Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices

TL;DR

This paper studies the automorphisms of a specific fine grading of the Lie algebra sl(n,C) induced by generalized Pauli matrices, revealing the normalizer group structure and analyzing grading-preserving contractions.

Contribution

It identifies the normalizer of the grading in sl(n,C) as SL(2, Z_n) and explores the structure of grading-preserving contractions, providing new insights into the symmetry group of this grading.

Findings

Normalizer of the grading is SL(2, Z_n)

The set of quadratic equations splits into two orbits

Detailed analysis for the case n=3

Abstract

We consider the grading of $sl(n,\mathbb{C})$ by the group $\Pi_n$ of generalized Pauli matrices. The grading decomposes the Lie algebra into $n^2-1$ one--dimensional subspaces. In the article we demonstrate that the normalizer of grading decomposition of $sl(n,\mathbb{C})$ in $\Pi_n$ is the group $SL(2, \mathbb{Z}_n)$, where $\mathbb{Z}_n$ is the cyclic group of order $n$. As an example we consider $sl(3,\mathbb{C})$ graded by $\Pi_3$ and all contractions preserving that grading. We show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in $\Pi_3$.