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arXiv · 1011.2174

Extending Structures II: The Quantum Version

A. L. Agore·G. Militaru
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Abstract

Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up to an isomorphism all Hopf algebras E that factorize through A and H: that is E is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in E with 1_{E} \in H and the multiplication map $A\otimes H \to E$ is bijective. The tool we use is a new product, we call it the unified product, in the construction of which A and H are connected by three coalgebra maps: two actions and a generalized cocycle. Both the crossed product of an Hopf algebra acting on an algebra and the bicrossed product of two Hopf algebras are special cases of the unified product. A Hopf algebra E factorizes through A and H if and only if E is isomorphic to a unified product of A and H. All such Hopf algebras E are classified up to an isomorphism that stabilizes A and H by a Schreier type classification theorem. A coalgebra version of lazy 1-cocycles as defined by Bichon and Kassel plays the key role in the classification theorem.

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A. L. Agore, G. Militaru. 2010-11-09. Extending Structures II: The Quantum Version. https://arxiv.org/abs/1011.2174

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