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., $(X,Y) \mapsto XY - YX$	$\! [M(a_1,b_1,c_1),M(a_2,b_2,c_2)] = M(0,a_1c_2 - a_2c_1,0)$

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)