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 cold smell of potato mould, the squelch and slap
Of soggy peat, the curt cuts of an edge
Through living roots awaken in my head.
But I’ve no spade to follow men like them.”
—Seamus Heaney (b. 1939)
“Young children make only the simple assumption: “This is life—you go along....” He stands ready to go along with whatever adults seem to want. He stands poised, trying to figure out what they want. The young child is almost at the mercy of adults—it is so important to him to please.”
—James L. Hymes, Jr. (20th century)