HNN-extensions of Leibniz algebras

Existentially closed Leibniz algebras and an embedding theorem

“…, for all a ∈ A and it implies that G is embedded in H. The HNN-extension of a group possesses an important position in algorithmic group theory which has been used for the proof of the embedding theorem, namely, that every countable group is embeddable into a group with two generators. Ladra et al [7] studied the same construction for Leibniz algebras (as well as their associative relatives, the so-called dialgebras) and proved that every Leibniz algebra embeds into any of its HNN-extensions. The main difference between the construction of HNN-extension for groups and algebras is that the concepts of subgroups and isomorphism are replaced by subalgebras and derivation, respectively.…”

“…The proof of the above theorem is based on the validity of Poincare-Birkhof-Witt theorem for Leibniz algebras which justifies the relation between construction of HNN-extension for dialgebras and HNN-extensions for Leibniz algebras. For an extensive proof see [7].…”

“…Our main tool in this work is the HNN-extensions of Leibniz algebras. It is worth pointing out that the concept of HNN-extension has been recently spread to some generalizations of Lie algebras, including Leibniz algebras, Lie superalgebras and Hom-generalization of Lie algebras in a series of papers [6], [7] and [12]. The paper is organized as follows.…”

HNN-extension of involutive multiplicative Hom-Lie algebras

“…They used HNN-extension in order to give a new proof for Shirshov's theorem [75], namely, a Lie algebra of finite or countable dimension can be embedded into a 2-generator Lie algebra. Moreover, the idea of HNN-extension has been recently spread to Leibniz algebras in [51] and Lie superalgebras in [50], which are respectively, non-antisymmetric and natural generalization of Lie algebras.…”

Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

“…It is worth saying Leibniz algebras were introduced by Bloh in [10] and Loday in [40], and they have many applications either in pure and applied mathematics or in physics. Because of this, many known results of the theory of Lie algebras as well as combinatorial group theory have been spread to Leibniz algebras during the last two decades, (see, for instance, [8,13,16,21,38] and [50]). It is well known that a Leibniz algebra decomposes into a semidirect sum of the solvable radical and a semisimple Lie algebra [8].…”