Let F be the free Leibniz algebra of rank two over a field K of characteristic zero freely generated by x 1 and x 2 . In this article we show that an endomorphism of F which preserves the orbit of a nontrivial element of F is an automorphism. Using this result, we determine some test elements of F .

In this work, we determine the structure of the automorphism group of the free metabelian Leibniz algebra of rank three over a field K of characteristic zero.

No abstract

Let Ln be the free metabelian Leibniz algebra generated by the set Xn = {x1,. .. , xn} over a field K of characteristic zero. This is the free algebra of rank n in the variety of solvable of class 2 Leibniz algebras. We call an element s(Xn) ∈ Ln symmetric if s(x σ(1) ,. .. , x σ(n)) = s(x1,. .. , xn) for each permutation σ of {1,. .. , n}. The set L Sn n of symmetric polynomials of Ln is the algebra of invariants of the symmetric group Sn. Let K[Xn] be the usual polynomial algebra with indeterminates from Xn. The description of the algebra K[Xn] Sn is well known, and the algebra (L n) Sn in the commutator ideal L n is a right K[Xn] Sn-module. We give explicit forms of elements of the K[Xn] Sn-module (L n) Sn. Additionally, we determine the description of the group Inn(L Sn n) of inner automorphisms of the algebra L Sn n. The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank n .

Let [Formula: see text] be a field of characteristic zero, [Formula: see text] and [Formula: see text] be two sets of variables, [Formula: see text] be the free metabelian Leibniz algebra generated by [Formula: see text], and [Formula: see text] be the commutative polynomial algebra generated by [Formula: see text] over the base field [Formula: see text]. Polynomials [Formula: see text] and [Formula: see text] are called symmetric if they satisfy [Formula: see text] and [Formula: see text], respectively, for all [Formula: see text]. The sets [Formula: see text] and [Formula: see text] of symmetric polynomials are the [Formula: see text]-invariant subalgebras of [Formula: see text] and [Formula: see text], respectively. The Leibniz subalgebra [Formula: see text] in the commutator ideal [Formula: see text] of [Formula: see text] is a right [Formula: see text]-module by the adjoint action. In this study, we provide a finite generating set for the right [Formula: see text]-module [Formula: see text]. In particular, we give free generating sets for [Formula: see text] and [Formula: see text] as [Formula: see text]-module and [Formula: see text]-module, respectively.