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