- Pin group
In
mathematics , the pin group is a certain subgroup of theClifford algebra associated to aquadratic space . It maps 2-to-1 to theorthogonal group , just as thespin group maps 2-to-1 to thespecial orthogonal group .In general the map from the Pin group to the orthogonal group is "not" onto or a
universal covering space , but if the quadratic form is definite, it is both.General definition
Definite form
The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it
double cover s the orthogonal group. The pin groups for a positive definite quadratic form and for its negative are not isomorphic, but the orthogonal groups are. [In fact, they are equal as subsets of GL("V"), not just isomorphic as abstract groups: an operator preserves a form if and only if it preserves the negative form.]In terms of the standard forms, , but .Using the "+" sign convention for Clifford algebras (where ), one writes:and these both map onto .
By contrast, we have the isomorphism [They are subalgebras of the different algebras , but they are equal as subsets of the vector spaces , and carry the same algebra structure, hence they are naturally identified.] and they are both the (unique)
universal cover of thespecial orthogonal group SO("n").Indefinite form
As topological group
Every connected
topological group has a unique universal cover as a topological space, which has a unique group structure as a central extension by the fundamental group. For a disconnected topological group, there is a unique universal cover of the identity component of the group, and one can take the same cover as topological spaces on the other components (which areprincipal homogeneous space s for the identity component) but the group structure on other components is not uniquely determined in general.The Pin and Spin groups are "particular" topological groups associated to the orthogonal and special orthogonal groups, coming from Clifford algebras: there are other similar groups, corresponding to other double covers or to other group structures on the other components, but they are not referred to as Pin or Spin groups, nor studied much.
Construction
The two pin groups correspond to the two central extensions:The group structure on (the connected component of determinant 1) is already determined; the group structure on the other component is determined up to the center, and thus has a ambiguity.
The two extensions are distinguished by whether the preimage of a reflection squares to , and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in , , so the square of the preimage of a reflection (which has determinant one) must be in the kernel of , so , and either choice determines a pin group (since all reflections are conjugate by an element of , which is connected, all reflections must square to the same value).
Concretely, in , has order 2, and the preimage of a subgroup is :if one repeats the same reflection twice, one gets the identity.
In , has order 4, and the preimage of a subgroup is :if one repeats the same reflection twice, one gets "a
rotation by 2π"—the non-trivial element of can be interpreted as "rotation by 2π" (every axis yields the same element).Low dimensions
In 2 dimensions, the distinction between and mirrors the distinction between the
dihedral group of a -gon and thedicyclic group of the cyclic group .In , the preimage of the dihedral group of an -gon, considered as a subgroup ,is the dihedral group of an -gon, ,while in , the preimage of the dihedral group isthe
dicyclic group .In 1 dimension, the pin groups are congruent to the first dihedral and dicyclic groups::
Center
Indefinite Pin groups
There are as many as eight different double covers of Spin(p,q), for , which correspond to the extensions of the center (which is either or ) by . Only two of them are taken to be pin groups, namely, those which admit the
Clifford algebra as a representation. They are called Pin(p,q) and Pin(q,p) respectively.Name
The name was introduced in M.F. Atiyah, R. Bott, A. Shapiro:"Clifford modules", Topology 3, suppl. 1 (1964), pp. 3-38, on page 3, line 17,where they state "This joke is due to J-P. Serre".It is a
back-formation from Spin: "Pin is to O("n") as Spin is to SO("n")", hence dropping the "S" from "Spin" yields "Pin". Further, the word "Pin" sounds like vulgar French slang when pronounced in French, which is alluded to by the name originating with (or being attributed to) Serre. [cite newsgroup
title = Re: Math jokes (dirty): Explanation
author = Pertti Lounesto
date = 04 Dec 1993 09:36:24 GMT
newsgroup = sci.math
id = LOUNESTO.93Dec4113624@dopey.hut.fi
url = http://groups.google.com/group/sci.math/tree/browse_frm/month/1993-12/f6a164fb29095c60?rnum=211&_done=%2Fgroup%2Fsci.math%2Fbrowse_frm%2Fmonth%2F1993-12%3Ffwc%3D2%26
accessdate = 2007-11-27French slang " [http://fr.wiktionary.org/wiki/pine pine] " means "penis", and further, saying that the "pin group has 2 parts" (the even part (Spin) and the odd part) suggests proximate anatomical comparisons.]
References
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