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:
[1] | Caenepeel, S.; Goyvaerts, I., Monoidal Hom-Hopf algebras, Commun. Algebra, 39, 6, 2216-2240 (2011) · Zbl 1255.16032 · doi:10.1080/00927872.2010.490800 |
[2] | Chen, Y. Y.; Wang, Z. W.; Zhang, L. Y., Quasi-triangular Hom-Lie bialgebras, J. Lie Theory, 22, 4, 1075-1089 (2012) · Zbl 1272.17026 |
[3] | Chen, Y. Y.; Zhang, L. Y., The category of Yetter-Drinfel”d Hom-modules and the quantum Hom-Yang-Baxter equation, J. Math. Phys., 55, 3, 031702 (2014) · Zbl 1292.16022 · doi:10.1063/1.4868964 |
[4] | Cohen, M.; Westreich, S., Determinants and symmetries in Yetter-Drinfeld categories, Appl. Categ. Struct., 6, 2, 267-289 (1998) · Zbl 0938.16032 · doi:10.1023/A:1008668314522 |
[5] | Cohen, M.; Westreich, S.; Zhu, S. L., Determinants, integrality and Noether”s theorem for quantum commutative algebras, Isr. J. Math., 96, 1, 185-222 (1996) · Zbl 0878.16019 · doi:10.1007/BF02785538 |
[6] | Gohr, A., On Hom-algebras with surjective twisting, J. Algebra, 324, 7, 1483-1491 (2010) · Zbl 1236.17003 · doi:10.1016/j.jalgebra.2010.05.003 |
[7] | Joyal, A.; Street, R., Braided tensor categories, Adv. Math., 102, 1, 20-78 (1993) · Zbl 0817.18007 · doi:10.1006/aima.1993.1055 |
[8] | Liu, L.; Sheng, B. L., Radford”s biproducts and Yetter-Drinfeld modules for monoidal Hom-Hopf algebras, J. Math. Phys., 55, 3, 031701 (2014) · Zbl 1306.16032 · doi:10.1063/1.4866760 |
[9] | Makhlouf, A.; Silvestrov, S., Hom-algebra structures, J. Gen. Lie Theory, 3, 2, 51-64 (2008) · Zbl 1184.17002 |
[10] | Makhlouf, A.; Silvestrov, S., Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, Journal of Generalized Lie Theory in Mathematics, Physics and Beyond, 189-206 (2009) · Zbl 1173.16019 |
[11] | Makhlouf, A.; Silvestrov, S., Hom-algebras and Hom-coalgebras, J. Algebra Appl., 09, 4, 553-589 (2010) · Zbl 1259.16041 · doi:10.1142/S0219498810004117 |
[12] | Makhlouf, A.; Panaite, F., Yetter-Drinfeld modules for Hom-bialgebras, J. Math. Phys., 55, 1, 013501 (2014) · Zbl 1292.16025 · doi:10.1063/1.4858875 |
[13] | Panaite, F.; Staic, M. D.; Oystaeyen, F. V., Pseudosymmetric braidings, twines and twisted algebras, J. Pure Appl. Algebra, 214, 6, 867-884 (2010) · Zbl 1207.16037 · doi:10.1016/j.jpaa.2009.08.008 |
[14] | Pareigis, B., Symmetric Yetter-Drinfeld categories are trivial, J. Pure Appl. Algebra, 155, 1, 91-91 (2001) · Zbl 0976.16030 · doi:10.1016/S0022-4049(99)00089-4 |
[15] | Sweedler, M. E., Hopf Algebras (1969) · Zbl 0194.32901 |
[16] | Wang, Z. W.; Chen, Y. Y.; Zhang, L. Y., The antipode and Drinfel”d double of Hom-Hopf algebras, Sci. Sin. Math., 42, 11, 1079-1093 (2012) · Zbl 1488.16090 · doi:10.1360/012011-138 |
[17] | Yau, D., The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras, J. Phys. A, 42, 16, 165202 (2009) · Zbl 1179.17001 · doi:10.1088/1751-8113/42/16/165202 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.