# Niltriangular matrix Lie ring:NT(3,p)

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(Redirected from Upper-triangular nilpotent matrix Lie ring:u(3,p))

Template:Prime-parametrized particular Lie ring

## Contents

## Definition

Let be a prime number. This Lie ring is a Lie ring of order defined either by a presentation or using matrices as follows.

### Definition by presentation

The presentation is as follows:

### Definition using matrices

This Lie ring is the Lie ring of strictly upper-triangular matrices over the prime field , 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., |

## Generalizations

The definition can be generalized to arbitrary fields, as well as to arbitrary unital rings.

## Particular cases

Prime number | Lie ring |
---|---|

2 | special linear Lie ring:sl(2,2) |

3 | niltriangular matrix Lie ring:NT(3,3) |

## Related groups

Group | Value |
---|---|

additive group | elementary abelian group of prime-cube order |

corresponding group via Baer correspondence for odd | upper-triangular unipotent matrix group:U(3,p). See Baer correspondence between U(3,p) and u(3,p) |