Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams

Abstract

Abstract Let 𝔤 ${{\mathfrak{g}}}$ be a simple Lie algebra. Let Aut ⁢ ( 𝔤 ) ${{\rm Aut}(\mathfrak{g})}$ be the group of all automorphisms on 𝔤 ${\mathfrak{g}}$ , and let Int ⁢ ( 𝔤 ) ${{\rm Int}(\mathfrak{g})}$ be its identity component. The outer automorphism group of 𝔤 ${\mathfrak{g}}$ is defined as Aut ⁢ ( 𝔤 ) / Int ⁢ ( 𝔤 ) ${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$ . If 𝔤 ${\mathfrak{g}}$ is complex and has Dynkin diagram D ${{\rm D}}$ , then Aut ⁢ ( 𝔤 ) / Int ⁢ ( 𝔤 ) ${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$ is isomorphic to Aut ⁢ ( D ) ${{\rm Aut}({\rm D})}$ . We provide an analogous result for the real case. For 𝔤 ${\mathfrak{g}}$ real, we let 𝔤 ${\mathfrak{g}}$ be represented by a painted diagram P ${{\rm P}}$ . Depending on whether the Cartan involution of 𝔤 ${\mathfrak{g}}$ belongs to Int ⁢ ( 𝔤 ) ${{\rm Int}(\mathfrak{g})}$ , we show that Aut ⁢ ( 𝔤 ) / Int ⁢ ( 𝔤 ) ${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$ is isomorphic to Aut ⁢ ( P ) ${{\rm Aut}({\rm P})}$ or Aut ⁢ ( P ) × ℤ 2 ${{\rm Aut}({\rm P})\times\mathbb{Z}_{2}}$ . This result extends to the outer automorphism groups of all real semisimple Lie algebras.