Some Results On Restrictied Leibniz Algebra Over Positive Characteristic

Leibniz algebra has been studied in many papers as generalization of Lie algebras since 1993. Many mathematicians and rescrchers have made the Leibniz algebra have a greatly progress. For example, they have got some useful results about the nilpotency, some properties and strcture theory of low dimentional Leibniz algebra underlying field F over zero characteristic. however, there are not much invcstigetion about modular Leibniz algebra. In this paper, we mainly consider the restrictability of Leibniz algebra over positive characteristic, as a analogous idea of modular Lie algebras. The main results are following.We firstly introduce a P-mapping on Leibniz algebra and give the defination of restricted Leibniz algebra consequently. And then give an universal example of restricted Leibniz algebra, which is not a restricted Lie algera. Two propositions about P-nilpotency and one theorem to determine the existence of P-mapping are as follows:Proposition1.2.3 Let I P L be a P-idcal of a restricted Leibniz algebra (L, [P]). Then L is P-nilpotent if and only if I and L/I are P-nilpotent.Proposition1.2.4 Let (L, [P]) be a fmite-dimentional restricted Leibniz algebra. Then there exists a unique P-ideal rad_P(L) such that:(1) radp(L) is P-nilpotent.(1) if I p L is P-nilpotent, then I rad_P(L).Theorem1.2.3 (N.Jacobson) Let (L, [P]) be a restricted Leibniz algebra. Suppose {e_j}_j∈J be a basis of L such that there are y_j ∈ L with (ade_j)P = ady_j. Then there exists exactly one P-mapping [P] : L → L such that e_j~[P] = y_j, .In addition, we also discuss the relation between restrictability and semidirect product and associative symmetric bilinear form. The main content are from Theorem 2.2.1 to Theorem 2.2.6.