P-solvable Restricted Lie Colour Algebras And The Basic Theory Of Associative Lie Colour Algebras

Lie colour algebras is a natural extension of Lie algebras and Lie superalgebras. And it is a topic of research and application in mathematics and physics.We introduce the definition of p-solvable restricted Lie colour algebras. We are concerned with the relationships between solvability and p-solvability of restricted Lie colour algebras, and obtain some properties of p-solvable restricted Lie colour algebras.We then study p-nilpotent restricted Lie colour algebras .Some necessary conditions and sufficient conditions of both p-nilpotent restricted Lie colour algebras and nilpotent restricted Lie colour algeras are given.On the other hand,we are concerned with the relationships between nilpotency and p-nilpotency of restricted Lie colour algeras.And we give some identities of associative Lie colour algebras and some properties of modularLie colour algebras.Colour derivations of modular Lie colour algebras are concerned.The main conclusions of this paper:Theorem 1 Let (L, [p]) be a restricted Lie colour algebra over F. then every inner colourivation of L is a restricted colourivation of L.Theorem 2 Let Pl(V) be the general linear Lie colour algebra of a finitedimensional ofΓ- graded vector space V over F.Suppose that there exist positive integers m and n such that (adA)^m(B) = 0 and Aⁿ(x) = 0 , where A∈Pl(V)_α,B∈Pl(V)_β,x∈V .Then A(adA)^m-1(B)A^n-1(x) =0 .Theorem 3 Let Pl(V) be the general linear Lie colour algebra of a finitedimensional ofΓ- graded vector space V over F. Suppose that there exists a positive integer m such that (adA)^m(B) = 0, where V_0A = {x∈V|Aⁱ(x) = 0,(?) i∈N} ,V_1A=∩_i=1^{∞}Aⁱ(V) ,A∈Pl(V)_α,B∈Pl(V)_β.Then the Fitting components V_0A,V_1A of V relative to A are invariant under B.Theorem 4 Let L be a Lie colour algebra over F. If D∈(Der_F(L))_α, x∈L_β,α,β∈Γ, then the following identities hold: where t = (1 - (α|α))/2...