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 most positive men are the most credulous.”
—Jonathan Swift (1667–1745)
“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)
“School divides life into two segments, which are increasingly of comparable length. As much as anything else, schooling implies custodial care for persons who are declared undesirable elsewhere by the simple fact that a school has been built to serve them.”
—Ivan Illich (b. 1926)