Purely Hom-Lie bialgebras

Abstract

In this paper, first we show that there is a Hom-Lie algebra structure on the set of $(\sigma ,\sigma )$-derivations of a commutative algebra. Then we construct dual representations of a representation of a Hom-Lie algebra. We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-$\mathcal{O}$-operators.

18 pages, to appear in Sci. China Math

References in corpus (1)

• Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras

Cited by in corpus (5)

• $\mathcal{O}$-Operators on Hom-Lie algebras
• On Hom-Gerstenhaber algebras and Hom-Lie algebroids
• Another approach to Hom-Lie bialgebras via Manin triples
• Sweedler duality for Hom-(co)algebras and Hom-(co)modules
• Weighted $\mathcal{O}$-operators on Hom-Lie triple systems