# Special linear Lie ring of degree two

## Definition

The special linear Lie algebra of degree two or special linear Lie ring of degree two over a field $k$, or more generally over a commutative unital ring $R$, is defined as the set of $2 \times 2$ matrices of trace zero with entries in $R$, where the addition is defined as matrix addition and the Lie bracket is defined as the commutator: $[A,B] := AB - BA$. This is denoted $sl(2,R)$ or $sl_2(R)$. In addition to being a Lie ring, this has the additional structure of an algebra over $R$ under scalar multiplication, and is thus a Lie algebra over $R$.

When $q$ is a prime power, $sl(2,q)$ is defined as $sl(2,F)$ where $F$ is the finite field (uniqe upto isomorphism) with $q$ elements.

The underlying set of the Lie ring is:

The underlying set of the group is:

$sl(2,R) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in R, a + d = 0 \right \}$.

Alternatively, it can be written as:

$sl(2,R) := \left \{ \begin{pmatrix} a & b \\ c & -a \\\end{pmatrix} \mid a,b,c\in R \right \}$. The addition is given by:

$\begin{pmatrix} a & b \\ c & -a \\\end{pmatrix} + \begin{pmatrix} a' & b' \\ c' & -a' \\\end{pmatrix} = \begin{pmatrix} a + a' & b + b' \\ c + c' & -a - a' \\\end{pmatrix}$.

The identity element is:

$\begin{pmatrix} 0 & 0 \\ 0 & 0\\\end{pmatrix}$.

The negative map is given by:

$- \begin{pmatrix} a & b \\ c & a \\\end{pmatrix} = \begin{pmatrix} -a & -b \\ -c & a \\\end{pmatrix}$

The Lie bracket is given by:

$\left[\begin{pmatrix} a & b \\ c & -a \\\end{pmatrix} , \begin{pmatrix} a' & b' \\ c' & -a' \\\end{pmatrix}\right] = \begin{pmatrix} bc' - b'c & 2(ab' - a'b) \\ 2(ca' - ac') & b'c - bc' \\\end{pmatrix}$

The scalar operation of $R$ is given by:

$\lambda \begin{pmatrix} a & b \\ c & -a \\\end{pmatrix} = \begin{pmatrix} \lambda a & \lambda b \\ \lambda c & -\lambda a \\\end{pmatrix}$

This Lie algebra is free as a $R$-module, with the following freely generating set:

$e := \begin{pmatrix} 0 & 1 \\ 0 & 0 \\\end{pmatrix}, \qquad f := \begin{pmatrix} 0 & 0 \\ 1 & 0 \\\end{pmatrix}, \qquad h := \begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$

## =Finite fields

Size of field Field Lie ring
2 field:F2 special linear Lie ring:sl(2,2)
3 field:F3 special linear Lie ring:sl(2,3)