Gr(?)bner-Shirshov Basis For Left Leibniz Algebra

Leibniz algebra is the non-commutative generalization of Lie algebra,and because its defining relations are different,we divide it into two classes:left Leibniz algebra and right Leibniz algebra.Although the defining relations of left and right Leibniz algebra are completely symmetric,but we use different order here,that is we use the deg-rlex order for left Leibniz algebra,so the computations of the Grobner-Shirshov basis of left and right Leibniz algebra are not same.In this thesis,we give a Grobner Shirshov basis for the left Leibniz algebra.Because direct computation is rather complicated,we divide the left Leibniz algebra into two subalgebras Li-left Leibniz algebra and L2-left Leibniz algebra,and then compute the Grobner-Shirshov basis of them separately and finally we compute the Grobner-Shirshov basis of the left Leibniz algebra.Also,since the Grobner-Shirshov basis here is infinite,we use induction to construct general formula for the relations.