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)