A note on extensions of nilpotent algebras of Type 2

TL;DR

This paper investigates complex Lie algebras with nilradicals of nilpotent type 2 using sl2(C)-representation theory, clarifying classifications and correcting previous errors, with implications for gauge theory models.

Contribution

It introduces a new approach to study nilpotent Lie algebras of type 2 and revises existing classifications, highlighting the role of specific free nilpotent algebras in physics.

Findings

Corrected classification of Lie algebras with nilradical L5;3

Identified L5;3 as a free nilpotent algebra of type 2

Connected algebraic structures to Yang-Mills gauge theories

Abstract

We propose the study and description of the structure of complex Lie algebras with nilradical a nilpotent Lie algebra of type 2 by using sl2(C)-representation theory. Our results will be applied to review the classification given in [1] (J. Geometry and Physics, 2011) of the Lie algebras with nilradical the quasiclassical algebra L5;3. A non-Lie algebra has been erroneously included in this classification. The 5-dimensional Lie algebra L5;3 is a free nilpotent algebra of type 2 and it is one of two free nilpotent algebras admitting an invariant metric. According to [, Ok98] quasiclassical algebras let construct consistent Yang-Mills gauge theories.