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| {{For|other uses|PIN Group (disambiguation)}}
| | == changes Wu Kai silver white god == |
| {{redirect|Pinor|other uses|Piñor}}
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| 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]].
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| 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 (and dimension is greater than 2), it is both.
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| The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of [[reflection through the origin]], generally denoted −''I''.
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| ==General definition==
| | <li>[http://borise.com/forum.php?mod=viewthread&tid=7556 http://borise.com/forum.php?mod=viewthread&tid=7556]</li> |
| {{See also|Clifford algebra#Spin_and_Pin_groups}}
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| | | <li>[http://www.qzmuseum.net/Review.asp?NewsID=139 http://www.qzmuseum.net/Review.asp?NewsID=139]</li> |
| ==Definite form==
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| [[Image:Spin-Pin-SO-O-definite.svg|right]] | | <li>[http://www.dfjds.cn/plus/feedback.php?aid=105 http://www.dfjds.cn/plus/feedback.php?aid=105]</li> |
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| The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it [[Double covering group|double cover]]s 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.<ref group="note">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.</ref>
| | </ul> |
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| 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 <math>v^2=Q(v) \in C\ell(V,Q)</math>), one writes
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| :<math>\mbox{Pin}_+(n) := \mbox{Pin}(n,0) \qquad \mbox{Pin}_-(n) := \mbox{Pin}(0,n)</math>
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| and these both map onto O(''n'') = O(''n'', 0) = O(0, ''n'').
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| By contrast, we have the natural isomorphism<ref group="note">They are subalgebras of the different algebras <math>C\ell(n,0) \not\cong C\ell(0,n)</math>, but they are equal as subsets of the vector spaces <math>C\ell(n,0) = C\ell(0,n) = \Lambda^* \mathbf{R}^n</math>, and carry the same algebra structure, hence they are naturally identified.</ref> Spin(''n'', 0) ≅ Spin(0, ''n'') and they are both the (unique) [[double cover]]{{dn|date=July 2012}} of the [[special orthogonal group]] SO(''n''), which is the (unique) [[universal cover]] for ''n'' ≥ 3.
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| ==Indefinite form==
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| {{Expand section|date=December 2009}}
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| There are as many as eight different double covers of [[generalized orthogonal group|O(''p'', ''q'')]], for ''p'', ''q'' ≠ 0, which correspond to the extensions of the center (which is either ''C''<sub>2</sub> × ''C''<sub>2</sub> or ''C''<sub>4</sub>) by ''C''<sub>2</sub>. Only two of them are pin groups—those that admit the [[Clifford algebra]] as a representation. They are called Pin(''p'', ''q'') and Pin(''q'', ''p'') respectively.
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| ==As topological group==
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| Every [[connected space|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 are [[principal homogeneous space]]s for the identity component) but the group structure on other components is not uniquely determined in general.
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| 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.
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| Recently, [[Andrzej Trautman]] <ref>{{cite journal|title=Double Covers of Pseudo-orthogonal Groups|author=A. Trautman|journal=Clifford Analysis and Its Applications, NATO Science Series,|volume=25|year=2001|pages=377–388}}</ref> found the set of all 32 inequivalent double covers of O(''p'') x O(''q''), the maximal compact subgroup of O(''p'', ''q'') and an explicit construction of 8 double covers of the same group O(''p'', ''q'').
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| ==Construction==
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| The two pin groups correspond to the two central extensions
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| :<math>1 \to \{\pm 1\} \to \mbox{Pin}_\pm(V) \to O(V) \to 1.</math>
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| The group structure on Spin(''V'') (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 ±1 ambiguity.
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| The two extensions are distinguished by whether the preimage of a reflection squares to ±1 ∈ Ker (Spin(''V'') → SO(''V'')), and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in O(''V''), ''r''<sup>2</sup> = 1, so the square of the preimage of a reflection (which has determinant one) must be in the kernel of Spin<sub>±</sub>(''V'') → SO(''V''), so <math>\tilde r^2 = \pm 1</math>, and either choice determines a pin group (since all reflections are conjugate by an element of SO(''V''), which is connected, all reflections must square to the same value).
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| Concretely, in Pin<sub>+</sub>, <math>\tilde r</math> has order 2, and the preimage of a subgroup {1, ''r''} is ''C''<sub>2</sub> × ''C''<sub>2</sub>: if one repeats the same [[Reflection (mathematics)|reflection]] twice, one gets the identity.
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| In Pin<sub>−</sub>, <math>\tilde r</math> has order 4, and the preimage of a subgroup {1, ''r''} is ''C''<sub>4</sub>: if one repeats the same reflection twice, one gets "a [[rotation]] by 2π"—the non-trivial element of Spin(''V'') → SO(''V'') can be interpreted as "rotation by 2π" (every axis yields the same element).
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| ===Low dimensions===
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| In 2 dimensions, the distinction between Pin<sub>+</sub> and Pin<sub>−</sub> mirrors the distinction between the [[dihedral group]] of a 2''n''-gon and the [[dicyclic group]] of the cyclic group ''C''<sub>2''n''</sub>.
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| In Pin<sub>+</sub>, the preimage of the dihedral group of an ''n''-gon, considered as a subgroup Dih<sub>''n''</sub> < O(2), is the dihedral group of an 2''n''-gon, Dih<sub>2''n''</sub> < Pin<sub>+</sub>(2), while in Pin<sub>−</sub>, the preimage of the dihedral group is the [[dicyclic group]] <math>\mbox{Dic}_n < \mbox{Pin}_-(2)</math>.
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| The resulting [[commutative square]] of subgroups for Spin(2), Pin<sub>+</sub>(2), SO(2), O(2) – namely ''C''<sub>2''n''</sub>, Dih<sub>2''n''</sub>, ''C<sub>n</sub>'', Dih<sub>''n''</sub> – is also obtained using the [[projective orthogonal group]] (going down from O by a 2-fold quotient, instead of up by a 2-fold cover) in the square SO(2), O(2), PSO(2), PO(2), though in this case it is also realized geometrically, as "the projectivization of a 2''n''-gon in the circle is an ''n''-gon in the projective line".
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| In 1 dimension, the pin groups are congruent to the first dihedral and dicyclic groups:
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| :<math>\begin{align}
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| \mbox{Pin}_+(1) &\cong C_2 \times C_2 = \mbox{Dih}_1\\
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| \mbox{Pin}_-(1) &\cong C_4 = \mbox{Dic}_1.
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| \end{align}</math>
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| ==Center==
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| The center is either (C2 × C2 or C4) by C2.
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| ==Name==
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| The name was introduced in {{Harv|Atiyah|Bott|Shapiro|1964|loc=page 3, line 17}}, where they state "This joke is due to [[Jean-Pierre Serre|J-P. Serre]]".
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| 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".
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| ==Notes==
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| {{Reflist|group=note}}
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| ==References==
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| {{Reflist}}
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| {{Refbegin}}
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| * {{Citation |first1 = M.F. | last1 = Atiyah | first2 = R. | last2 = Bott | first3 = A. | last3 = Shapiro | authorlink1=Michael Atiyah | authorlink2=Raoul Bott| title = Clifford modules | journal = [[Topology (journal)|Topology]] | volume = 3, suppl. 1 | year = 1964 | pages = 3–38 }}
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| {{Refend}}
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Pin Group}}
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| [[Category:Lie groups]]
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changes Wu Kai silver white god
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'Then I'll look.' Luo Feng burning to the ultimate god, Wu Kai moment also instilled into gods.
changes Wu Kai silver white god, instant breath rose, Luo Feng Wu Kai God already muster a second ルイヴィトン ダミエ バッグ tier, one time strength skyrocketed, with opponents is holding the ghost knife.
'human!' body bathed in the glow of the Scarlet Beast Brady circles one after the right hand outstretched, as if each grab ルイヴィトン 新作 2014 hold an endless ルイヴィトン ケース universe like, then the right hand one finger changes, suddenly a swirl channels appear directly In the golden waves. It is followed by a finger of the left hand is also changing, and ルイヴィトン 財布 新作 again a swirl channel was born.
two channel walls
swirl channel is like ルイヴィトン キャップ a mirror that reflects everything outside.
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