Skip to main content
MathOverflow

Return to Answer

A formula simplified
Source Link
Mikhail Borovoi
Mikhail Borovoi
  • 14.1k
  • 2
  • 31
  • 71

Maybe I will just answer the specific question the OP is interested in by giving a reference of sorts (in case it is helpful to the OP).

In Bourbaki Éléments de mathématique: groupes et algèbres de Lie. Chapitre 9 Exercise 3 one is asked to prove that $\mathrm{Out}(PSU(n))\cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$.

It seems to me that in this case one can show this explicitly. The non-trivial non-inner automorphism is represented by the Cartan involution $A\mapsto ((A)^{-1})^T$$A\mapsto (A^{-1})^T$ on $PSL(n,\mathbb{C})$ which simplifies to $A\mapsto \overline{A}$ on $PSU(n)$.

Maybe I will just answer the specific question the OP is interested in by giving a reference of sorts (in case it is helpful to the OP).

In Bourbaki Éléments de mathématique: groupes et algèbres de Lie. Chapitre 9 Exercise 3 one is asked to prove that $\mathrm{Out}(PSU(n))\cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$.

It seems to me that in this case one can show this explicitly. The non-trivial non-inner automorphism is represented by the Cartan involution $A\mapsto ((A)^{-1})^T$ on $PSL(n,\mathbb{C})$ which simplifies to $A\mapsto \overline{A}$ on $PSU(n)$.

Maybe I will just answer the specific question the OP is interested in by giving a reference of sorts (in case it is helpful to the OP).

In Bourbaki Éléments de mathématique: groupes et algèbres de Lie. Chapitre 9 Exercise 3 one is asked to prove that $\mathrm{Out}(PSU(n))\cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$.

It seems to me that in this case one can show this explicitly. The non-trivial non-inner automorphism is represented by the Cartan involution $A\mapsto (A^{-1})^T$ on $PSL(n,\mathbb{C})$ which simplifies to $A\mapsto \overline{A}$ on $PSU(n)$.

Source Link
Sean Lawton
Sean Lawton
  • 8.5k
  • 3
  • 46
  • 78

Maybe I will just answer the specific question the OP is interested in by giving a reference of sorts (in case it is helpful to the OP).

In Bourbaki Éléments de mathématique: groupes et algèbres de Lie. Chapitre 9 Exercise 3 one is asked to prove that $\mathrm{Out}(PSU(n))\cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$.

It seems to me that in this case one can show this explicitly. The non-trivial non-inner automorphism is represented by the Cartan involution $A\mapsto ((A)^{-1})^T$ on $PSL(n,\mathbb{C})$ which simplifies to $A\mapsto \overline{A}$ on $PSU(n)$.