Special linear Lie ring of degree two
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:
|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)|