Niltriangular matrix Lie ring:NT(3,p)

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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:

\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)