Root Systems and Lie Theory
Irreducible root systems classify a number of related objects in Lie theory, notably the
- simple Lie groups (see the list of simple Lie groups), including the
- simple complex Lie groups;
- their associated simple complex Lie algebras; and
- simply connected complex Lie groups which are simple modulo centers.
In each case, the roots are non-zero weights of the adjoint representation.
In the case of a simply connected simple compact Lie group G with maximal torus T, the root lattice can naturally be identified with Hom(T, T) and the coroot lattice with Hom(T, T); see Adams (1983).
For connections between the exceptional root systems and their Lie groups and Lie algebras see E8, E7, E6, F4, and G2.
|
||||
Read more about this topic: Root System
Famous quotes containing the words root, systems, lie and/or theory:
“A radical generally meant a man who thought he could somehow pull up the root without affecting the flower. A conservative generally meant a man who wanted to conserve everything except his own reason for conserving anything.”
—Gilbert Keith Chesterton (1874–1936)
“I am beginning to suspect all elaborate and special systems of education. They seem to me to be built up on the supposition that every child is a kind of idiot who must be taught to think.”
—Anne Sullivan (1866–1936)
“A box of teak, a box of sandalwood,
A brass-ringed spyglass in a case,
A coin, leaf-thin with many polishings,
Last kingdom of a gold forgotten face,
These lie about the room....”
—Philip Larkin (1922–1986)
“The theory of rights enables us to rise and overthrow obstacles, but not to found a strong and lasting accord between all the elements which compose the nation.”
—Giuseppe Mazzini (1805–1872)