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Crossed Modules and Quantum Groups in Braided Categories

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Abstract

Let A be a Hopf algebra in a braided category \(\mathcal{C}\). Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category \(\mathcal{D}\mathcal{Y}\left( \mathcal{C} \right)_A^A \) of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group \(\left( {A,\bar A,\mathcal{R}} \right)\) the corresponding braided category of modules \(\mathcal{C}_{\mathcal{O}\left( {A,\bar A} \right)} \) is identified with a full subcategory in \(\mathcal{D}\mathcal{Y}\left( \mathcal{C} \right)_A^A \). The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.

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  1. Institute for Theoretical Physics, Metrologichna str., 14-b, Kiev 143, 252143, Ukraine. e-mail

    Yu. N. Bespalov

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Bespalov, Y.N. Crossed Modules and Quantum Groups in Braided Categories. Applied Categorical Structures 5, 155–204 (1997). https://doi.org/10.1023/A:1008674524341

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