# Subring structure of upper-triangular nilpotent matrix Lie ring:u(3,p)

Template:Lie ring-specific information

This article discusses the structure of Lie subrings of the Lie ring upper-triangular nilpotent matrix Lie ring:u(3,p), which is also a Lie algebra over . This is is the Lie ring whose elements are matrices over the prime field , with 0s on and below the diagonal, i.e., matrices of the form:

The addition is defined as matrix addition and the Lie bracket is defined as where the product is matrix multiplication. Explicitly:

Operation | How it's defined (short version) | How it's defined (formula) |
---|---|---|

Addition | Matrix addition | |

Identity for addition | Zero matrix | |

Negative for addition | Negative as matrix | |

Lie bracket | Commutator as matrices, i.e., |

Note that when is odd, there is a Baer correspondence between U(3,p) and u(3,p), i.e., a correspondence between this and a corresponding group prime-cube order group:U(3,p), which can be defined as the group of upper-triangular unipotent matrices over . The Baer correspondence relates subrings and subgroups, so the subring structure matches the subgroup structure of prime-cube order group:U(3,p).

When , the Baer correspondence does not work, so the subring structure of differs from the subgroup structure of , which is dihedral group:D8 (see subgroup structure of dihedral group:D8).