \(A\otimes B\) as a coalgebra, and such that \(A\otimes K\) and \(K\otimes B\) are subalgebras, trivially isomorphic to \(A\) and \(B\). When \(A\) and \(B\) are given, all bicrossed products of \(A\) and \(B\) can be described in terms of actions of \(A\) over \(B\) and of \(B\) over \(A\). When one of these actions is trivial, the bicrossed product is said to be a smash product.
In this paper, bicrossed product of the Taft algebra \(T_{m^2}(q)\) and of a group algebra \(K[G]\) are studied, where \(q\) is a \(m\)-th primitive root of unity and \(G\) is a group which is generated by elements of finite order. It is proved that any bicrossed product between \(T_{m^2}(q)\) and \(K[G]\) is in fact a smash product. All bicrossed products are then given when \(G\) belongs to a family of semidirect products of cyclic groups, including the dihedral groups.
Generally, the classification of these smash products is strongly related to the automorphism group of \(G\). This classification is done when \(G\) is a dihedral group \(D_{2n}\), leading to arithmetical considerations on \(m\) and \(n\).

MSC:

16T05 Hopf algebras and their applications
16S40 Smash products of general Hopf actions
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References:

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