History
The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem). He used them in his attempt to classify all simple Lie algebras over the field of complex numbers. Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F4. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.
Killing investigated the structure of a Lie algebra, by considering (what is now called) a Cartan subalgebra . Then he studied the roots of the characteristic polynomial, where . Here a root is considered as a function of, or indeed as an element of the dual vector space . This set of roots form a root system inside, as defined above, where the inner product is the Killing form.
Read more about this topic: Root System
Famous quotes containing the word history:
“Systematic philosophical and practical anti-intellectualism such as we are witnessing appears to be something truly novel in the history of human culture.”
—Johan Huizinga (1872–1945)
“There has never been in history another such culture as the Western civilization M a culture which has practiced the belief that the physical and social environment of man is subject to rational manipulation and that history is subject to the will and action of man; whereas central to the traditional cultures of the rivals of Western civilization, those of Africa and Asia, is a belief that it is environment that dominates man.”
—Ishmael Reed (b. 1938)
“There is no example in history of a revolutionary movement involving such gigantic masses being so bloodless.”
—Leon Trotsky (1879–1940)