Abstract
An element of a free Leibniz algebra is called Jordan if it belongs to a free Leibniz-Jordan subalgebra. Elements of the Jordan commutant of a free Leibniz algebra are called weak Jordan. We prove that an element of a free Leibniz algebra over a field of characteristic 0 is weak Jordan if and only if it is left-central. We show that free Leibniz algebra is an extension of a free Lie algebra by left-center. We find the dimensions of the homogeneous components of the Jordan commutant and the base of its multilinear part. We find criterion for an element of free Leibniz algebra to be Jordan.
Highlights
Let K be a field of characteristic 0 and K = K t1, t2, . . . be a free magma, i.e., a space of non-associative non-commutative polynomials with generators t1, t2, . . . . An ideal I of K is called T -ideal if for any f (t1, . . . , tk) ∈ I and for any endomorphism φ of K, f (φ(t1), . . . , φ(tk)) ∈ I
Let F (X) be the free Leibniz algebra defined on a set of generators X = {xi|i ∈ [q]}
The aim of this paper is to prove that the left-center of a free Leibniz algebra
Summary
Let F (X) be the free Leibniz algebra defined on a set of generators X = {xi|i ∈ [q]}. Let F (X) be a free Leibniz algebra over a field K of characteristic 0 generated by a set X = {xi|i ∈ [q]}. For the case of Leibniz algebras, Theorem 1.1 gives us the following criterion for weak Jordan elements: any element a ∈ F (X) is weak Jordan if and only if z is left-central. Opposite to the associative case, tetrads for Leibniz algebras are weak Jordan elements. When the degree is 5, two identities appear for Leibniz algebras under the Lie commutator These facts and other properties of left-central elements can be found in [3] and [1].
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