Niltriangular matrix Lie ring:NT(3,p)

Definition

Let $p$ be a prime number. This Lie ring is a Lie ring of order $p^{3}$ defined either by a presentation or using matrices as follows.

Definition by presentation

The presentation is as follows:

$\langle a,b,c\mid pa=pb=pc=0,[a,c]=b,[a,b]=[b,c]=0\rangle$

Definition using matrices

This Lie ring is the Lie ring of strictly upper-triangular matrices over the prime field $\mathbb {F} _{p}$, i.e., matrices of the form:

${\begin{pmatrix}0&a&b\\0&0&c\\0&0&0\\\end{pmatrix}}$

The addition is defined as matrix addition and the Lie bracket is defined as $[X,Y]=XY-YX$ where the product is matrix multiplication. Explicitly:

Operation	How it's defined (short version)	How it's defined (formula)
Addition	Matrix addition	$\!M(a_{1},b_{1},c_{1})+M(a_{2},b_{2},c_{2})=M(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})$
Identity for addition	Zero matrix	$\!M(0,0,0)$
Negative for addition	Negative as matrix	$\!-M(a,b,c)=M(-a,-b,-c)$
Lie bracket	Commutator as matrices, i.e., $(X,Y)\mapsto XY-YX$	$\![M(a_{1},b_{1},c_{1}),M(a_{2},b_{2},c_{2})]=M(0,a_{1}c_{2}-a_{2}c_{1},0)$

Generalizations

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

Particular cases

Prime number $p$	Lie ring $u(3,p)$
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 $p$	upper-triangular unipotent matrix group:U(3,p). See Baer correspondence between U(3,p) and u(3,p)