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HNN-extension of Lie superalgebras

Manuel Ladra, Pilar Páez-Guillán, Chia Zargeh

TL;DR

This work constructs HNN-extensions for Lie superalgebras and proves that each Lie superalgebra embeds into any of its HNN-extensions, using Gröbner–Shirshov basis theory and the Composition–Diamond lemma. The main method replaces subgroup/monomorphism data from groups with subalgebra/derivation data to build the extension and establish embeddability via a careful analysis of Lie compositions. A central result is that every Lie superalgebra embeds into its HNN-extension; as an application, any countable-dimensional Lie superalgebra embeds into a two-generator Lie superalgebra, in line with classical embedding theorems in group theory. The approach provides a conceptual parallel to known results for Lie algebras and extends them to the superalgebra setting, with explicit constructions and derivations guiding the embedding proofs.

Abstract

We construct HNN-extensions of Lie superalgebras and prove that every Lie superalgebra embeds into any of its HNN-extensions. Then as an application we show that any Lie superalgebra with at most countable dimension embeds into a two-generator Lie superalgebra.

HNN-extension of Lie superalgebras

TL;DR

This work constructs HNN-extensions for Lie superalgebras and proves that each Lie superalgebra embeds into any of its HNN-extensions, using Gröbner–Shirshov basis theory and the Composition–Diamond lemma. The main method replaces subgroup/monomorphism data from groups with subalgebra/derivation data to build the extension and establish embeddability via a careful analysis of Lie compositions. A central result is that every Lie superalgebra embeds into its HNN-extension; as an application, any countable-dimensional Lie superalgebra embeds into a two-generator Lie superalgebra, in line with classical embedding theorems in group theory. The approach provides a conceptual parallel to known results for Lie algebras and extends them to the superalgebra setting, with explicit constructions and derivations guiding the embedding proofs.

Abstract

We construct HNN-extensions of Lie superalgebras and prove that every Lie superalgebra embeds into any of its HNN-extensions. Then as an application we show that any Lie superalgebra with at most countable dimension embeds into a two-generator Lie superalgebra.
Paper Structure (4 sections, 6 theorems, 32 equations)

This paper contains 4 sections, 6 theorems, 32 equations.

Table of Contents

  1. Preliminaries
  2. Gröbner-Shirshov bases theory for Lie superalgebras
  3. Construction of the HNN-extension
  4. Application

Key Result

Lemma 2.5

Let $u$ and $v$ be super-Lyndon-Shirshov words such that $v$ is contained in $u$ as a subword. Write $u=avb$ with $a,b\in T^{\ast}$. Then there is an arrangement of brackets $[u]=(a[v]b)$ on $u$ such that $[v]$ is a super-Lyndon-Shirshov monomial, $\overline{[u]}=u$ and the leading coefficient of $[

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: B2
  • Definition 2.6
  • Lemma 2.7: S1
  • ...and 6 more