Graded contractions of the Pauli graded sl(3,C)

TL;DR

This paper classifies all solutions to contraction equations of the Pauli graded sl(3,C), identifying numerous non-isomorphic solvable Lie algebras and providing a comprehensive overview of their structure and symmetries.

Contribution

It provides a complete classification of contraction solutions for the Pauli grading of sl(3,C), including parametric families and symmetry analysis, advancing the understanding of Lie algebra contractions.

Findings

175 non-parametric solution classes

13 parametric solution families

88 indecomposable solvable Lie algebras of dimension 8

Abstract

The Lie algebra $sl(3,\C)$ is considered in the basis of generalized Pauli matrices. Corresponding grading is the Pauli grading here. It is one of the four gradings of the algebra which cannot be further refined. The set $\es$ of 48 contraction equations for 24 contraction parameters is solved. Our main tools are the symmetry group of the Pauli grading of $sl(3,\C)$, which is essentially the finite group $SL(2,\Z_3)$, and the induced symmetry of the system $\es$. A list of all equivalence classes of solutions of the contraction equations is provided. Among the solutions, 175 equivalence classes are non-parametric and 13 solutions depend on one or two continuous parameters, providing a continuum of equivalence classes and subsequently continuum of non-isomorphic Lie algebras. Solutions of the contraction equations of Pauli graded $sl(3,\bC)$ are identified here as specific solvable Lie algebras of dimensions up to 8. Earlier algorithms for identification of Lie algebras, given by their structure constants, had to be made more efficient in order to distinguish non-isomorphic Lie algebras encountered here. Resulting Lie algebras are summarized in tabular form. There are 88 indecomposable solvable Lie algebras of dimension 8, 77 of them being nilpotent. There are 11 infinite sets of parametric Lie algebras which still deserve further study.