Symmetries and the \(u\)-condition in Hom-Yetter-Drinfeld categories. (English) Zbl 1306.16034
Summary: Let \((H,S,\alpha)\) be a monoidal Hom-Hopf algebra and \(^H_H\mathcal{HYD}\) the Hom-Yetter-Drinfeld category over \((H,\alpha)\). Then in this paper, we first find sufficient and necessary conditions for \(^H_H\mathcal{HYD}\) to be symmetric and pseudosymmetric, respectively. Second, we study the \(u\)-condition in \(^H_H\mathcal{HYD}\) and show that the Hom-Yetter-Drinfeld module \((H,\text{adjoint},\Delta,\alpha)\) (resp., \((H,m,\text{coadjoint},\alpha)\)) satisfies the \(u\)-condition if and only if \(S^2=id\). Finally, we prove that \(^H_H\mathcal{HYD}\) over a triangular (resp., cotriangular) Hom-Hopf algebra contains a rich symmetric subcategory.{
©2014 American Institute of Physics}
©2014 American Institute of Physics}
MSC:
| 16T05 | Hopf algebras and their applications |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 17A30 | Nonassociative algebras satisfying other identities |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
Keywords:
monoidal Hom-Hopf algebras; Hom-Yetter-Drinfeld modules; Hom-Yetter-Drinfeld categories; braided monoidal categories; symmetric categories; quasitriangular Hopf algebras; coquasitriangular Hopf algebrasReferences:
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