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Expanding Lie (super)algebras through abelian semigroups

Fernando Izaurieta, Eduardo Rodríguez, Patricio Salgado

TL;DR

This work introduces S-expansion, a general framework that builds new Lie (super)algebras by tensoring an original algebra g with a finite abelian semigroup S. It defines resonant subalgebras and 0_S-reductions to systematically extract physically relevant substructures, and connects Maurer–Cartan form expansions to S-expansion through the S_E^(N) semigroups. The authors apply the method to osp(32|1), obtaining the M algebra, a D'Auria–Fré–like superalgebra, and a novel d=11 algebra via a Z4–based construction, while also deriving invariant tensors essential for CS and transgression formulations. Collectively, the results provide a versatile toolkit for constructing and analyzing higher-dimensional gauge algebras relevant to 11D supergravity and related theories, with clear pathways to Lagrangian formulations. The work suggests broad generalizations, including nonstandard semigroups and broader reduction schemes, offering a structured approach to exploring the landscape of extended symmetries in theoretical physics.

Abstract

We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662 (2003) 185]. The basic idea consists in considering the direct product between an abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S \times g are given. We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D'Auria-Fre-like Superalgebra, are considered. Finally, we find explicit, non-trace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of Supergravity theories in arbitrary space-time dimensions.

Expanding Lie (super)algebras through abelian semigroups

TL;DR

This work introduces S-expansion, a general framework that builds new Lie (super)algebras by tensoring an original algebra g with a finite abelian semigroup S. It defines resonant subalgebras and 0_S-reductions to systematically extract physically relevant substructures, and connects Maurer–Cartan form expansions to S-expansion through the S_E^(N) semigroups. The authors apply the method to osp(32|1), obtaining the M algebra, a D'Auria–Fré–like superalgebra, and a novel d=11 algebra via a Z4–based construction, while also deriving invariant tensors essential for CS and transgression formulations. Collectively, the results provide a versatile toolkit for constructing and analyzing higher-dimensional gauge algebras relevant to 11D supergravity and related theories, with clear pathways to Lagrangian formulations. The work suggests broad generalizations, including nonstandard semigroups and broader reduction schemes, offering a structured approach to exploring the landscape of extended symmetries in theoretical physics.

Abstract

We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662 (2003) 185]. The basic idea consists in considering the direct product between an abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S \times g are given. We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D'Auria-Fre-like Superalgebra, are considered. Finally, we find explicit, non-trace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of Supergravity theories in arbitrary space-time dimensions.

Paper Structure

This paper contains 23 sections, 5 theorems, 131 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semigroups
  4. Reduced Lie Algebras
  5. The $S$-Expansion Procedure
  6. $S$-Expansion for an Arbitrary Semigroup $S$
  7. Maurer--Cartan Forms Power Series Algebra Expansion as an $S$-Expansion
  8. $S$-Expansion Subalgebras
  9. Resonant Subalgebras for an Arbitrary Semigroup $S$
  10. $S=S_{\mathrm{E}}$ Resonant Subalgebras and MC Expanded Algebras
  11. Case when $\mathfrak{g} = V_{0} \oplus V_{1}$, with $V_{0}$ being a Subalgebra and $V_{1}$ a Symmetric Coset
  12. Case when $\mathfrak{g}$ fulfills the Weimar-Woods Conditions
  13. Case when $\mathfrak{g}=V_{0}\oplus V_{1}\oplus V_{2}$ is a Superalgebra
  14. $S$-Expansions of $\mathfrak{osp}\left( 32|1\right)$ and $d=11$ Superalgebras
  15. The M Algebra
  16. ...and 8 more sections

Key Result

Theorem 3.1

Let $S=\left\{ \lambda_{\alpha}\right\}$ be an abelian semigroup with $2$-selector $K_{\alpha \beta}^{\space \gamma}$ and $\mathfrak{g}$ a Lie (super)algebra with basis $\left\{ \bm{T}_{A}\right\}$ and structure constants $C_{AB}^{\space C}$. Denote a basis element of the direct product $S \times \

Figures (8)

  • Figure 1: $S_{\mathrm{E}}^{(3)}$-expansion of an algebra $\mathfrak{g} = V_{0} \oplus V_{1}$, where $V_{0}$ is a subalgebra and $V_{1}$ a symmetric coset. (a) The gray region corresponds to the full $S_{\mathrm{E}}^{(3)}$-expanded algebra, $\mathfrak{G} = S_{\mathrm{E}}^{(3)} \times \mathfrak{g}$. (b) The shaded area here depicts a resonant subalgebra $\mathfrak{G}_{\mathrm{R}}$. (c) The gray region now shows the $0_{S}$-reduction of the resonant subalgebra $\mathfrak{G}_{\mathrm{R}}$.
  • Figure 2: (a) The shaded region shows a $S_{\mathrm{E}}^{(4)}$ resonant subalgebra when $\mathfrak{g} = V_{0} \oplus V_{1} \oplus V_{2} \oplus V_{3}$ satisfies the Weimar-Woods conditions. (b) The $0_{S}$-reduction of this resonant subalgebra removes all sectors of the form $0_{S} \times \mathfrak{g}$. This corresponds to the case $\mathcal{G}(4,4,4,4)$ in the context of Azcarraga Et Al.
  • Figure 3: (a) The shaded area corresponds to an $S_{\mathrm{E}}^{(4)}$ resonant subalgebra of $\mathfrak{G} = S_{\mathrm{E}}^{(4)} \times \mathfrak{g}$ when $\mathfrak{g}$ is a superalgebra. (b) The gray region shows the $0_{S}$-reduction of the resonant subalgebra $\mathfrak{G}_{\mathrm{R}}$. This corresponds to $\mathcal{G}(4,3,4)$ in the context of Azcarraga Et Al.
  • Figure 4: The M algebra as an $S_{\mathrm{E}}^{\left( 2 \right)}$-expansion of $\mathfrak{osp} \left( 32|1 \right)$. (a) A resonant subalgebra of the $S_{\mathrm{E}}^{\left( 2 \right)}$-expanded algebra $\mathfrak{G} = S_{\mathrm{E}}^{\left( 2 \right)} \times \mathfrak{osp} \left( 32|1 \right)$ is shown in the shaded region. (b) The M algebra itself (gray area) is obtained after $0_{S}$-reducing the resonant subalgebra.
  • Figure 5: A D'Auria--Fré-like Superalgebra regarded here as an $S_{\mathrm{E}}^{\left( 3 \right)}$-expansion of $\mathfrak{osp} \left( 32|1 \right)$. (a) A resonant subalgebra of the $S_{\mathrm{E}}^{\left( 3\right)}$-expanded algebra $\mathfrak{G} = S_{\mathrm{E}}^{\left( 3 \right)} \times \mathfrak{osp} \left( 32|1 \right)$ is shown in the shaded region. (b) A Superalgebra similar to the ones introduced by D'Auria and Fré in DAuria-Fre is obtained after $0_{S}$-reducing the resonant subalgebra.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 4.1
  • Theorem 4.2
  • Definition 4.3
  • ...and 3 more