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 Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F4.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:
“Don’t give your opinions about Art and the Purpose of Life. They are of little interest and, anyway, you can’t express them. Don’t analyse yourself. Give the relevant facts and let your readers make their own judgments. Stick to your story. It is not the most important subject in history but it is one about which you are uniquely qualified to speak.”
—Evelyn Waugh (1903–1966)
“No one is ahead of his time, it is only that the particular variety of creating his time is the one that his contemporaries who are also creating their own time refuse to accept.... For a very long time everybody refuses and then almost without a pause almost everybody accepts. In the history of the refused in the arts and literature the rapidity of the change is always startling.”
—Gertrude Stein (1874–1946)
“At present cats have more purchasing power and influence than the poor of this planet. Accidents of geography and colonial history should no longer determine who gets the fish.”
—Derek Wall (b. 1965)