Fine grading of $sl(p^2,\mathbb{C})$ generated by tensor product of generalized Pauli matrices and its symmetries

Abstract

Study of the normalizer of the MAD-group corresponding to a finegrading offers the most important tool for describing symmetries in the system of non-linear equations connected with contraction of a Lie algebra. One fine grading that is always present in any Lie algebra $sl(n,\mathbb{C})$ is the Pauli grading. The MAD-group corresponding to it is generated by generalized Pauli matrices. For such MAD-group, we already know its normalizer; its quotient group is isomorphic to the Lie group $Sl(2,\mathbb{Z}_n)\times v\mathbb{Z}_2$. In this paper, we deal with a more complicated situation, namely that the fine grading of $sl(p^2, \mathbb{C})$ is given by a tensor product of the Pauli matrices of the same order $p$, $p$ being a prime. We describe the normalizer of the corresponding MAD-group and we show that its quotient group is isomorphic to $Sp(4,\mathbb{Z}_p)\times\mathbb{Z}_2$.