Niltriangular matrix Lie ring:NT(3,p)

Template:Prime-parametrized particular Lie ring

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:

$⟨ a, b, c ∣ p a = p b = p c = 0, [a, c] = b, [a, b] = [b, c] = 0 ⟩$

Definition using matrices

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

$(\begin{matrix} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{matrix})$

The addition is defined as matrix addition and the Lie bracket is defined as $[X, Y] = X Y - Y X$ 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., $(X, Y) \mapsto X Y - Y X$	$[M (a_{1}, b_{1}, c_{1}), M (a_{2}, b_{2}, c_{2})] = M (0, a_{1} c_{2} - a_{2} c_{1}, 0)$

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)