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New Lie algebras over the group $\mathbb Z_2^3$

Francisco Cuenca Carrégalo, Cristina Draper

TL;DR

This work expands the landscape of Lie algebras by developing and exploiting generalized group algebras over the grading group $G=\mathbb{Z}_2^3$ and combining them with graded contractions. It provides explicit constructions realizing $\mathfrak{so}_8$, $\mathfrak{so}_7$, and $\mathfrak{g}_2$ as Lie algebras over $G$, including a self-contained, octonion-free description of $\mathfrak{g}_2$ via subalgebras and triality. The authors classify graded contractions for the $\mathbb{Z}_2^3$-gradings attached to $\mathfrak{d}_4$, $\mathfrak{b}_3$, and $\mathfrak{g}_2$, demonstrating that contracted algebras remain generalized group algebras; this yields at least 60 distinct Lie algebras in dimensions 16, 24, and 32 plus several infinite families. The results reveal a rich, computationally tractable zoo of reductive, solvable, and nilpotent Lie algebras linked by explicit twists, boosting tools for constructing and manipulating Lie algebras with graded-symmetry structure. By connecting generalized group algebras to graded contractions and octonionic structures, the work opens pathways to explore larger exceptional algebras (e.g., $F_4$) within a unified, computable framework.

Abstract

A new structure, based on joining copies of a group by means of a \emph{twist}, has recently been considered to describe the brackets of the two exceptional real Lie algebras of type $G_2$ in a highly symmetric way. In this work we show that these are not isolated examples, providing a wide range of Lie algebras which are generalized group algebras over the group $\mathbb{Z}_2^3$. On the one hand, some orthogonal Lie algebras are quite naturally generalized group algebras over such group. On the other hand, previous classifications on graded contractions can be applied to this context getting many more examples, involving solvable and nilpotent Lie algebras of dimensions 32, 28, 24, 21, 16 and 14.

New Lie algebras over the group $\mathbb Z_2^3$

TL;DR

This work expands the landscape of Lie algebras by developing and exploiting generalized group algebras over the grading group and combining them with graded contractions. It provides explicit constructions realizing , , and as Lie algebras over , including a self-contained, octonion-free description of via subalgebras and triality. The authors classify graded contractions for the -gradings attached to , , and , demonstrating that contracted algebras remain generalized group algebras; this yields at least 60 distinct Lie algebras in dimensions 16, 24, and 32 plus several infinite families. The results reveal a rich, computationally tractable zoo of reductive, solvable, and nilpotent Lie algebras linked by explicit twists, boosting tools for constructing and manipulating Lie algebras with graded-symmetry structure. By connecting generalized group algebras to graded contractions and octonionic structures, the work opens pathways to explore larger exceptional algebras (e.g., ) within a unified, computable framework.

Abstract

A new structure, based on joining copies of a group by means of a \emph{twist}, has recently been considered to describe the brackets of the two exceptional real Lie algebras of type in a highly symmetric way. In this work we show that these are not isolated examples, providing a wide range of Lie algebras which are generalized group algebras over the group . On the one hand, some orthogonal Lie algebras are quite naturally generalized group algebras over such group. On the other hand, previous classifications on graded contractions can be applied to this context getting many more examples, involving solvable and nilpotent Lie algebras of dimensions 32, 28, 24, 21, 16 and 14.
Paper Structure (12 sections, 12 theorems, 43 equations)

This paper contains 12 sections, 12 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Some orthogonal algebras which are generalized group algebras
  3. Generalized group algebras which are Lie algebras
  4. The orthogonal Lie algebra of size 8 as generalized group algebra
  5. The series $\mathfrak{g}_2\subset \mathfrak{b}_3\subset \mathfrak{d}_4$ as Lie algebras over $\mathbb{Z}_2^3$
  6. Lie algebras over $G$ and the Killing form
  7. Generalized group algebras and irreducible representations
  8. generalized group algebras coming from graded contractions
  9. Preliminaries on graded contractions
  10. Graded contractions of $\Gamma_{\mathfrak{d}_4}$, $\Gamma_{\mathfrak{b}_3}$ and $\Gamma_{\mathfrak{g}_2}$
  11. Graded contractions on generalized group algebras
  12. Some conclusions.

Key Result

Proposition 2.3

Let $V=\mathbb{F}^4$ and $G=\mathbb{Z}_2^3$. Then $\mathfrak{L}=V^\sigma[G]$ is a generalized group algebra for the map $\sigma\colon G\times G \rightarrow \mathrm{Bil}(V\times V,V)$, $\sigma(g_i,g_j)\equiv \sigma_{i,j}$ given by, for any $i\in I$, if $r=(a_1,a_2,a_3,a_4)$, $s=(b_1,b_2,b_3,b_4)$, $\sigma_{0i}(r,s)=\sigma_{i0}(r,s)=\sigma_{00}(r,s)=\sigma_{ii}(r,s)=0$ and Furthermore the center $

Theorems & Definitions (28)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • ...and 18 more