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:
“Blessed be the inventor of photography! I set him above even the inventor of chloroform! It has given more positive pleasure to poor suffering humanity than anything else that has “cast up” in my time or is like to—this art by which even the “poor” can possess themselves of tolerable likenesses of their absent dear ones. And mustn’t it be acting favourably on the morality of the country?”
—Jane Welsh Carlyle (1801–1866)
“Our roots are in the dark; the earth is our country. Why did we look up for blessing—instead of around, and down? What hope we have lies there. Not in the sky full of orbiting spy-eyes and weaponry, but in the earth we have looked down upon. Not from above, but from below. Not in the light that blinds, but in the dark that nourishes, where human beings grow human souls.”
—Ursula K. Le Guin (b. 1929)
“The Cairo conference ... is about a complicated web of education and employment, consumption and poverty, development and health care. It is also about whether governments will follow where women have so clearly led them, toward safe, simple and reliable choices in family planning. While Cairo crackles with conflict, in the homes of the world the orthodoxies have been duly heard, and roundly ignored.”
—Anna Quindlen (b. 1952)