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:
“Men just don’t “get” that the reason to become involved is for ourselves. Doing more with our children won’t simply make women happier or keep them “off our backs,” but will create a deeper, more positive connection with the kids.”
—Ron Taffel (20th century)
“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)
“We all enter the world with fairly simple needs: to be protected, to be nurtured, to be loved unconditionally, and to belong.”
—Louise Hart (20th century)