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)
“To the young mind, every thing is individual, stands by itself. By and by, it finds how to join two things, and see in them one nature; then three, then three thousand; and so, tyrannized over by its own unifying instinct, it goes on tying things together, diminishing anomalies, discovering roots running underground, whereby contrary and remote things cohere, and flower out from one stem.”
—Ralph Waldo Emerson (1803–1882)
“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)