Hom-Jordan Bialgebras And Six-dimensional Phase Spaces

Jordan algebra is a very important class of non-associative algebra,the notion of Jordan algebra is first given by physicist P.Jordan in the 1930 ' s.Jordan algebra is widely used in the field of mathematics and quantum mechanics.In recent years,Hom-algebra has become a hot spot in research,and has attracted extensive attention.Left-symmetric algebra is a very important class of non-associative algebra.Hom-Jordan algebra can be obtained by the algebraic deformation of Jordan algebra.S-equation is an analogue of the Classical YangBaxter equation,we can obtain the 1-coboundary left-symmetric bialgebra by using the solution of S-equation.Phase space is isomorphic to the left-symmetric bialgebra.This paper mainly studies the construction of Hom-Jordan bialgebras and six-dimensional phase spaces corresponding to the left-symmetric algebras on Heisenberg algebra.This paper is constructed as follows:The first part will introduce some basic theories,including the definition of Jordan algebra,the definition of left-symmetric algebras and left-symmetric bialgebra,S-equation on the left-symmetric algebra,the classification of the left-symmetric algebra on Heisenberg algebra and so on.The second part will give the definition of the representation of Hom-Jordan algebra.We find a method of constructing Hom-Jordan algebra on the direct sum of Hom-Jordan algebra and its representation space,and get the equivalent condition that the duality of the representation of the Hom-Jordan algebra which is still a represention,and the duality of the regular representation of Hom-Jordan algebra is still a representation.Then the conditions of the compatible Hom-Jordan algebra are obtained.The third part will give the definition of the matched pair of Hom-Jordan algebra.We find a method of constructing Hom-Jordan algebra on the direct sum of two Hom-Jordan algebras.At last,the method of constructing Hom-Jordan bialgebra is given by matched pair,and we will give the definition and equivalent conditions of Manin triple on Hom-Jordan algebra.The forth part will use the classification of left-symmetric algebras on three-dimensional Heisenberg algebra and the method of constructing left-symmetric bialgebra,then get 1-boundary left-symmetric bialgebra in six dimension corresponding to the left-symmetric algebra on Heisenberg algebra,and list it in form.