Group of unit quaternions
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This group is denoted and is defined in a number of equivalent ways.
As the group of unit quaternions
Denote by the division ring of Hamiltonian quaternions. The group we are interested in is the multiplicative subgroup of comprising those unit quaternions satisfying . Note that (and are allowed to be equal). Explicitly, the multiplication is given by:
The identity element is:
The inverse is given by:
As the special unitary group
The group has the following structures:
- It is a real Lie group (note that it is not a complex Lie group).
- It is a linear algebraic group over the field of real numbers (note that it is not algebraic over the complex numbers).
- It is a topological group.
|dimension of an algebraic group||3||As : As : (note that is a group only for )|
|dimension of a real Lie group||3|| As : |
As : (note that is a group only for )
Abstract group properties
|abelian group||No||Follows from center of general linear group is group of scalar matrices over center|
|nilpotent group||No||Follows from special linear group is quasisimple|
|solvable group||No||Follows from special linear group is quasisimple|
|simple group||No||Has a proper nontrivial center, also has normal subgroup .|
|almost simple group||No||Has a nontrivial center.|
|quasisimple group||Yes||It is a perfect group and its quotient by its center is SO(3,R) which is simple.|
|almost quasisimple group||Yes||Follows from being quasisimple.|
|divisible group||Yes||For any element of the group and any natural number , the element has a root in the group.|
Topological/Lie group properties
The topology here is the subspace topology from the Euclidean topology on the set of all matrices, which is identified with the Euclidean space .
|connected topological group||Yes||easy to see from geometric description as .|
|compact group||Yes||Easy to see from geometric description as .|