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:
“No man ever looks at the world with pristine eyes. He sees it edited by a definite set of customs and institutions and ways of thinking.”
—Ruth Benedict (1887–1948)
“The legislator should direct his attention above all to the education of youth; for the neglect of education does harm to the constitution. The citizen should be molded to suit the form of government under which he lives. For each government has a peculiar character which originally formed and which continues to preserve it. The character of democracy creates democracy, and the character of oligarchy creates oligarchy.”
—Aristotle (384–323 B.C.)