Outer Automorphism Groups of Complex and Real Simple Lie Algebras
The preceding interpretation of outer automorphisms as symmetries of a Dynkin diagram follows from the general fact, that for a complex or real simple Lie algebra, the automorphism group is a semidirect product of and, i.e., the short exact sequence
splits. In the complex simple case, this is a classical result, whereas for real simple Lie algebras, this fact has been proven as recently as 2010.
Read more about this topic: Outer Automorphism Group
Famous quotes containing the words outer, groups, complex, real, simple and/or lie:
“The heartless and enormous Outer Black ...”
—Robert Frost (1874–1963)
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)
“In ordinary speech the words perception and sensation tend to be used interchangeably, but the psychologist distinguishes. Sensations are the items of consciousness—a color, a weight, a texture—that we tend to think of as simple and single. Perceptions are complex affairs that embrace sensation together with other, associated or revived contents of the mind, including emotions.”
—Jacques Barzun (b. 1907)
“Somewhere along the line of development we discover who we really are, and then we make our real decision for which we are responsible. Make that decision primarily for yourself because you can never really live anyone else’s life not even your child’s. The influence you exert is through your own life and what you become yourself.”
—Eleanor Roosevelt (1884–1962)
“If, then, we would indeed restore mankind ... let us first be as simple and well as Nature ourselves, dispel the clouds which hang over our own brows, and take up a little life into our pores. Do not stay to be an overseer of the poor, but endeavor to become one of the worthies of the world.”
—Henry David Thoreau (1817–1862)
“No head knows where its rest is
Or may lie down with reason
When war’s usurping claws
Shall take heart escheat....”
—Allen Tate (1899–1979)