The construction of braided tensor categories from Hopf braces

ABSTRACT

Let $(H,\cdot ,\circ )$ be a Hopf brace. We first show that the category of left compatible modules over $(H,\cdot ,\circ )$ is a tensor category. Next, we show that its Drinfeld centre is equivalent to the category ${}_{\mathbb{H}}^{H^{\prime }}\mathcal{Y}\mathcal{D}$ of left Yetter–Drinfeld modules over $(H,\cdot ,\circ )$ as a braided tensor category. Finally, we show that Radford's biproduct $(R\mathrm{♯}H,\cdot )$ and $(R\mathrm{♯}{H}_{\circ },\circ )$ constitute a Hopf brace, where R is a Hopf algebra in ${}_{\mathbb{H}}^{H^{\prime }}\mathcal{Y}\mathcal{D}$. This presents some new method to construct Hopf braces.

Keywords:

• Hopf braces
• Yang–Baxter equation
• Braided tensor categories
• Yetter–Drinfeld modules
• Hopf algebras

2010 Mathematics Subject Classifications:

• 16T05
• 16T20
• 18M15

Acknowledgments

The author would like to thank some referee for their constructive suggestions and valuable comments, and Dr P. Stefanelli for sending me the paper [12]. This work was partially supported by the FNS of China (Grant No.11302066).

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by National Natural Science Foundation of China [grant number 11302066].