On ideals and contraideals in Leibniz algebras

Abstract

A subalgebra S of a Leibniz algebra L is called a contraideal, if an ideal, generated by S coincides with L. We study the Leibniz algebras, whose subalgebras are either an ideal or a contraideal. Let L be an algebra over a field F with the binary operations + and [ , ]. Then L is called a Leibniz algebra (more precisely, a left Leibniz algebra), if it satisfies the following identity: [[a, b], c] = [a, [b, c]] – [b, [a, c]] for all a, b, c ∈ L. We will also use another form of this identity: [a, [b, c]] = [[a, b], c] + [b, [a, c]]. Leibniz algebras are generalizations of Lie algebras. As usual, a subspace A of a Leibniz algebra L is called a subalgebra, if [x,y] ∈ A for all elements x, y Î A. A subalgebra A is called a left (respectively right) ideal of L, if [y,x] ∈ A (respectively, [x,y] ∈ A) for every x ∈ A, y ∈ L. In other words, if A is a left (respectively, right) ideal, then [L, A] ≤ A (respectively, [A, L] ≤ A). A subalgebra A of L is called an ideal of L (more precisely, a twosided ideal), if it is both a left ideal and a right ideal, that is, [y, x], [x, y] ∈ A for every x ∈ A, y ∈ L. A subalgebra A of L is called an contraideal of L, if AL = L. The theory of Leibniz algebras has been developed quite intensively, but very uneven. However, there are problems natural for any algebraic structure that were not previously considered for Leibniz algebras. We have received a complete description of the Leibniz algebras, which are not Lie algebras, whose subalgebras are an ideal or a contraideal. We also obtain a description of Lie algebras, whose subalgebras are ideals or contraideals up to simple Lie algebras.