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:
“People who talk about revolution and class struggle without referring explicitly to everyday life, without understanding what is subversive about love and what is positive in the refusal of constraints, such people have a corpse in their mouth.”
—Raoul Vaneigem (b. 1934)
““Jim,” she said earnestly, “if I was put down there in the middle of the night, I could find my way all over that little town; and along the river to the next town, where my grandmother lived. My feet remember all the little paths through the woods, and where the big roots stick out to trip you. I ain’t never forgot my own country.””
—Willa Cather (1873–1947)
“no thread
Of cloudy silver sprinkles in your gown
Its venom of renown, and on your head
No crown is simpler than the simple hair.”
—Wallace Stevens (1879–1955)