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Concept
Pin group
Résumé
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group.
In general the map from the Pin group to the orthogonal group is not surjective or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both.
The non-trivial element of the kernel is denoted which should not be confused with the orthogonal transform of reflection through the origin, generally denoted
Clifford algebra#Spin and Pin groups
Let be a vector space with a non-degenerate quadratic form . The pin group is the subset of the Clifford algebra consisting of elements of the form , where the are vectors such that . The spin group is defined similarly, but with restricted to be even; it is a subgroup of the pin group.
In this article, is always a real vector space. When has basis vectors satisfying and the pin group is denoted Pin(p, q).
Geometrically, for vectors with , is the reflection of a vector across the hyperplane orthogonal to . More generally, an element of the pin group acts on vectors by transforming to , which is the composition of k reflections. Since every orthogonal transformation can be expressed as a composition of reflections (the Cartan–Dieudonné theorem), it follows that this representation of the pin group is a homomorphism from the pin group onto the orthogonal group. This is often called the twisted adjoint representation. The elements ±1 of the pin group are the elements which map to the identity , and every element of O(p, q) corresponds to exactly two elements of Pin(p, q).
The pin group of a definite form maps onto the orthogonal group, and each component is simply connected (in dimension 3 and higher): 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.
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