[Paper Review] Moufang symmetry V. Triple closure

Eugen Paal|ArXiv.org|Feb 27, 2008

Mathematics and Applications5 references3 citations

TL;DR

This paper investigates the triple closure property of infinitesimal translations in analytic Moufang loops, showing that the left, right, and multiplication operators form Lie triple systems under the Loos bracket. The key result establishes an equivalence between reductivity and triple closure, linking Mal’tsev algebras to Lie triple systems via the Yamaguti and Loos brackets.

ABSTRACT

Triple closure of the infinitesimal translations of an analytic Moufang loop is inquired. This property is equivalent to reductivity and relates Mal'tsev algebras to the Lie triple systems.

Motivation & Objective

• To investigate the triple closure property of infinitesimal translations in local analytic Moufang loops.
• To establish the equivalence between reductivity and triple closure in the context of Moufang loops.
• To connect Mal’tsev algebras and Lie triple systems through the structure of infinitesimal translations.
• To analyze the role of the Loos bracket in encoding the algebraic closure properties of the translation operators.
• To demonstrate that the tangent algebra of an analytic Moufang loop is a Lie triple system under the Loos bracket.

Proposed method

• Uses the Yamaguti brackets and the Loos bracket to define ternary operations on the tangent algebra of a Moufang loop.
• Applies the reductivity conditions (2.1a–c) to relate the double Lie bracketing of translation operators to the Loos bracket.
• Employs the Jacobi identity for Lie algebras to derive structural identities for the Loos bracket.
• Establishes the triple closure relations (2.4a–c) by combining reductivity and the definition of the Loos bracket.
• Derives the fundamental identities of the Loos bracket (2.6a–c) using the Jacobi identity and closure properties.
• Applies triality to extend results from the left translation algebra to the right and multiplication algebras.

Experimental results

Research questions

• RQ1 Does the triple closure property hold for the infinitesimal translations of an analytic Moufang loop?
• RQ2 How is the triple closure property related to the reductivity of the Moufang loop’s tangent algebra?
• RQ3 What algebraic structure do the left, right, and multiplication operators form under the Loos bracket?
• RQ4 How do the Loos brackets satisfy the identities of a Lie triple system?
• RQ5 What is the role of the Yamaguti brackets in connecting Mal’tsev algebras to Lie triple systems?

Key findings

• The infinitesimal translations $L_x$, $R_x$, and $M_x$ satisfy the triple closure conditions $[[L_x, L_y], L_z] = L_{\{x,y,z\}}$, and similarly for $R$ and $M$, if and only if the reductivity conditions hold.
• The vector spaces $\mathfrak{L}$, $\mathfrak{R}$, and $\mathcal{M}$ are closed under double Lie bracketing, thus forming Lie triple systems.
• The Loos bracket $\{x,y,z\}$ satisfies the skew-symmetry $\{x,y,z\} = -\{y,x,z\}$ and the cyclic identity $\{x,y,z\} + \{y,z,x\} + \{z,x,y\} = 0$.
• The Loos bracket satisfies the fundamental identity $\{x,y,\{z,w,v\}\} = \{\{x,y,z\},w,v\} + \{z,\{x,y,w\},v\} + \{z,w,\{x,y,v\}\}$, confirming its structure as a Lie triple system.
• The tangent algebra $\Gamma$ of an analytic Moufang loop is a Lie triple system under the Loos bracket, as shown by Corollary 2.6.
• The structure relations (2.5a–c) confirm that $\mathfrak{L} + [\mathfrak{L}, \mathfrak{L}]$ forms a symmetric space algebra, with analogous results for $\mathfrak{R}$ and $\mathcal{M}$ via triality.