Hom-Structures And Rota-Baxter Operators On The Twisted Heisenberg Lie Algebras

The Hom-algebra structures arose first in quasi-deformation of Lie algebras of vector fields.Discrete modifications of vector fields via twisted derivations lead to Hom-Lie and quasi-Hom-Lie structures in which the Jacobi condition is twisted.Deformation theory of algebras is now one of the important branches of algebras theory.Hom-Lie algebras are considered to be a quantum deformation of Lie alge-bras.In recent years,Hom-structure on Lie algebras and Lie superalgebras become one of the important research topics in Lie theory and attracted more and more attention of scholars.In this paper,we compute all the multiplicative and the(not necessarily multiplicative)Hom-structures on the twisted Heisenberg Lie algebras by Hom-Jacobi identity.Rota-Baxter operators were introduced to solve certain analytic and combi-natorial problems and applied to many areas in mathematics and mathematical physics.For example,as a solution of the classical condition of Yang-Baxter equa-tion it is in the form of operators that appeared in the application of Lie algebras,and it plays an important role in the integration system.The Rota-Baxter opera-tors with any weight non-zero can be obtained by the Rota-Baxter operators with weight one,so only the Rota-Baxter operators with weight zero and one on the 4-dimensional twisted Heisenberg Lie algebras are described in this paper.