The simply connected cover of the orthogonal group. Its restriction along the inclusion of the special orthogonal group is the Spin group. Hence the -group is “like the Spin group, but including reflections”.
Conventions as in (Varadarajan 04, section 5.3).
We write for the corresponding quadratic form.
Since the tensor algebra is naturally -graded, the Clifford algebra is naturally -graded.
Let be the -dimensional Cartesian space with its canonical scalar product. Write for the complexification of its Clifford algebra.
There exists a unique complex representation
of the algebra of smallest dimension
The Pin group of a quadratic vector space, def. 1, is the subgroup of the group of units in the Clifford algebra
on those elements which are multiples of elements with .
The Spin group is the further subgroup of on those elements which are even number multiples of elements with .
Specifically, “the” Spin group is
|group||symbol||universal cover||symbol||higher cover||symbol|
|orthogonal group||Pin group||Tring group|
|special orthogonal group||Spin group||String group|
|anti de Sitter group|
|Poincaré group||Poincaré spin group|
|super Poincaré group|
A standard textbook account is