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 $Sp(4,{\mathbb{Z}}_p)\times {\mathbb{Z}}_2$.

References in corpus (2)

• Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices
• Graded contractions of the Pauli graded sl(3,C)

Cited by in corpus (8)

• On Clifford groups in quantum computing
• Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra
• Symmetries of the finite Heisenberg group for composite systems
• Symmetries of finite Heisenberg groups for multipartite systems
• Weyl groups of the fine gradings on $\e_6$
• A classification of finite quantum kinematics
• Symmetries of finite Heisenberg groups for k-partite systems
• The Weyl group of the fine grading of $sl(n,\mathbb{C})$ associated with tensor product of generalized Pauli matrices