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)
“The cold smell of potato mould, the squelch and slap
Of soggy peat, the curt cuts of an edge
Through living roots awaken in my head.
But I’ve no spade to follow men like them.”
—Seamus Heaney (b. 1939)
“‘Tis the gift to be simple ‘tis the gift to be free
‘Tis the gift to come down where you ought to be
And when we find ourselves in the place just right
‘Twill be in the valley of love and delight.”
—Unknown. ‘Tis the Gift to Be Simple.
AH. American Hymns Old and New, Vols. I–II. Vol. I, with music; Vol. II, notes on the hymns and biographies of the authors and composers. Albert Christ-Janer, Charles W. Hughes, and Carleton Sprague Smith, eds. (1980)