Coloring Lie Algebra Structure

Lie superalgebras are the generalization of Lie algebras, and they are closely related with physics. During the last few years, the theory of Lie superalgebras has seen a remarkable evolution, both in mathematics and physics. Lie superalgebras can be calledZ2 - graded Lie algebras. If we replace the two-graded group Z2 with the generalcommutative group, we will obtain the definition of Lie color superalgebras.In this paper, we mainly study Lie color algebras; discuss the algebraical structure of Lie color algebras. First, we give the definition of Lie color superalgebras using the symmetric bicharacter on a finite commutative group, and also we introduce some fundamental notions about Lie color superalgebras.In the second part, we prove an equivalent condition. In use of this condition the study of Lie color superalgebras can be reduced into the study of general Lie algebras and their representation theories. According to this equivalent condition we construct some kinds of Lie color superalgebras subsequently.In the last part, we further generalize left symmetric algebra and left symmetric structure on Lie algebras into Lie color algebras. By the study of two kinds of affine representations of Lie color algebras, we give some necessary and (or) sufficient conditions to the question whether there is any left color symmetric structure on a given Lie color algebra. Finally, we prove that if there exit left color symmetric structure on some kind of Lie color algebra, then its first co-homology group is not vanishing.