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:
“Whoever influences the child’s life ought to try to give him a positive view of himself and of his world. The child’s future happiness and his ability to cope with life and relate to others will depend on it.”
—Bruno Bettelheim (20th century)
“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)
“Cannot a plain man live and think no harm,
But that his simple truth must be abused
With silken, sly, insinuating Jacks?”
—William Shakespeare (1564–1616)