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:
“God is a foreman with certain definite views
Who orders life in shifts of work and leisure.”
—Seamus Heaney (b. 1939)
“The playing adult steps sideward into another reality; the playing child advances forward to new stages of mastery....Child’s play is the infantile form of the human ability to deal with experience by creating model situations and to master reality by experiment and planning.”
—Erik H. Erikson (20th century)