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

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

2003-11-10

arxiv.org/abs/math-ph/0311015v1

J. Math. Phys. 43 (2002), 1083-1094

Physics

Mathematical Physics

Scientific paper

10.1063/1.1430046

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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