Editing 3-sphere (section)

==Group structure==
When considered as the set of [[unit quaternion]]s, {{math|''S''<sup>3</sup>}} inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, {{math|''S''<sup>3</sup>}} takes on the structure of a [[group (mathematics)|group]]. Moreover, since quaternionic multiplication is [[smooth function|smooth]], {{math|''S''<sup>3</sup>}} can be regarded as a real [[Lie group]]. It is a [[nonabelian group|nonabelian]], [[Compact space|compact]] Lie group of dimension 3. When thought of as a Lie group, {{math|''S''<sup>3</sup>}} is often denoted {{math|[[symplectic group|Sp(1)]]}} or {{math|U(1, '''H''')}}.

It turns out that the only [[hypersphere|spheres]] that admit a Lie group structure are {{math|[[unit circle|''S''<sup>1</sup>]]}}, thought of as the set of unit [[complex number]]s, and {{math|''S''<sup>3</sup>}}, the set of unit quaternions (The degenerate case {{math|''S''<sup>0</sup>}} which consists of the real numbers 1 and −1 is also a Lie group, albeit a 0-dimensional one). One might think that {{math|''S''<sup>7</sup>}}, the set of unit [[octonion]]s, would form a Lie group, but this fails since octonion multiplication is [[associative|nonassociative]]. The octonionic structure does give {{math|''S''<sup>7</sup>}} one important property: ''[[parallelizability]]''. It turns out that the only spheres that are parallelizable are {{math|''S''<sup>1</sup>}}, {{math|''S''<sup>3</sup>}}, and {{math|''S''<sup>7</sup>}}.

By using a [[matrix (mathematics)|matrix]] representation of the quaternions, {{math|'''H'''}}, one obtains a matrix representation of {{math|''S''<sup>3</sup>}}. One convenient choice is given by the [[Pauli matrices]]:
:<math>x_1+ x_2 i + x_3 j + x_4 k \mapsto \begin{pmatrix}\;\;\,x_1 + i x_2 & x_3 + i x_4 \\ -x_3 + i x_4 & x_1 - i x_2\end{pmatrix}.</math>
This map gives an [[injective]] [[algebra homomorphism]] from {{math|'''H'''}} to the set of 2&nbsp;×&nbsp;2 complex matrices. It has the property that the [[absolute value]] of a quaternion {{mvar|q}} is equal to the [[square root]] of the [[determinant]] of the matrix image of {{mvar|q}}.

The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the [[special unitary group]] {{math|SU(2)}}. Thus, {{math|''S''<sup>3</sup>}} as a Lie group is [[isomorphic]] to {{math|SU(2)}}.

Using our Hopf coordinates {{math|(''η'', ''ξ''<sub>1</sub>, ''ξ''<sub>2</sub>)}} we can then write any element of {{math|SU(2)}} in the form
:<math>\begin{pmatrix}
e^{i\,\xi_1}\sin\eta & e^{i\,\xi_2}\cos\eta \\
-e^{-i\,\xi_2}\cos\eta & e^{-i\,\xi_1}\sin\eta
\end{pmatrix}.</math>

Another way to state this result is if we express the matrix representation of an element of {{math|SU(2)}} as an exponential of a linear combination of the Pauli matrices. It is seen that an arbitrary element {{math|''U'' ∈ SU(2)}} can be written as
:<math>U=\exp \left( \sum_{i=1}^3\alpha_i J_i\right).</math><ref>{{Cite book |last=Schwichtenberg |first=Jakob |title=Physics from symmetry |date=2015 |publisher=Springer |isbn=978-3-319-19201-7 |location=Cham |oclc=910917227}}</ref>
The condition that the determinant of {{mvar|U}} is +1 implies that the coefficients {{math|''α''<sub>1</sub>}} are constrained to lie on a 3-sphere.