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:
“It is easy and dismally enervating to think of opposition as merely perverse or actually evil—far more invigorating to see it as essential for honing the mind, and as a positive good in itself. For the day that moral issues cease to be fought over is the day the word “human” disappears from the race.”
—Jill Tweedie (b. 1936)
“The roots of the grass strain,
Tighten, the earth is rigid, waits—he is waiting—
And suddenly, and all at once, the rain!”
—Archibald MacLeish (1892–1982)
“It is not the simple statement of facts that ushers in freedom; it is the constant repetition of them that has this liberating effect. Tolerance is the result not of enlightenment, but of boredom.”
—Quentin Crisp (b. 1908)