A construction of the Hom-Yetter–Drinfeld category
Volume 137 / 2014
Colloquium Mathematicum 137 (2014), 43-65
MSC: 16T05, 81R50.
DOI: 10.4064/cm137-1-4
Abstract
In continuation of our recent work about smash product Hom-Hopf algebras [Colloq. Math. 134 (2014)], we introduce the Hom-Yetter–Drinfeld category $_H^H{\mathbb {YD}}$ via the Radford biproduct Hom-Hopf algebra, and prove that Hom-Yetter–Drinfeld modules can provide solutions of the Hom-Yang–Baxter equation and $_H^H{\mathbb {YD}}$ is a pre-braided tensor category, where $(H, \beta , S)$ is a Hom-Hopf algebra. Furthermore, we show that $(A^{\natural }_{\diamond } H,\alpha \otimes \beta )$ is a Radford biproduct Hom-Hopf algebra if and only if $(A,\alpha )$ is a Hom-Hopf algebra in the category $_H^H{\mathbb {YD}}$. Finally, some examples and applications are given.
