Positive Roots and Simple Roots
Given a root system Φ we can always choose (in many ways) a set of positive roots. This is a subset of Φ such that
- For each root exactly one of the roots, – is contained in .
- For any two distinct such that is a root, .
If a set of positive roots is chosen, elements of are called negative roots.
An element of is called a simple root if it cannot be written as the sum of two elements of . The set of simple roots is a basis of with the property that every vector in is a linear combination of elements of with all coefficients non-negative, or all coefficients non-positive. For each choice of positive roots, the corresponding set of simple roots is the unique set of roots such that the positive roots are exactly those that can be expressed as a combination of them with non-negative coefficients, and such that these combinations are unique.
Read more about this topic: Root System
Famous quotes containing the words positive, roots and/or simple:
“The learner always begins by finding fault, but the scholar sees the positive merit in everything.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“The Plains are not forgiving. Anything that is shallow—the easy optimism of a homesteader; the false hope that denies geography, climate, history; the tree whose roots don’t reach ground water—will dry up and blow away.”
—Kathleen Norris (b. 1947)
“And would you be a poet
Before you’ve been to school?
Ah, well! I hardly thought you
So absolute a fool.
First learn to be spasmodic—
A very simple rule.
For first you write a sentence,
And then you chop it small;
Then mix the bits, and sort them out
Just as they chance to fall:
The order of the phrases makes
No difference at all.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)