Vtyjkyuk

The goal is to write down explicit expressions of inner / outer automorphism of SU($n$), for $ngeq 2$.

We know that SU(2) has an SO(3) ($supseteq mathbbZ_2$)-inner automorphism,

while SU(n) has a $mathbbZ_2$-outer automorphism. For simply connected simple Lie groups, the outer automorphisms come from the automorphisms of the Dynkin diagram. See also the discussion in MO.

• For SU(2), we can write the group element as
  $$ g_textSU(2) = expleft(thetasum_k=1^3 i t_k fracsigma_k2right) $$
  where $(t_1,t_2,t_3)$ forms a unit vector [effectively pointing in some direction on a unit 2-sphere $S^2$], and $sigma_k$ are Pauli matrices:
  beginalign
  sigma_1 &=
  beginpmatrix
  0&1\
  1&0
  endpmatrix \
  sigma_2 &=
  beginpmatrix
  0&-i\
  i&0
  endpmatrix \
  sigma_3 &=
  beginpmatrix
  1&0\
  0&-1
  endpmatrix ,.
  endalign
  Notice that any group element on $SU(2)$ can be parametrized by some $theta$ and $(t_1,t_2,t_3)$. Also $theta$ has a periodicity $[0,4 pi)$.

The inner automorphism is given by,
$$
x g_textSU(2) x^-1=
expleft(thetasum_k=1^3 (-i) t_k fracsigma_k^T2right)
expleft(thetasum_k=1^3 (-i) t_k fracsigma_k^*2right)
=g_textSU(2)^*.
$$
where
$$x=e^ifracpi 2sigma_2 = isigma_2= beginpmatrix
0&1\
-1&0
endpmatrix in textSU(2),$$

• For SU($n$), $n>2$,

Do we have a simple expression of $g_textSU(n)$?

(It looks like the answer given here in ME by Anon is negative. But the Refs here Ref 1, Ref 2, Ref 3 writing down suggestive expressions
$$ g_textSU(n) = expleft(thetasum_k=1^n^2-1 i t_k fraclambda_k2right)??? $$

So the outer automorphism of SU(n) simply sends $g_textSU(n)$ to its complex conjugation
$$
g_textSU(n) to g_textSU(n)^*?
$$