Special linear Lie ring:sl(2,2)
This article is about a particular Lie ring, viz., a Lie ring unique upto isomorphism
View a complete list of particular Lie rings
Definition
This Lie ring can be defined in the following equivalent ways:
- It is the special linear Lie ring of degree two over the field of two elements: the Lie ring of matrices of trace zero over the field of two elements. It is defined by the following presentation:
![{\displaystyle \langle e,f,h\mid e+e=f+f=h+h=0,[e,f]=h,[h,e]=[h,f]=0\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/44809b724e6aed937051015eb76f40c4cda802cd)
- It is the Lie ring of strictly upper-triangular matrices over the field of two elements, with the Lie bracket given by the commutator of two matrices .
![{\displaystyle [A,B]:=AB-BA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863994f9f066fdc4ae4679dd7aad6dc461724a90)
Arithmetic functions
| Function | Value | Explanation |
|---|---|---|
| order of a Lie ring | 8 | |
| nilpotency class of a Lie ring | 2 | |
| derived length of a Lie ring | 2 | |
| Frattini length of a Lie ring | 2 |
Lie ring properties
| Property | Satisfied | Explanation |
|---|---|---|
| abelian Lie ring | No | |
| nilpotent Lie ring | Yes | |
| solvable Lie ring | Yes | |
| simple Lie ring | No |