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:
“After one look at this planet any visitor from outer space would say “I WANT TO SEE THE MANAGER.””
—William Burroughs (b. 1914)
“Writers and politicians are natural rivals. Both groups try to make the world in their own images; they fight for the same territory.”
—Salman Rushdie (b. 1947)
“I have met charming people, lots who would be charming if they hadn’t got a complex about the British and everyone has pleasant and cheerful manners and I like most of the American voices. On the other hand I don’t believe they have any God and their hats are frightful. On balance I prefer the Arabs.”
—Freya Stark (1893–1993)
“In my craft or sullen art
Exercised in the still night
When only the moon rages
And the lovers lie abed
With all their griefs in their arms,”
—Dylan Thomas (1914–1953)