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)
“Black creeps from root to root,
each leaf
cuts another leaf on the grass,
shadow seeks shadow,
then both leaf
and leaf-shadow are lost.”
—Hilda Doolittle (1886–1961)
“We find ourselves under the government of a system of political institutions, conducing more essentially to the ends of civil and religious liberty, than any of which the history of former times tells us.”
—Abraham Lincoln (1809–1865)