Editing Pauli matrices (section)

== SU(2) ==
The group [[SU(2)]] is the [[Lie group]] of [[unitary matrix|unitary]] {{math|2 × 2}} matrices with unit determinant; its [[Lie algebra]] is the set of all {{math|2 × 2}} anti-Hermitian matrices  with trace 0.  Direct calculation, as above, shows that the [[Lie algebra]] <math>\mathfrak{su}_2</math> is the three-dimensional real algebra [[linear span|spanned]] by the set {{math|{''iσ{{sub|k}}''}<nowiki/>}}. In compact notation,
:<math> \mathfrak{su}(2) = \operatorname{span} \{\; i\,\sigma_1\, ,\; i\,\sigma_2\, , \; i\,\sigma_3 \;\}.</math>

As a result, each {{math|''iσ{{sub|j}}''}} can be seen as an [[Lie group#The Lie algebra associated with a Lie group|infinitesimal generator]] of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper [[Representation theory of SU(2)|representation of {{math|su(2)}}]], as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is {{math|1=''λ'' = {{sfrac|1|2}},}} so that
:<math> \mathfrak{su}(2) = \operatorname{span} \left\{\frac{\,i\,\sigma_1\,}{2}, \frac{\,i\,\sigma_2\,}{2}, \frac{\,i\,\sigma_3\,}{2} \right\}.</math>

As SU(2) is a compact group, its [[Cartan decomposition]] is trivial.

=== SO(3) ===
The Lie algebra <math> \mathfrak{su}(2)</math> is [[isomorphism|isomorphic]] to the Lie algebra <math> \mathfrak{so}(3)</math>, which corresponds to the Lie group [[Rotation group SO(3)|SO(3)]], the [[group (mathematics)|group]] of [[rotation]]s in three-dimensional space. In other words, one can say that the {{math|''iσ{{sub|j}}''}} are a realization (and, in fact, the lowest-dimensional realization) of ''infinitesimal'' rotations in three-dimensional space. However, even though <math> \mathfrak{su}(2)</math> and <math> \mathfrak{so}(3)</math> are isomorphic as Lie algebras, {{math|SU(2)}} and {{math|SO(3)}} are not isomorphic as Lie groups. {{math|SU(2)}} is actually a [[Double covering group|double cover]] of {{math|SO(3)}}, meaning that there is a two-to-one group homomorphism from {{math|SU(2)}} to {{math|SO(3)}}, see [[Rotation group SO(3)#Connection between SO(3) and SU(2)|relationship between SO(3) and SU(2)]].

=== Quaternions ===
{{main|Spinor#Three dimensions}}
The real linear span of {{math|{''I'', ''iσ''{{sub|1}}, ''iσ''{{sub|2}}, ''iσ''{{sub|3}}<nowiki>}</nowiki>}} is isomorphic to the real algebra of [[quaternions]], <math>\mathbb{H}</math>, represented by the span of the basis vectors <math> \left\{\; \mathbf{1}, \, \mathbf{i}, \, \mathbf{j}, \, \mathbf{k} \;\right\} .</math> The isomorphism from <math>\mathbb{H}</math> to this set is given by the following map (notice the reversed signs for the Pauli matrices):
<math display=block>
  \mathbf{1} \mapsto I, \quad
  \mathbf{i} \mapsto - \sigma_2\sigma_3 = - i\,\sigma_1, \quad
  \mathbf{j} \mapsto - \sigma_3\sigma_1 = - i\,\sigma_2, \quad
  \mathbf{k} \mapsto - \sigma_1\sigma_2 = - i\,\sigma_3.
</math>

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,<ref>
{{cite book
 |first=Mikio |last=Nakahara
 |year=2003
 |title=Geometry, Topology, and Physics
 |edition=2nd  |page=[https://books.google.com/books?id=cH-XQB0Ex5wC&pg=PR22 xxii]
 |publisher=CRC Press
 |isbn=978-0-7503-0606-5
 |url=https://books.google.com/books?id=cH-XQB0Ex5wC&q=%22Pauli+matrices%22+OR+%22Pauli+matrix%22
 |via=Google Books
}}
</ref>
:<math>
  \mathbf{1} \mapsto I, \quad
  \mathbf{i} \mapsto i\,\sigma_3 \, , \quad
  \mathbf{j} \mapsto i\,\sigma_2 \, , \quad
  \mathbf{k} \mapsto i\,\sigma_1 ~ .
</math>

As the set of [[versor]]s {{math|''U'' ⊂ <math>\mathbb{H}</math>}} forms a group isomorphic to {{math|SU(2)}}, {{mvar|U}} gives yet another way of describing {{math|SU(2)}}. The two-to-one homomorphism from {{math|SU(2)}} to {{math|SO(3)}} may be given in terms of the Pauli matrices in this formulation.