HNN-Extension of Lie Superalgebras

Operadic approach to HNN-extensions of Leibniz algebras

“…For a group G with an isomorphism φ between two of its subgroups A and B, H is an extension of G with an element t ∈ H, such that t −1 at = φ(a) for every a ∈ A. The group H is presented by H = G, t | t −1 at = φ(a), a ∈ A and it implies that G is embedded in H. The concept of HNN-extension was constructed for (restricted) Lie algebras in independent works by Lichtman and Shirvani [4] and Wasserman [19], and it has recently been extended to generalized versions of Lie algebras, namely, Leibniz algebras, Lie superalgebras and Hom-setting of Lie algebras in [10], [11] and [18], respectively. As an application of HNN-extensions, Wasserman in [19] obtained some analogous results to group theory and proved that Markov properties of finitely presented Lie algebras are undecidable.…”

“…Leibniz algebras are non-antisymmetric generalization of Lie algebras introduced by Bloh [2] and Loday [13], [14], and they have many applications in both pure and applied mathematics and in physics. Because of this, many known results of the theory of Lie algebras as well as combinatorial group theory have been extended to Leibniz algebras in the last two decades (see, for instance, [1], [10] and [20]. )…”

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.…”

A Normal form for HNN-extension of Dialgebras

“…Neumann and H. Neumann [5] who showed that if A 1 and A 2 are isomorphic subgroups of a group S, then it is possible to find a group H containing S such that A 1 and A 2 are conjugate to each other in H and such that S is embeddable in H. HNN-extension was investigated for Lie algebras in independent works of Lichtman and Shirvani [5] and Wasserman [12]. HNN-extension has been recently spread to other algebraic structures such as Leibniz algebras by Ladra, Shahryari and Zargeh in [6] and Lie superalgebras by Ladra, Guilan and Zargeh in [7]. The HNN-extension for dialgebras; the generalization of the Lie bracket produces Leibniz algebras (see Loday [9]), was first constructed in [6] as an approach for construction of HNN-extension of Leibniz algebras.…”