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:
“Property is the fruit of labor—property is desirable—is a positive good in the world.”
—Abraham Lincoln (1809–1865)
“Look at this poet William Carlos Williams: he is primitive and native, and his roots are in raw forest and violent places; he is word-sick and place-crazy. He admires strength, but for what? Violence! This is the cult of the frontier mind.”
—Edward Dahlberg (1900–1977)
“Meantime the education of the general mind never stops. The reveries of the true and simple are prophetic. What the tender poetic youth dreams, and prays, and paints today, but shuns the ridicule of saying aloud, shall presently be the resolutions of public bodies, then shall be carried as grievance and bill of rights through conflict and war, and then shall be triumphant law and establishment for a hundred years, until it gives place, in turn, to new prayers and pictures.”
—Ralph Waldo Emerson (1803–1882)