Outer Automorphism Groups of Complex Lie Groups
Let G now be a connected reductive group over an algebraically closed field. Then any two Borel subgroups are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel subgroup. Associated to the Borel subgroup is a set of simple roots, and the outer automorphism may permute them, while preserving the structure of the associated Dynkin diagram. In this way one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out(G).
D4 has a very symmetric Dynkin diagram, which yields a large outer automorphism group of Spin(8), namely Out(Spin(8)) = S3; this is called triality.
Read more about this topic: Outer Automorphism Group
Famous quotes containing the words outer, groups, complex and/or lie:
“I know, it must have been my imagination, but it makes me realize how desperately alone the Earth is. Hanging in space like a speck of food floating in the ocean. Sooner or later to be swallowed up by some creature floating by.... Time will tell, Dr. Mason. We can only wait and wonder. Wonder how, wonder when.”
—Tom Graeff. Young astronomer, Teenagers from Outer Space, after just seeing the invading spaceship through his telescope, and dismissing it (1959)
“screenwriter
Policemen so cherish their status as keepers of the peace and protectors of the public that they have occasionally been known to beat to death those citizens or groups who question that status.”
—David Mamet (b. 1947)
“It’s a complex fate, being an American, and one of the responsibilities it entails is fighting against a superstitious valuation of Europe.”
—Henry James (1843–1916)
“O, the world hath not a sweeter creature! She might lie by an emperor’s side and command him tasks.”
—William Shakespeare (1564–1616)