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 Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F4.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:
“History is not what you thought. It is what you can remember. All other history defeats itself.
In Beverly Hills ... they don’t throw their garbage away. They make it into television shows.
Idealism is the despot of thought, just as politics is the despot of will.”
—Mikhail Bakunin (1814–1876)
“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)
“They are a sort of post-house,where the Fates
Change horses, making history change its tune,
Then spur away o’er empires and o’er states,
Leaving at last not much besides chronology,
Excepting the post-obits of theology.”
—George Gordon Noel Byron (1788–1824)