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More examples of bicrossproduct and double cross product Hopf algebras. (English) Zbl 0725.17015

The author had previously defined bicrossproducts and double cross products of Hopf algebras. Here he constructs examples where the factors are certain quantum groups coming from solutions of the quantum Yang-Baxter equations. He also describes iterated double cross products of quantum groups. These particular quantum groups are constructed using a suitable notion of mutually dual Hopf algebras and a certain dual quantum group to the Hopf algebra given directly by a solution of the quantum Yang-Baxter equation. This dual pairing leads to the notion of a weak Hopf algebra with weak antipode. The algebraic ideas here are motivated by the author’s approach to quantum mechanics combined with gravity.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B38 Yang-Baxter equations and Rota-Baxter operators
16T20 Ring-theoretic aspects of quantum groups
16T25 Yang-Baxter equations
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