[Paper Review] Solution of Contraction Equations for the Pauli Grading of sl(3,C)

Jiří Hrivnák|ArXiv.org|Oct 12, 2004

Spectral Theory in Mathematical Physics2 references3 citations

TL;DR

This paper solves the system of contraction equations for the Pauli grading of the complex special linear algebra $\mathfrak{sl}(3,\mathbb{C})$, employing the symmetry group of the grading to reduce case analysis and classify all graded contractions. The key contribution is a complete enumeration of 144 non-isomorphic Lie algebras of dimension 8 obtained via graded contractions under the Pauli grading, with explicit matrix representations provided in the appendix.

ABSTRACT

We consider the Pauli grading of the Lie algebra sl(3,C) and use a concept of graded contractions to construct non-isomorphic Lie algebras of dimension 8, while preserving the Pauli grading. We show how the symmetry group of a grading simplifies the solution of contraction equations. We present the list of all 180 non-equivalent solutions of non-linear contraction system.

Motivation & Objective

• To systematically solve the system of contraction equations for the Pauli grading of $\mathfrak{sl}(3,\mathbb{C})$, a challenging case due to limited coarsenings.
• To classify all graded contractions of $\mathfrak{sl}(3,\mathbb{C})$ under the Pauli grading, yielding new Lie algebras not isomorphic to those from other gradings.
• To develop and apply a general method using the symmetry group of the grading to reduce computational complexity in case-by-case analysis.
• To provide explicit matrix representations of all solutions in the appendix for further analysis of the resulting Lie algebras.

Proposed method

• The paper employs the symmetry group of the Pauli grading, isomorphic to a finite matrix group, to classify and reduce equivalent solutions of the contraction system.
• It uses normalization of contraction matrices to eliminate redundant solutions and ensure consistency in the classification.
• The contraction system $\mathcal{S}_3$ is simplified by exploiting group action and invariance under automorphisms of the grading.
• An algorithm is developed to evaluate all solutions by systematically analyzing the structure of the grading and its invariants.
• The method relies on the fact that equivalent solutions under the symmetry group are identified and grouped, reducing the number of distinct cases to evaluate.
• Explicit matrix representations of all 144 solutions are computed and listed in the appendix, with 24 solutions having 23 zeros and 120 having 22 zeros.

Experimental results

Research questions

• RQ1 What is the complete set of graded contractions of $\mathfrak{sl}(3,\mathbb{C})$ under the Pauli grading, and how can they be classified?
• RQ2 How can the symmetry group of the Pauli grading be used to simplify the solution of the contraction equations?
• RQ3 What is the number and structure of non-isomorphic Lie algebras of dimension 8 that arise from graded contractions of $\mathfrak{sl}(3,\mathbb{C})$ under the Pauli grading?
• RQ4 Why is the solution of the contraction system for the Pauli grading more complex than for other gradings, such as the toroidal grading?
• RQ5 What is the role of the symmetry group in reducing the number of distinct solutions and ensuring completeness?

Key findings

• The paper identifies 144 distinct solutions to the contraction system $\mathcal{S}_3$ for the Pauli grading of $\mathfrak{sl}(3,\mathbb{C})$.
• Among the 144 solutions, 120 have 22 zeros and 24 have 23 zeros in their matrix representations.
• The solutions include two trivial cases: the zero matrix and the identity matrix, both of which are explicitly listed.
• The symmetry group of the Pauli grading is used to group equivalent solutions, significantly reducing the number of cases to evaluate.
• The method successfully classifies all graded contractions under the Pauli grading, yielding Lie algebras that are non-isomorphic to those obtained from other gradings like the toroidal grading.
• The appendix provides explicit matrix representations of all 144 solutions, enabling further analysis of the resulting Lie algebras.