# 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 , or more generally over a commutative unital ring , is defined as the set of matrices of trace zero with entries in , where the addition is defined as matrix addition and the Lie bracket is defined as the commutator: . This is denoted or . In addition to being a Lie ring, this has the additional structure of an algebra over under scalar multiplication, and is thus a Lie algebra over .

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

The underlying set of the Lie ring is:

The underlying set of the group is:

.

Alternatively, it can be written as:

. The addition is given by:

.

The identity element is:

.

The negative map is given by:

The Lie bracket is given by:

The scalar operation of is given by:

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

## Particular cases

## =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) |