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

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 ${\Pi }_3$.

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