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)
“He who sins easily, sins less. The very power
Renders less vigorous the roots of evil.”
—Ovid (Publius Ovidius Naso)
“The Americans never use the word peasant, because they have no idea of the class which that term denotes; the ignorance of more remote ages, the simplicity of rural life, and the rusticity of the villager have not been preserved among them; and they are alike unacquainted with the virtues, the vices, the coarse habits, and the simple graces of an early stage of civilization.”
—Alexis de Tocqueville (1805–1859)