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| {{Group theory sidebar}}
| | My name is Joanne Humphreys but everybody calls me Joanne. I'm from Austria. I'm studying at the high school (2nd year) and I play the Bass Guitar for 7 years. Usually I choose songs from my famous films :). <br>I have two brothers. I love Leaf collecting and pressing, watching TV (2 Broke Girls) and Roller Derby.<br><br>my web blog - [http://www.youtube.com/watch?v=OwB43bJoEfk work from home] |
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| In [[mathematics]] the '''spin group''' Spin(''n'') <ref>{{Cite book | last1=Lawson | first1=H. Blaine | last2=Michelsohn | first2=Marie-Louise | author-link2=Marie-Louise Michelsohn| title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 | postscript=<!--None-->}} page 14</ref><ref>{{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}} page 15</ref> is the [[covering space|double cover]] of the [[special orthogonal group]] SO(''n'')=SO(''n'',R), such that there exists a [[short exact sequence]] of [[Lie group]]s
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| :<math>1 \to \mathbf{Z}_2 \to \operatorname{Spin}(n) \to \operatorname{SO}(n) \to 1.</math>
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| As a Lie group Spin(''n'') therefore shares its dimension, ''n'' (''n'' − 1)/2, and its [[Lie algebra]] with the special orthogonal group. For ''n'' > 2, Spin(''n'') is [[simply connected]] and so coincides with the [[universal cover]] of [[special orthogonal group|SO(''n'')]].
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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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| Spin(''n'') can be constructed as a [[subgroup]] of the invertible elements in the [[Clifford algebra]] ''C''ℓ(''n'').
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| ==Accidental isomorphisms==
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| In low dimensions, there are [[isomorphism]]s among the classical Lie groups called ''[[accidental isomorphism]]s''. For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups, owing to low dimensional isomorphisms between the [[root system]]s (and corresponding isomorphisms of [[Dynkin diagram]]s) of the different families of [[simple Lie algebra]]s. Specifically, we have
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| :Spin(1) = [[Orthogonal group|O(1)]]
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| :Spin(2) = [[U(1)]] = [[Special orthogonal group|SO(2)]] which acts on ''z'' in '''R'''<sup>2</sup> by double phase rotation ''z'' ↦ ''u''<sup>2</sup>''z''
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| :Spin(3) = [[Symplectic group|Sp(1)]] = [[Special unitary group|SU(2)]], corresponding to <math>B_1 \cong A_1.</math>
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| :Spin(4) = SU(2) × SU(2), corresponding to <math>D_2 \cong A_1 \times A_1.</math>
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| :Spin(5) = [[Symplectic group|Sp(2)]], corresponding to <math>B_2 \cong C_2.</math>
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| :Spin(6) = [[Special unitary group|SU(4)]], corresponding to <math>D_3 \cong A_3.</math>
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| There are certain vestiges of these isomorphisms left over for ''n'' = 7, 8 (see [[Spin(8)]] for more details). For higher ''n'', these isomorphisms disappear entirely.
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| ==Indefinite signature==
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| In [[signature (quadratic form)|indefinite signature]], the spin group Spin(''p'', ''q'') is constructed through [[Clifford algebra]]s in a similar way to standard spin groups. It is a connected [[covering group|double cover]] of SO<sub>0</sub>(''p'', ''q''), the [[connected component of the identity]] of the [[indefinite orthogonal group]] SO(''p'', ''q'') (there are a variety of conventions on the connectedness{{Clarify|date=January 2012}}<!-- what a crap? the topology of a Lie group is defined unambiguously. the only thing to fall under conventions can be an extent of the group, does or does not it contain particular elements --> of Spin(''p'', ''q''); in this article, it is taken to be connected for {{math|1=''p'' + ''q'' > 2}} ). As in definite signature, there are some accidental isomorphisms in low dimensions:
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| :Spin(1, 1) = [[General linear group|GL(1, '''R''')]]
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| :Spin(2, 1) = [[SL2(R)|SL(2, '''R''')]]
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| :Spin(3, 1) = [[Special linear group|SL(2,'''C''')]]
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| :Spin(2, 2) = [[SL2(R)|SL(2, '''R''')]] × [[SL2(R)|SL(2, '''R''')]]
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| :Spin(4, 1) = [[Symplectic group|Sp(1, 1)]] | |
| :Spin(3, 2) = [[Symplectic group|Sp(4, '''R''')]]
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| :Spin(5, 1) = [[Special linear group|SL(2, '''H''')]]
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| :Spin(4, 2) = [[Special unitary group|SU(2, 2)]]
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| :Spin(3, 3) = [[Special linear group|SL(4, '''R''')]]
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| Note that Spin(''p'', ''q'') = Spin(''q'', ''p'').
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| ==Topological considerations==
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| [[Connected space|Connected]] and [[simply connected]] Lie groups are classified by their Lie algebra. So if ''G'' is a connected Lie group with a simple Lie algebra, with ''G''′ the [[universal cover]] of ''G'', there is an inclusion
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| :<math> \pi_1 (G) \subset Z(G'), </math>
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| with ''Z''(''G''′) the [[center (group theory)|center]] of ''G''′. This inclusion and the Lie algebra <math>\mathfrak{g}</math> of ''G'' determine ''G'' entirely (note that it is not the fact that <math>\mathfrak{g}</math> and π<sub>1</sub>(''G'') determine ''G'' entirely; for instance SL(2, '''R''') and PSL(2, '''R''') have the same Lie algebra and same fundamental group '''Z''', but are not isomorphic).
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| The definite signature Spin(''n'') are all [[simply connected]] for ''n'' > 2 , so they are the universal coverings for SO(''n'').
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| In indefinite signature, Spin(''p'', ''q'') is not connected, and in general the [[identity component]], Spin<sub>0</sub>(''p'', ''q''), is not simply connected, thus it is not a universal cover. The fundamental group is most easily understood by considering the [[maximal compact subgroup]] of SO(''p'', ''q'') , which is SO(''p'') × SO(''q''), and noting that rather than being the product of the 2-fold covers (hence a 4-fold cover), Spin(''p'', ''q'') is the "diagonal" 2-fold cover – it is a 2-fold quotient of the 4-fold cover. Explicitly, the maximal compact connected subgroup of Spin(''p'', ''q'') is
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| :Spin(''p'') × Spin(''q'')/{(1, 1), (−1, −1)}.
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| This allows us to calculate the [[fundamental groups]] of Spin(''p'', ''q''), taking ''p'' ≥ ''q'':
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| :<math>\pi_1(\mbox{Spin}(p,q)) = \begin{cases}
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| \{0\} & (p,q)=(1,1) \mbox{ or } (1,0) \\
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| \{0\} & p > 2, q = 0,1 \\
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| \mathbf{Z} & (p,q)=(2,0) \mbox{ or } (2,1) \\
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| \mathbf{Z} \times \mathbf{Z} & (p,q) = (2,2) \\
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| \mathbf{Z} & p > 2, q=2 \\
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| \mathbf{Z}_2 & p, q >2\\
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| \end{cases}</math>
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| Thus once ''p'', ''q'' > 2 the fundamental group is '''Z'''<sub>2</sub>, as it is a 2-fold quotient of a product of two universal covers.
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| The maps on fundamental groups are given as follows. For ''p'', ''q'' > 2, this implies that the map π<sub>1</sub>(Spin(''p'', ''q'')) → π<sub>1</sub>(SO(''p'', ''q'')) is given by 1 ∈ '''Z'''<sub>2</sub> going to (1,1) ∈ '''Z'''<sub>2</sub> × '''Z'''<sub>2</sub>. For ''p'' = 2, ''q'' > 2 , this map is given by 1 ∈ '''Z''' → (1,1) ∈ '''Z''' × '''Z'''<sub>2</sub>. And finally, for ''p'' = ''q'' = 2 , (1,0) ∈ '''Z''' × '''Z''' is sent to (1,1) ∈ '''Z''' × '''Z''' and (0, 1) is sent to (1, −1).
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| ==Center==
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| The center of the spin groups (complex and real) are given as follows:<ref>{{Harv |Varadarajan |2004 | loc = [http://books.google.com/books?id=sZ1-G4hQgIIC&pg=PA208 p. 208]}}</ref>
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| :<math>\begin{align}
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| Z(\operatorname{Spin}(n,\mathbf{C})) &= \begin{cases}
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| \mathbf{Z}_2 & n = 2k+1\\
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| \mathbf{Z}_4 & n = 4k+2\\
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| \mathbf{Z}_2 \oplus \mathbf{Z}_2 & n = 4k\\
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| \end{cases} \\
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| Z(\operatorname{Spin}(p,q)) &= \begin{cases}
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| \mathbf{Z}_2 & n = 2k+1,\\
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| \mathbf{Z}_2 & n = 2k, \text{ and } p, q \text{ odd}\\
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| \mathbf{Z}_4 & n = 2k, \text{ and } p, q \text{ even}\\
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| \end{cases}
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| \end{align}</math>
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| ==Quotient groups==
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| [[Quotient group]]s can be obtained from a spin group by quotienting out by a subgroup of the center, with the spin group then being a [[covering group]] of the resulting quotient, and both groups having the same Lie algebra.
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| Quotienting out by the entire center yields the minimal such group, the [[projective special orthogonal group]], which is [[centerless]], while quotienting out by {±1} yields the special orthogonal group – if the center equals {±1} (namely in odd dimension), these two quotient groups agree. If the spin group is simply connected (as Spin(''n'') is for ''n'' > 2), then Spin is the ''maximal'' group in the sequence, and one has a sequence of three groups,
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| :Spin(''n'') → SO(''n'') → PSO(''n''),
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| splitting by parity yields:
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| :Spin(2''n'') → SO(2''n'') → PSO(2''n''),
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| :Spin(2''n''+1) → SO(2''n''+1) = PSO(2''n''+1),
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| which are the three [[compact real form]]s (or two, if SO = PSO ) of the [[compact Lie algebra]] <math>\mathfrak{so} (n, \mathbf{R}).</math>
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| The [[homotopy group]]s of the cover and the quotient are related by the [[long exact sequence of a fibration]], with discrete fiber (the fiber being the kernel) – thus all homotopy groups for ''k'' > 1 are equal, but π<sub>0</sub> and π<sub>1</sub> may differ.
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| For ''n'' > 2, Spin(''n'') is [[simply connected]] (π<sub>0</sub> = π<sub>1</sub> = {1} is trivial), so SO(''n'') is connected and has fundamental group '''Z'''<sub>2</sub> while PSO(''n'') is connected and has fundamental group equal to the center of Spin(''n'').
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| In indefinite signature the covers and homotopy groups are more complicated – Spin(''p'', ''q'') is not simply connected, and quotienting also affects connected components. The analysis is simpler if one considers the maximal (connected) compact SO(''p'') × SO(''q'') ⊂ SO(''p'', ''q'') and the [[component group]] of Spin(''p'', ''q'').
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| ==Discrete subgroups==
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| Discrete subgroups of the spin group can be understood by relating them to discrete subgroups of the special orthogonal group (rotational [[point group]]s).
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| Given the double cover Spin(''n'') → SO(''n''), by the [[lattice theorem]], there is a [[Galois connection]] between subgroups of Spin(''n'') and subgroups of SO(''n'') (rotational point groups): the image of a subgroup of Spin(''n'') is a rotational point group, and the preimage of a point group is a subgroup of Spin(''n''), and the [[closure operator]] on subgroups of Spin(''n'') is multiplication by {±1}. These may be called "binary point groups"; most familiar is the 3-dimensional case, known as [[binary polyhedral group]]s.
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| Concretely, every binary point group is either the preimage of a point group (hence denoted 2''G'', for the point group ''G''), or is an index 2 subgroup of the preimage of a point group which maps (isomorphically) onto the point group; in the latter case the full binary group is abstractly <math>C_2 \times G</math> (since {±1} is central). As an example of these latter, given a cyclic group of odd order <math>C_{2k+1}</math> in SO(''n''), its preimage is a cyclic group of twice the order, <math>C_{4k+2} \cong C_{2k+1} \times C_2,</math> and the subgroup <math>C_{2k+1} < \operatorname{Spin}(n)</math> maps isomorphically to <math>C_{2k+1} < \operatorname{SO}(n).</math>
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| Of particular note are two series:
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| * higher [[binary tetrahedral group]]s, corresponding to the 2-fold cover of symmetries of the ''n''-simplex.
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| *:This group can also be considered as the [[covering groups of the alternating and symmetric groups|double cover of the symmetric group]], <math>2\cdot A_n \to A_n,</math> with the alternating group being the (rotational) symmetry group of the ''n''-simplex.
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| * higher [[binary octahedral group]]s, corresponding to the 2-fold covers of the [[hyperoctahedral group]] (symmetries of the [[hypercube]], or equivalently of its dual, the [[cross-polytope]]).
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| For point groups that reverse orientation, the situation is more complicated, as there are two [[pin group]]s, so there are two possible binary groups corresponding to a given point group.
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| ==Complex case==
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| {{Main|Spin_structure#Spinc_structures}}
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| The spin<sup>''c''</sup> group is defined by the [[exact sequence]]
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| :<math>1 \to \mathbb{Z}_2 \to {\mathrm {Spin}}^{\mathbb C}(n) \to {\mathrm {SO}}(n)\times {\mathrm U}(1) \to 1.</math>
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| This has important applications in 4-manifold theory and [[Seiberg-Witten theory]].
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| ==See also==
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| {{Colbegin}}
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| * [[Clifford algebra]]
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| * [[Clifford analysis]]
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| * [[Spinor]]
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| * [[Spinor bundle]]
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| * [[Spin structure]]
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| * [[Anyon]]
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| * [[Orientation entanglement]]
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| * [[Complex Spin Group]]
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| {{Colend}}
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| ===Related groups===
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| * [[Pin group]] Pin(''n'') – two-fold cover of [[orthogonal group]], O(''n'')
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| * [[Metaplectic group]] Mp(2''n'') – two-fold cover of [[symplectic group]], Sp(2''n'')
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * {{Cite book | last1=Lawson | first1=H. Blaine | last2=Michelsohn | first2=Marie-Louise | author-link2=Marie-Louise Michelsohn | title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 | postscript=<!--None-->}}
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| * {{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}}
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| * {{Cite book | last1=Karoubi | first1=Max|title=K-Theory | publisher=Springer | isbn=978-3-540-79889-7 | year=2008 |pages=210-214| postscript=<!--None-->}}
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| * {{Cite book
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| | publisher = AMS Bookstore
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| | isbn = 978-0-8218-3574-6
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| | last = Varadarajan
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| | first = V. S.
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| | title = Supersymmetry for mathematicians
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| | date = 2004
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| }}
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| {{DEFAULTSORT:Spin Group}}
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| [[Category:Lie groups]]
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| [[Category:Topology of Lie groups]]
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| [[Category:Spinors]]
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