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)
“If church prelates, past or present, had even an inkling of physiology they’d realise that what they term this inner ugliness creates and nourishes the hearing ear, the seeing eye, the active mind, and energetic body of man and woman, in the same way that dirt and dung at the roots give the plant its delicate leaves and the full-blown rose.”
—Sean O’Casey (1884–1964)
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)