Matrix Representations of Derivations for Low-Dimensional Mock-Lie Algebras

TL;DR

This paper develops a systematic approach to the derivation structure of Mock-Lie algebras by deriving explicit matrix representations of derivations for dimensions up to four. It offers a detailed classification: a 2D case with a simple triangular form, two 3D types corresponding to distinct isomorphism classes, and five 4D types (A–E) corresponding to specific direct-sum decompositions, each accompanied by precise coefficient constraints. The results illuminate how the derivation algebra $\mathrm{Der}(L)$ acts via basis-dependent matrices, shedding light on the internal symmetries and constraints of Mock-Lie algebras. This classification advances understanding of low-dimensional Mock-Lie structures and provides a concrete toolkit for exploring their automorphisms, enveloping constructions, and representations. The methods and explicit matrices are poised to facilitate further algebraic investigations and potential applications in related Jordan-Lie frameworks.

Abstract

In this work, we study the matrix representation of derivations for Mock-Lie algebras with dimensions up to four. Using matrix methods, we examine their structure and properties, showing how these derivations help us better understand the algebraic nature of Mock-Lie algebras.

Theorems & Definitions (12)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• Theorem 3.1
• proof
• Theorem 3.2
• proof