Journal of Lie Theory 22 (2012), No. 4, 1075--1089
Copyright Heldermann Verlag 2012
Quasi-triangular Hom-Lie Bialgebras
Yuanyuan Chen
College of Science, Nanjing Agricultural University, Nanjing 210095, P. R. China
Zhongwei Wang
College of Science, Nanjing Agricultural University, Nanjing 210095, P. R. China
Liangyun Zhang
College of Science, Nanjing Agricultural University, Nanjing 210095, P. R. China
zlyun@njau.edu.cn
Recently certain twisted Lie algebras, so-called Hom-Lie algebras, and their duals have been considered in the literature. In this paper we investigate boundary and quasi-triangular Hom-Lie bialgebras further. In particular, we characterize the quasi-triangularity of boundary Hom-Lie bialgebras in terms of both a certain Hom-Lie algebra morphism and a certain Hom-Lie coalgebra morphism. We also give a necessary and sufficient condition for a given Hom-Lie algebra and a given 2-tensor to admit a coboundary Hom-Lie bialgebra structure. Finally, we generalize the Drinfeld double of a Lie bialgebra to Hom-Lie bialgebras and discuss the dual codouble.
Keywords: Hom-Lie algebra, Hom-Lie bialgebra, quasi-triangular Hom-Lie bialgebra, (co)double Hom-Lie bialgebra.
MSC: 16W30, 17B99, 17B37
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