Decompositions And (σ,τ)-Derivations Of Lie Color Algebras

As the natural generalization of Lie algebras and Lie superalgebras,Lie color algebras have many applications in mathematical physics and have attracted widespread attention.In this thesis,we mainly study decompositions and(σ,τ)-derivations of Lie color algebras.It consists of three chapters.Chapter 1 is introduction.This chapter introduces the development of Lie color algebras,the progress of derivations and generalized derivations of Lie color algebras,and some related results of complete Lie algebras.Chapter 2 is dedicated to decompositions of Lie color algebras.Firstly,we study decompositions of perfect Lie color algebras and prove that every perfect and complete Lie color algebra can be uniquely decomposed into the direct sum of indecomposable perfect and complete ideals.Secondly,we study perfect and complete decompositions of Lie color algebras,and prove that every perfect and complete Lie color algebra has a decomposition of direct sum of the maximal perfect and complete ideals and characteristic ideals.Finally,we study constructions of perfect and complete Lie color algebras and show that derivations of Lie color algebras are perfect and complete.In particular,if Lie color algebras are indecomposable,then derivations are simply perfect and complete.Chapter 3 is concerned with(σ,τ)-derivations of Lie color algebras.Firstly,we research some properties of(σ,τ)-derivations and the relationship with derivations of Lie color algebras.In particular,if a(σ,σ)-derivation is invertible,by properties of left-symmetric color algebras,we can prove that a left-multiplication structure related to the(σ,τ)-derivations of Lie color algebras can induce a left-symmetric color algebra.Secondly,we study relationships between(α,β,γ)-derivations,quasiderivations and centroids of Lie color algebras.Finally,the interior of G-derivations of Lie color algebras is studied,and when G is an infinite cyclic group,its corresponding Hilbert series are calculated.