Editing Special unitary group (section)

== Properties ==
The special unitary group {{math|SU(''n'')}} is a strictly real [[Lie group]] (vs. a more general [[complex Lie group]]). Its dimension as a [[manifold|real manifold]] is {{math|''n''<sup>2</sup> − 1}}. Topologically, it is [[compact space|compact]] and [[simply connected]].<ref>{{harvnb|Hall|2015}}, Proposition 13.11</ref> Algebraically, it is a [[simple Lie group]] (meaning its [[Lie algebra]] is simple; see below).<ref>{{cite book |author-link=Brian Garner Wybourne |author=Wybourne, B.G. |year=1974 |title=Classical Groups for Physicists |publisher=Wiley-Interscience |isbn=0471965057}}</ref>

The [[center of a group|center]] of {{math|SU(''n'')}} is isomorphic to the [[cyclic group]] <math>\mathbb{Z}/n\mathbb{Z}</math>, and is composed of the diagonal matrices {{math|''ζ'' ''I''}} for {{math|''ζ''}} an {{math|''n''}}th root of unity and {{math|''I''}} the {{math|''n'' × ''n''}} identity matrix.

Its [[outer automorphism group]] for {{math|''n'' ≥ 3}} is <math>\mathbb{Z}/2\mathbb{Z},</math> while the outer automorphism group of {{math|SU(2)}} is the [[trivial group]].

A [[maximal torus]] of [[Algebraic torus#Split rank of a semisimple group|rank]] {{math|''n'' − 1}} is given by the set of diagonal matrices with determinant {{math|1}}. The [[Weyl group#The Weyl group of a connected compact Lie group|Weyl group]] of {{math|SU(''n'')}} is the [[symmetric group]] {{math|''S<sub>n</sub>''}}, which is represented by [[generalized permutation matrix#Signed permutation group|signed permutation matrices]] (the signs being necessary to ensure that the determinant is {{math|1}}).

The [[Lie algebra]] of {{math|SU(''n'')}}, denoted by <math>\mathfrak{su}(n)</math>, can be identified with the set of [[traceless]] [[antiHermitian|anti‑Hermitian]] {{math|''n'' × ''n''}} complex matrices, with the regular [[commutator]] as a Lie bracket. [[Particle physics|Particle physicists]] often use a different, equivalent representation: The set of traceless [[Hermitian]] {{math|''n'' × ''n''}} complex matrices with Lie bracket given by {{math|−''i''}} times the commutator.