Group of unit quaternions

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is denoted $S^{3},SU(2),S^{0}(\mathbb {H} ),S^{1}(\mathbb {C} )$ and is defined in a number of equivalent ways.

As the group of unit quaternions

Denote by $\mathbb {H}$ the division ring of Hamiltonian quaternions. The group we are interested in is the multiplicative subgroup of $\mathbb {H} ^{*}$ comprising those unit quaternions $a+b\mathbf {i} +c\mathbf {j} +d\mathbf {k}$ satisfying $a^{2}+b^{2}+c^{2}+d^{2}=1$. Note that $a,b,c,d\in \mathbb {R}$ (and are allowed to be equal). Explicitly, the multiplication is given by:

$(a_{1}+b_{1}\mathbf {i} +c_{1}\mathbf {j} +d_{1}\mathbf {k} )(a_{2}+b_{2}\mathbf {i} +c_{2}\mathbf {j} +d_{2}\mathbf {k} )=(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2})+(a_{1}b_{2}+a_{2}b_{1}+c_{1}d_{2}-c_{2}d_{1})\mathbf {i} +(a_{1}c_{2}+a_{2}c_{1}+d_{1}b_{2}-d_{2}b_{1})\mathbf {j} +(a_{1}d_{2}+a_{2}d_{1}+b_{1}c_{2}-b_{2}c_{1})\mathbf {k}$

The identity element is:

$1+0\mathbf {i} +0\mathbf {j} +0\mathbf {k}$

The inverse is given by:

$(a+b\mathbf {i} +c\mathbf {j} +d\mathbf {k} )^{-1}=a-b\mathbf {i} -c\mathbf {j} -d\mathbf {k}$

As the special unitary group

The group can also be defined as the special unitary group of degree two over the field of complex numbers. It is denoted $SU(2)$ or $SU(2,\mathbb {C} )$.

Structures

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.

Arithmetic functions

Function	Value	Similar groups	Explanation
dimension of an algebraic group	3		As $SU(n,\mathbb {C} ),n=2$: $n^{2}-1=2^{2}-1=3$As $S^{n},n=3$: $n=3$ (note that $S^{n}$ is a group only for $n=0,1,3$)
dimension of a real Lie group	3		As $SU(n,\mathbb {C} ),n=2$: $n^{2}-1=2^{2}-1=3$ As $S^{n},n=3$: $n=3$ (note that $S^{n}$ is a group only for $n=0,1,3$)

Group properties

Abstract group properties

Property	Satisfied?	Explanation
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 $SL(2,\mathbb {R} )$.
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 $n$, the element has a $n^{th}$ 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 $\mathbb {R} ^{4}$.

Property	Satisfied?	Explanation
connected topological group	Yes	easy to see from geometric description as $S^{3}$.
compact group	Yes	Easy to see from geometric description as $S^{3}$.

Linear representation theory

Further information: Linear representation theory of group of unit quaternions