Dual Root System and Coroots
See also: Langlands dual groupIf Φ is a root system in V, the coroot αV of a root α is defined by
The set of coroots also forms a root system ΦV in V, called the dual root system (or sometimes inverse root system). By definition, αV V = α, so that Φ is the dual root system of ΦV. The lattice in V spanned by ΦV is called the coroot lattice. Both Φ and ΦV have the same Weyl group W and, for s in W,
If Δ is a set of simple roots for Φ, then ΔV is a set of simple roots for ΦV.
Read more about this topic: Root System
Famous quotes containing the words dual, root and/or system:
“Thee for my recitative,
Thee in the driving storm even as now, the snow, the winter-day
declining,
Thee in thy panoply, thy measur’d dual throbbing and thy beat
convulsive,
Thy black cylindric body, golden brass and silvery steel,”
—Walt Whitman (1819–1892)
“The lack of money is the root of all evil.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“A person, seasoned with a just sense of the imperfections of natural reason, will fly to revealed truth with the greatest avidity: while the haughty Dogmatist, persuaded that he can erect a compleat system of Theology by the mere help of philosophy, disdains any further aid, and rejects this adventitious instructor.”
—David Hume (1711–1776)