Definite Form
The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it double covers the orthogonal group. The pin groups for a positive definite quadratic form Q and for its negative −Q are not isomorphic, but the orthogonal groups are.
In terms of the standard forms, O(n, 0) = O(0,n), but Pin(n, 0) and Pin(0, n) are not isomorphic. Using the "+" sign convention for Clifford algebras (where ), one writes
and these both map onto O(n) = O(n, 0) = O(0, n).
By contrast, we have the natural isomorphism Spin(n, 0) ≅ Spin(0, n) and they are both the (unique) double cover of the special orthogonal group SO(n), which is the (unique) universal cover for n ≥ 3.
Read more about this topic: Pin Group
Famous quotes containing the words definite and/or form:
“Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff to any degree of fineness; but, nevertheless, what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulae will not get a definite result out of loose data.”
—Thomas Henry Huxley (1825–1895)
“Quite generally, the familiar, just because it is familiar, is not cognitively understood. The commonest way in which we deceive either ourselves or others about understanding is by assuming something as familiar, and accepting it on that account; with all its pros and cons, such knowing never gets anywhere, and it knows not why.... The analysis of an idea, as it used to be carried out, was, in fact, nothing else than ridding it of the form in which it had become familiar.”
—Georg Wilhelm Friedrich Hegel (1770–1831)