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:
“Our role is to support anything positive in black life and destroy anything negative that touches it. You have no other reason for being. I don’t understand art for art’s sake. Art is the guts of the people.”
—Elma Lewis (b. 1921)
“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 a curious emotion, this certain homesickness I have in mind. With Americans, it is a national trait, as native to us as the rollercoaster or the jukebox. It is no simple longing for the home town or country of our birth. The emotion is Janus-faced: we are torn between a nostalgia for the familiar and an urge for the foreign and strange. As often as not, we are homesick most for the places we have never known.”
—Carson McCullers (1917–1967)