THE DIAGONAL MAP IN HOMOLOGY OF LEIBNIZ ALGEBRAS

Let k be a field and let g be a Leibniz algebra over k. The diagonal map g --> g + g induces a graded linear map HL(*) (g) --> HL(*) (g + g) where HL(*) stands for Leibniz homology. By using the Kunneth style formula of [5], we obtain a graded linear map Phi : HL(*) (g) --> HL(*) (g) * HL(*) (g) = k + HL(*)($) over bar ($) over bar (g) + HL(*)($) over bar ($) over bar (g) + (HL(*)($) over bar ($) over bar (g) x HL(*)($) over bar ($) over bar (g)) + ..., where HL(*)($) over bar ($) over bar (g) = +(p greater than or equal to 1) HL(p) (g). Let Delta be the projection of Phi onto the first factor HL(*)($) over bar ($) over bar (g) x HL(*)($) over bar ($) over bar (g) then Delta defines a coproduct on HL(*)($) over bar ($) over bar (g). We will first see how strongly related to a cup-product in Leibniz cohomology Delta is. Next, we will give a complete description of Phi according to Delta.