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:
“I will go root away
The noisome weeds which without profit suck
The soil’s fertility from wholesome flowers.”
—William Shakespeare (1564–1616)
“Not out of those, on whom systems of education have exhausted their culture, comes the helpful giant to destroy the old or to build the new, but out of unhandselled savage nature, out of terrible Druids and Berserkirs, come at last Alfred and Shakespeare.”
—Ralph Waldo Emerson (1803–1882)
“Some collaboration has to take place in the mind between the woman and the man before the art of creation can be accomplished. Some marriage of opposites has to be consummated. The whole of the mind must lie wide open if we are to get the sense that the writer is communicating his experience with perfect fullness.”
—Virginia Woolf (1882–1941)
“Don’t confuse hypothesis and theory. The former is a possible explanation; the latter, the correct one. The establishment of theory is the very purpose of science.”
—Martin H. Fischer (1879–1962)