8667:
8263:
8662:{\displaystyle {\begin{aligned}\operatorname {Z} (\operatorname {Spin} (n,\mathbf {C} ))&={\begin{cases}\mathrm {Z} _{2}&n=2k+1\\\mathrm {Z} _{4}&n=4k+2\\\mathrm {Z} _{2}\oplus \mathrm {Z} _{2}&n=4k\\\end{cases}}\\\operatorname {Z} (\operatorname {Spin} (p,q))&={\begin{cases}\mathrm {Z} _{2}&p{\text{ or }}q{\text{ odd}}\\\mathrm {Z} _{4}&n=4k+2,{\text{ and }}p,q{\text{ even}}\\\mathrm {Z} _{2}\oplus \mathrm {Z} _{2}&n=4k,{\text{ and }}p,q{\text{ even}}\\\end{cases}}\end{aligned}}}
6221:
52:
5865:
7250:
1501:
9066:
6216:{\displaystyle \pi _{1}({\mbox{SO}}(p,q))={\begin{cases}0&(p,q)=(1,1){\mbox{ or }}(1,0)\\\mathbb {Z} _{2}&p>2,q=0,1\\\mathbb {Z} &(p,q)=(2,0){\mbox{ or }}(2,1)\\\mathbb {Z} \times \mathbb {Z} &(p,q)=(2,2)\\\mathbb {Z} &p>2,q=2\\\mathbb {Z} _{2}&p,q>2\\\end{cases}}}
2146:
4674:
7411:
7077:
4821:
7615:
2071:
944:
1416:
8966:
8157:
9327:
1737:
1248:
1346:
7257:
3518:
7064:
3773:
8080:
2519:
7932:
4551:
correspond to the physics notion of fermions; the even subspace corresponds to the bosons. The representations of the action of the spin group on the spinor space can be built in a relatively straightforward fashion.
2079:
7850:
4575:
4429:
2995:
6901:
3409:
3950:
5644:
7994:
6983:
4877:. In physics, the Spin group is appropriate for describing uncharged fermions, while the Spin group is used to describe electrically charged fermions. In this case, the U(1) symmetry is specifically the
4316:
8695:, 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(
6829:
4720:
2782:
1579:
8268:
3164:
8821:
2852:
7519:
6474:
6422:
2669:
3214:
3565:
8236:
6627:
5184:
5113:
3830:
3675:
7715:
7245:{\displaystyle \cdots \rightarrow \pi _{k}({\text{SO}}(n-1))\rightarrow \pi _{k}({\text{SO}}(n))\rightarrow \pi _{k}(S^{n-1})\rightarrow \pi _{k-1}({\text{SO}}(n-1))\rightarrow \cdots }
1612:
9377:
9205:
7662:
4525:
3289:
3062:
2625:
2428:
2386:; the resulting spinor fields can be seen to be anti-commuting as a by-product of the Clifford algebra construction. This anti-commutation property is also key to the formulation of
3257:
2709:
1922:
1834:
6682:
4736:
2904:
2460:
2264:
1866:
983:
4219:
3630:
9243:
1792:
4007:
3091:
1648:
7462:
6761:
6711:
6577:
5302:
5243:
8085:
5715:
5683:
4488:
4266:
3896:
2010:
4461:
2564:
2368:
3030:
2593:
497:
472:
435:
2000:
2322:
1958:
3714:
3342:
2194:
9172:), 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
4095:
2293:
872:
3111:
2872:
1496:{\displaystyle \operatorname {Cl} (V)=\operatorname {Cl} ^{0}\oplus \operatorname {Cl} ^{1}\oplus \operatorname {Cl} ^{2}\oplus \cdots \oplus \operatorname {Cl} ^{n}}
9061:{\displaystyle \ldots \rightarrow {\text{Fivebrane}}(n)\rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow {\text{SO}}(n)\rightarrow {\text{O}}(n)}
7770:
7743:
7514:
7488:
6500:
4066:
4036:
1404:
1101:. Strictly speaking, the spin group describes a fermion in a zero-dimensional space; however, space is not zero-dimensional, and so the spin group is used to define
8184:
6993:
6532:
5034:
4844:
3860:
1378:
799:
4161:
4549:
4339:
4239:
4138:
1544:
1524:
4903:. For instance, there are isomorphisms between low-dimensional spin groups and certain classical Lie groups, owing to low-dimensional isomorphisms between the
9252:
1656:
1190:
1266:
8908:) is not simply connected, and quotienting also affects connected components. The analysis is simpler if one considers the maximal (connected) compact
3417:
3722:
6910:
2141:{\displaystyle \operatorname {Cl} ^{\text{even}}=\operatorname {Cl} ^{0}\oplus \operatorname {Cl} ^{2}\oplus \operatorname {Cl} ^{4}\oplus \cdots }
8006:
2468:
7858:
357:
9743:
9588:
9558:
7775:
4669:{\displaystyle 1\to \mathrm {Z} _{2}\to \operatorname {Spin} ^{\mathbf {C} }(n)\to \operatorname {SO} (n)\times \operatorname {U} (1)\to 1.}
5815:) 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(
9106:
Discrete subgroups of the spin group can be understood by relating them to discrete subgroups of the special orthogonal group (rotational
307:
7406:{\displaystyle \pi _{2}(S^{n-1})\rightarrow \pi _{1}({\text{SO}}(n-1))\rightarrow \pi _{1}({\text{SO}}(n))\rightarrow \pi _{1}(S^{n-1}).}
4344:
2909:
8688:
6834:
5372:
3358:
4527:. The (complexified) Clifford algebra acts naturally on this space; the (complexified) spin group corresponds to the length-preserving
2160:
being an even number. The restriction to the even subspace is key to the formation of two-component (Weyl) spinors, constructed below.
8830:
3905:
1134:
792:
302:
9682:
9617:
7937:
4274:
5787:), is not simply connected, thus it is not a universal cover. The fundamental group is most easily understood by considering the
6766:
5587:
4685:
2714:
9767:
9644:
4490:. It is straightforward to see that the spinors anti-commute, and that the product of a spinor and anti-spinor is a scalar.
718:
9071:
The tower is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing
9580:
9157:) is multiplication by {±1}. These may be called "binary point groups"; most familiar is the 3-dimensional case, known as
1549:
1049:
785:
5425:
there are two connected components. As in definite signature, there are some accidental isomorphisms in low dimensions:
3116:
2390:. The Clifford algebra and the spin group have many interesting and curious properties, some of which are listed below.
1258:
9076:
8779:
2787:
6431:
6379:
5376:
1138:
402:
216:
2630:
3259:, that the multiplication is continuous, and the group axioms are satisfied with inversion being continuous, making
3169:
9550:
8680:
can be obtained from a spin group by quotienting out by a subgroup of the center, with the spin group then being a
5332:
3526:
8197:
6582:
5136:
4874:
6630:
5065:
3781:
3642:
2371:
7667:
9471:
9369:
6904:
5788:
4985:
4816:{\displaystyle \operatorname {Spin} ^{\mathbf {C} }(V)=\left(\operatorname {Spin} (V)\times S^{1}\right)/\sim }
1795:
1584:
1034:
837:
600:
334:
211:
99:
9175:
7620:
4503:
3262:
3035:
2598:
2401:
2398:
The spin groups can be constructed less explicitly but without appealing to
Clifford algebras. As a manifold,
3222:
2674:
1874:
1807:
9542:
9403:
9158:
6655:
4899:
2877:
2433:
2206:
1839:
1166:
956:
4196:
2566:
is a set with two elements, and one can be chosen without loss of generality to be the identity. Call this
1149:(where one of a pair of entangled, virtual fermions falls past the event horizon, and the other does not).
3573:
750:
540:
9210:
7610:{\displaystyle 0\rightarrow \pi _{1}({\text{SO}}(n-1))\rightarrow \pi _{1}({\text{SO}}(n))\rightarrow 0,}
1761:
9407:
5654:
5520:
5504:
5261:
5059:
3958:
3067:
1620:
1045:
624:
7418:
6716:
2148:
is the subspace generated by elements that are the product of an even number of vectors. That is, Spin(
2066:{\displaystyle \operatorname {Spin} (V)=\operatorname {Pin} (V)\cap \operatorname {Cl} ^{\text{even}},}
6687:
6537:
5267:
5208:
9072:
5696:
5664:
5510:
5494:
5450:
5430:
4466:
4244:
856:
564:
552:
170:
104:
8490:
8318:
5915:
3865:
2370:. This anti-commutation turns out to be of importance in physics, as it captures the spirit of the
9762:
9709:
9461:
8774:
7069:
5807:), and noting that rather than being the product of the 2-fold covers (hence a 4-fold cover), Spin(
4434:
2524:
2383:
2327:
1407:
1106:
1097:. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the
1078:
139:
34:
3000:
2569:
480:
455:
418:
6637:
5840:
5772:
4912:
1963:
1126:
950:
939:{\displaystyle 1\to \mathbb {Z} _{2}\to \operatorname {Spin} (n)\to \operatorname {SO} (n)\to 1.}
124:
96:
4531:. There is a natural grading on the exterior algebra: the product of an odd number of copies of
2301:
1930:
9422:
For point groups that reverse orientation, the situation is more complicated, as there are two
3683:
3314:
2166:
9739:
9678:
9650:
9640:
9613:
9584:
9554:
9502:
9441:
9130:
8766:
4071:
3345:
2269:
1146:
1118:
695:
529:
372:
266:
9164:
Concretely, every binary point group is either the preimage of a point group (hence denoted 2
3096:
2857:
9772:
9510:
9492:
9436:
9150:
8957:
8863:
7749:
7722:
7493:
7467:
6479:
5749:
5562:
5484:
5478:
5348:
5202:
5055:
4962:
4882:
4680:
4498:
4190:
4045:
4015:
3353:
1383:
1158:
1070:
1026:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
134:
8162:
6505:
5012:
4829:
3838:
9207:(since {±1} is central). As an example of these latter, given a cyclic group of odd order
9126:
8953:
8929:
6425:
5574:
5558:
1354:
1142:
1122:
1110:
1030:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
8152:{\displaystyle {\text{SO}}(n)\rightarrow {\text{SO}}(1,n)^{\uparrow }\rightarrow H^{n},}
4143:
9456:
9415:
9145:) is a rotational point group, and the preimage of a point group is a subgroup of Spin(
8826:
8681:
8677:
5352:
4908:
4567:
4561:
4534:
4324:
4224:
4123:
1529:
1509:
1173:
1130:
1102:
833:
771:
707:
397:
377:
314:
279:
200:
190:
175:
160:
114:
91:
9756:
9451:
8239:
2387:
2379:
1114:
690:
612:
446:
319:
185:
9322:{\displaystyle \mathrm {C} _{4k+2}\cong \mathrm {Z} _{2k+1}\times \mathrm {Z} _{2},}
2200:, then the quotient above endows the space with a natural anti-commuting structure:
1732:{\displaystyle \operatorname {Cl} ^{2}={\mathfrak {spin}}(V)={\mathfrak {spin}}(n),}
9520:
9091:
8900:
In indefinite signature the covers and homotopy groups are more complicated – Spin(
8191:
4528:
2463:
1243:{\displaystyle \mathrm {T} V=\mathbb {R} \oplus V\oplus (V\otimes V)\oplus \cdots }
545:
244:
233:
180:
155:
150:
109:
80:
43:
8833:, with discrete fiber (the fiber being the kernel) – thus all homotopy groups for
1341:{\displaystyle \operatorname {Cl} (V)=\mathrm {T} V/\left(v\otimes v-q(v)\right),}
9669:
2378:. A precise formulation is out of scope, here, but it involves the creation of a
9426:, so there are two possible binary groups corresponding to a given point group.
9107:
8187:
4904:
4894:
4878:
4722:
of the
Clifford algebra, and specifically, it is the subgroup generated by Spin(
4172:
1751:
1008:
993:
813:
3513:{\displaystyle \left(e_{i}e_{j}\cdots e_{k}\right)^{t}=e_{k}\cdots e_{j}e_{i}.}
8692:
7059:{\displaystyle {\text{SO}}(n-1)\rightarrow {\text{SO}}(n)\rightarrow S^{n-1}.}
5049:
4928:
3768:{\displaystyle \alpha \colon \operatorname {Cl} (V)\to \operatorname {Cl} (V)}
1184:
by a two-sided ideal. The tensor algebra (over the reals) may be written as
712:
440:
1048:
is denoted −1, which should not be confused with the orthogonal transform of
9654:
9484:
9423:
9411:
8075:{\displaystyle \pi ({\text{SO}}(1,n)^{\uparrow })\cong \pi ({\text{SO}}(n))}
6988:
2514:{\displaystyle p:\operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)}
1802:
825:
533:
17:
9395:, with the alternating group being the (rotational) symmetry group of the
7927:{\displaystyle {\text{SO}}(3)\cong \mathbb {RP} ^{3}\cong S^{3}/\{\pm 1\}}
51:
4101:; this follows from the anti-commuting property of the Clifford algebra.
1098:
1066:
829:
70:
8687:
Quotienting out by the entire center yields the minimal such group, the
8684:
of the resulting quotient, and both groups having the same Lie algebra.
2462:. Its multiplication law can be defined by lifting as follows. Call the
5316:
3902:). With this notation, an explicit double covering is the homomorphism
2375:
1157:
Construction of the Spin group often starts with the construction of a
1094:
1090:
412:
326:
7845:{\displaystyle \pi _{1}({\text{SO}}(n))\cong \pi _{1}({\text{SO}}(3))}
3219:
It can then be shown that this definition is independent of the paths
9446:
8893:) is connected and has fundamental group equal to the center of Spin(
5468:
5460:
5440:
6241:, as it is a 2-fold quotient of a product of two universal covers.
4424:{\displaystyle \eta _{k}=\left(e_{2k-1}-ie_{2k}\right)/{\sqrt {2}}}
2990:{\displaystyle \gamma (t)=p(\gamma _{a}(t))\cdot p(\gamma _{b}(t))}
1758:, and in this way can be shown to be isomorphic to the Lie algebra
9466:
6896:{\displaystyle {\text{Stab}}_{{\text{SO}}(n)}(v)={\text{SO}}(n-1)}
3404:{\displaystyle t:\operatorname {Cl} (V)\to \operatorname {Cl} (V)}
4965:= {+1, −1} the orthogonal group of dimension zero.
4175:
are constructed, given this formalism. Given a real vector space
4981:
3945:{\displaystyle \operatorname {Pin} (V)\to \operatorname {O} (V)}
1093:
to describe the symmetries of (electrically neutral, uncharged)
8710:
group in the sequence, and one has a sequence of three groups,
5308:
There are certain vestiges of these isomorphisms left over for
7989:{\displaystyle \pi _{1}({\text{SO}}(3))\cong \mathbb {Z} _{2}}
9141:) (rotational point groups): the image of a subgroup of Spin(
6978:{\displaystyle {\text{SO}}(n)/{\text{SO}}(n-1)\cong S^{n-1}.}
5756: > 2, so they are the universal coverings of SO(
4311:{\displaystyle V\otimes \mathbf {C} =W\oplus {\overline {W}}}
4097:
corresponds a reflection across the hyperplane orthogonal to
949:
The group multiplication law on the double cover is given by
7254:
and we concentrate on a section at the end of the sequence:
9720:
9713:
8651:
8437:
6209:
6824:{\displaystyle {\text{Orbit}}_{{\text{SO}}(n)}(v)=S^{n-1}}
6502:
as the three smallest have familiar underlying manifolds:
5639:{\displaystyle \pi _{1}(G)\subset \operatorname {Z} (G'),}
4715:{\displaystyle \operatorname {Cl} (V)\otimes \mathbf {C} }
2777:{\displaystyle \gamma _{a}(0)=\gamma _{b}(0)={\tilde {e}}}
9372:, corresponding to the 2-fold cover of symmetries of the
6244:
The maps on fundamental groups are given as follows. For
5569:
is a connected Lie group with a simple Lie algebra, with
4873:
This has important applications in 4-manifold theory and
5198:) the two-by-two matrices with quaternionic coefficients
9637:
Supersymmetry for mathematicians : an introduction
5737:) have the same Lie algebra and same fundamental group
1406:. The resulting space is finite dimensional, naturally
9249:), its preimage is a cyclic group of twice the order,
7068:
Then
Theorem 4.41 in Hatcher tells us that there is a
6068:
5957:
5883:
5565:
Lie groups are classified by their Lie algebra. So if
4507:
27:
Double cover Lie group of the special orthogonal group
9255:
9213:
9178:
8969:
8782:
8266:
8200:
8165:
8088:
8009:
7940:
7861:
7778:
7752:
7725:
7670:
7623:
7522:
7496:
7470:
7421:
7260:
7080:
6996:
6913:
6837:
6769:
6719:
6690:
6658:
6585:
6540:
6508:
6482:
6434:
6382:
5868:
5699:
5667:
5590:
5270:
5257:) the four-by-four matrices with complex coefficients
5211:
5139:
5068:
5015:
4832:
4739:
4688:
4578:
4537:
4506:
4469:
4437:
4347:
4327:
4277:
4247:
4227:
4199:
4146:
4126:
4074:
4048:
4018:
3961:
3908:
3868:
3841:
3784:
3725:
3686:
3645:
3576:
3529:
3420:
3361:
3317:
3265:
3225:
3172:
3119:
3099:
3070:
3038:
3003:
2912:
2880:
2860:
2790:
2717:
2677:
2633:
2601:
2572:
2527:
2471:
2436:
2404:
2330:
2304:
2272:
2209:
2169:
2082:
2013:
1966:
1933:
1877:
1842:
1810:
1764:
1659:
1623:
1587:
1552:
1532:
1512:
1419:
1386:
1357:
1269:
1193:
959:
875:
483:
458:
421:
5133:
Spin(4) = SU(2) × SU(2), corresponding to
4221:. It can be written as the direct sum of a subspace
2196:
are an orthonormal basis of the (real) vector space
1574:{\displaystyle \operatorname {Cl} ^{0}=\mathbf {R} }
9639:. Providence, R.I.: American Mathematical Society.
9376:-simplex; this group can also be considered as the
5771:) is not necessarily connected, and in general the
1141:coordinates), which in turn provides a footing for
1125:; the spin connection can simplify calculations in
9321:
9237:
9199:
9079:for the homotopy group to be removed. Killing the
9060:
8815:
8661:
8230:
8178:
8151:
8074:
7988:
7926:
7844:
7764:
7737:
7709:
7656:
7609:
7508:
7482:
7456:
7405:
7244:
7058:
6977:
6895:
6823:
6755:
6705:
6676:
6621:
6571:
6526:
6494:
6468:
6416:
6215:
5709:
5677:
5638:
5351:in a similar way to standard spin groups. It is a
5296:
5237:
5178:
5107:
5028:
4838:
4815:
4714:
4668:
4543:
4519:
4482:
4455:
4423:
4333:
4310:
4260:
4233:
4213:
4155:
4132:
4089:
4060:
4030:
4001:
3944:
3890:
3854:
3824:
3767:
3708:
3669:
3624:
3559:
3512:
3403:
3336:
3283:
3251:
3208:
3159:{\displaystyle {\tilde {\gamma }}(0)={\tilde {e}}}
3158:
3105:
3085:
3056:
3024:
2989:
2898:
2866:
2846:
2776:
2703:
2663:
2619:
2587:
2558:
2513:
2454:
2422:
2362:
2316:
2287:
2258:
2188:
2140:
2065:
1994:
1952:
1916:
1860:
1828:
1786:
1731:
1642:
1606:
1573:
1538:
1518:
1495:
1398:
1372:
1340:
1242:
977:
938:
491:
466:
429:
9523:String(n) – the next group in the Whitehead tower
8829:of the cover and the quotient are related by the
8816:{\displaystyle {\mathfrak {so}}(n,\mathbf {R} ).}
2847:{\displaystyle \gamma _{a}(1)=a,\gamma _{b}(1)=b}
6469:{\displaystyle \pi _{1}(\operatorname {SO} (n))}
6417:{\displaystyle \pi _{1}(\operatorname {SO} (n))}
2664:{\displaystyle a,b\in \operatorname {Spin} (n)}
1868:'s Clifford group of all elements of the form
6424:can be more directly derived using results in
3567:by linearity. It is an antihomomorphism since
3209:{\displaystyle a\cdot b={\tilde {\gamma }}(1)}
1172:. The Clifford algebra is the quotient of the
3560:{\displaystyle a,b\in \operatorname {Cl} (V)}
793:
8:
9406:, corresponding to the 2-fold covers of the
8231:{\displaystyle {\text{SO}}(1,n)^{\uparrow }}
7921:
7912:
6622:{\displaystyle SO(3)\cong \mathbb {RP} ^{3}}
5693:entirely (note that it is not the case that
5179:{\displaystyle D_{2}\cong A_{1}\times A_{1}}
3331:
3318:
3311:) can be given explicitly, as follows. Let
2550:
2544:
2183:
2170:
1746:being a real vector space of real dimension
1410:(as a vector space), and can be written as
8257:, (complex and real) are given as follows:
5108:{\displaystyle B_{1}\cong C_{1}\cong A_{1}}
4171:It is worth reviewing how spinor space and
3825:{\displaystyle \alpha (v)=-v,\quad v\in V,}
3670:{\displaystyle a\in \operatorname {Cl} (V)}
9610:Riemannian Geometry and Geometric Analysis
8885:) is connected and has fundamental group Z
8242:(the proper orthochronous Lorentz group).
7710:{\displaystyle \pi _{1}({\text{SO}}(n-1))}
6642:
3064:is a double cover, there is a unique lift
1380:is the quadratic form applied to a vector
1129:. The spin connection in turn enables the
800:
786:
238:
64:
29:
9677:. Cambridge: Cambridge University Press.
9310:
9305:
9286:
9281:
9262:
9257:
9254:
9220:
9215:
9212:
9185:
9180:
9177:
9044:
9027:
9010:
8993:
8976:
8968:
8802:
8784:
8783:
8781:
8643:
8629:
8606:
8601:
8591:
8586:
8576:
8562:
8533:
8528:
8518:
8510:
8499:
8494:
8485:
8414:
8409:
8399:
8394:
8363:
8358:
8327:
8322:
8313:
8295:
8267:
8265:
8222:
8201:
8199:
8170:
8164:
8140:
8127:
8106:
8089:
8087:
8055:
8037:
8016:
8008:
7980:
7976:
7975:
7954:
7945:
7939:
7907:
7901:
7888:
7884:
7881:
7880:
7862:
7860:
7825:
7816:
7792:
7783:
7777:
7751:
7724:
7684:
7675:
7669:
7637:
7628:
7622:
7581:
7572:
7542:
7533:
7521:
7495:
7469:
7439:
7426:
7420:
7385:
7372:
7348:
7339:
7309:
7300:
7278:
7265:
7259:
7213:
7198:
7176:
7163:
7139:
7130:
7100:
7091:
7079:
7041:
7020:
6997:
6995:
6960:
6933:
6928:
6914:
6912:
6873:
6845:
6844:
6839:
6836:
6809:
6777:
6776:
6771:
6768:
6718:
6697:
6693:
6692:
6689:
6657:
6613:
6609:
6606:
6605:
6584:
6563:
6539:
6507:
6481:
6439:
6433:
6387:
6381:
6183:
6179:
6178:
6146:
6145:
6102:
6101:
6094:
6093:
6067:
6028:
6027:
5988:
5984:
5983:
5956:
5910:
5882:
5873:
5867:
5701:
5700:
5698:
5669:
5668:
5666:
5595:
5589:
5323:, these isomorphisms disappear entirely.
5288:
5275:
5269:
5229:
5216:
5210:
5170:
5157:
5144:
5138:
5099:
5086:
5073:
5067:
5020:
5014:
4831:
4805:
4794:
4745:
4744:
4738:
4707:
4687:
4605:
4604:
4591:
4586:
4577:
4536:
4508:
4505:
4470:
4468:
4436:
4414:
4409:
4395:
4370:
4352:
4346:
4326:
4298:
4284:
4276:
4248:
4246:
4226:
4206:
4198:
4145:
4125:
4073:
4047:
4017:
3990:
3960:
3907:
3882:
3867:
3846:
3840:
3783:
3724:
3694:
3685:
3644:
3613:
3603:
3590:
3575:
3528:
3501:
3491:
3478:
3465:
3454:
3441:
3431:
3419:
3360:
3325:
3316:
3264:
3243:
3230:
3224:
3186:
3185:
3171:
3145:
3144:
3121:
3120:
3118:
3098:
3072:
3071:
3069:
3037:
3002:
2969:
2938:
2911:
2879:
2859:
2823:
2795:
2789:
2763:
2762:
2744:
2722:
2716:
2695:
2682:
2676:
2632:
2600:
2574:
2573:
2571:
2532:
2526:
2470:
2435:
2403:
2354:
2341:
2329:
2303:
2271:
2250:
2240:
2224:
2214:
2208:
2177:
2168:
2126:
2113:
2100:
2087:
2081:
2054:
2012:
1977:
1965:
1938:
1932:
1905:
1892:
1882:
1876:
1841:
1809:
1766:
1765:
1763:
1702:
1701:
1674:
1673:
1664:
1658:
1625:
1624:
1622:
1607:{\displaystyle \operatorname {Cl} ^{1}=V}
1592:
1586:
1566:
1557:
1551:
1531:
1511:
1487:
1468:
1455:
1442:
1418:
1385:
1356:
1296:
1288:
1268:
1206:
1205:
1194:
1192:
958:
888:
884:
883:
874:
485:
484:
482:
460:
459:
457:
423:
422:
420:
9200:{\displaystyle \mathrm {C} _{2}\times G}
9090:), one obtains the infinite-dimensional
8238:is the identity component of the proper
7657:{\displaystyle \pi _{1}({\text{SO}}(n))}
4520:{\displaystyle \textstyle {\bigwedge }W}
3523:This can be extended to all elements of
3284:{\displaystyle \operatorname {Spin} (n)}
3057:{\displaystyle \operatorname {Spin} (n)}
2620:{\displaystyle \operatorname {Spin} (n)}
2423:{\displaystyle \operatorname {Spin} (n)}
2156:), given above, with the restriction to
9533:
4679:It is a multiplicative subgroup of the
4463:and the complex conjugate spinors span
4104:This gives a double covering of both O(
3775:which on degree 1 elements is given by
3252:{\displaystyle \gamma _{a},\gamma _{b}}
2704:{\displaystyle \gamma _{a},\gamma _{b}}
1917:{\displaystyle v_{1}v_{2}\cdots v_{k},}
1829:{\displaystyle \operatorname {Pin} (V)}
356:
122:
32:
9604:
9602:
9600:
9577:Dirac Operators in Riemannian Geometry
8003:The same argument can be used to show
6677:{\displaystyle \operatorname {SO} (n)}
5661:′. This inclusion and the Lie algebra
4931:and the general understanding that Cl(
4897:among the classical Lie groups called
4562:Spin structure § SpinC structures
3639:) can then be defined as all elements
2899:{\displaystyle \operatorname {SO} (n)}
2455:{\displaystyle \operatorname {SO} (n)}
2259:{\displaystyle e_{i}e_{j}=-e_{j}e_{i}}
1861:{\displaystyle \operatorname {Cl} (V)}
1085:Motivation and physical interpretation
978:{\displaystyle \operatorname {SO} (n)}
358:Classification of finite simple groups
4214:{\displaystyle V\otimes \mathbf {C} }
3898:, which is an antiautomorphism of Cl(
7:
9630:
9628:
8186:is the upper sheet of a two-sheeted
3625:{\displaystyle (ab)^{t}=b^{t}a^{t}.}
1077:). A distinct article discusses the
9414:, or equivalently of its dual, the
9378:double cover of the symmetric group
9364:Of particular note are two series:
9238:{\displaystyle \mathrm {Z} _{2k+1}}
8788:
8785:
8689:projective special orthogonal group
8250:The center of the spin groups, for
5702:
5670:
5373:connected component of the identity
4907:(and corresponding isomorphisms of
4730:. Alternately, it is the quotient
2595:. Then to define multiplication in
2004:The spin group is then defined as
1787:{\displaystyle {\mathfrak {so}}(n)}
1770:
1767:
1742:where the last is a short-hand for
1712:
1709:
1706:
1703:
1684:
1681:
1678:
1675:
1635:
1632:
1629:
1626:
1011:with the special orthogonal group.
9719:Grothendieck's "torsion index" is
9306:
9282:
9258:
9216:
9181:
8831:long exact sequence of a fibration
8602:
8587:
8529:
8495:
8445:
8410:
8395:
8359:
8323:
8271:
5613:
4645:
4587:
4002:{\displaystyle \rho (a)v=ava^{*},}
3927:
3086:{\displaystyle {\tilde {\gamma }}}
2152:) consists of all elements of Pin(
1643:{\displaystyle {\mathfrak {spin}}}
1289:
1195:
1069:of the invertible elements in the
25:
7457:{\displaystyle \pi _{k}(S^{n})=0}
6756:{\displaystyle v=(1,0,\cdots ,0)}
4566:The Spin group is defined by the
4140:gives the same transformation as
8803:
8296:
7415:Corollary 4.9 in Hatcher states
6706:{\displaystyle \mathbb {R} ^{n}}
6636:The proof uses known results in
6572:{\displaystyle SO(2)\cong S^{1}}
5839:This allows us to calculate the
5297:{\displaystyle D_{3}\cong A_{3}}
5238:{\displaystyle B_{2}\cong C_{2}}
4746:
4708:
4606:
4285:
4207:
1567:
1145:, as well as a formalization of
50:
6987:Geometrically, this provides a
5710:{\displaystyle {\mathfrak {g}}}
5678:{\displaystyle {\mathfrak {g}}}
4947:) and so on, one then has that
4911:) of the different families of
4483:{\displaystyle {\overline {W}}}
4261:{\displaystyle {\overline {W}}}
3809:
1044:The non-trivial element of the
9738:. Springer. pp. 210–214.
9055:
9049:
9041:
9038:
9032:
9024:
9021:
9015:
9007:
9004:
8998:
8990:
8987:
8981:
8973:
8807:
8793:
8475:
8472:
8460:
8451:
8303:
8300:
8286:
8277:
8223:
8219:
8206:
8133:
8128:
8124:
8111:
8103:
8100:
8094:
8069:
8066:
8060:
8052:
8043:
8038:
8034:
8021:
8013:
7968:
7965:
7959:
7951:
7873:
7867:
7839:
7836:
7830:
7822:
7806:
7803:
7797:
7789:
7704:
7701:
7689:
7681:
7651:
7648:
7642:
7634:
7598:
7595:
7592:
7586:
7578:
7565:
7562:
7559:
7547:
7539:
7526:
7445:
7432:
7397:
7378:
7365:
7362:
7359:
7353:
7345:
7332:
7329:
7326:
7314:
7306:
7293:
7290:
7271:
7236:
7233:
7230:
7218:
7210:
7191:
7188:
7169:
7156:
7153:
7150:
7144:
7136:
7123:
7120:
7117:
7105:
7097:
7084:
7034:
7031:
7025:
7017:
7014:
7002:
6950:
6938:
6925:
6919:
6890:
6878:
6867:
6861:
6856:
6850:
6799:
6793:
6788:
6782:
6763:. The orbit of this vector is
6750:
6726:
6713:, in particular on the vector
6671:
6665:
6598:
6592:
6553:
6547:
6521:
6515:
6463:
6460:
6454:
6445:
6411:
6408:
6402:
6393:
6138:
6126:
6120:
6108:
6086:
6074:
6064:
6052:
6046:
6034:
5975:
5963:
5953:
5941:
5935:
5923:
5904:
5901:
5889:
5879:
5763:In indefinite signature, Spin(
5630:
5619:
5607:
5601:
5319:for more details). For higher
4784:
4778:
4761:
4755:
4701:
4695:
4660:
4657:
4651:
4639:
4633:
4624:
4621:
4615:
4597:
4582:
4084:
4078:
3971:
3965:
3939:
3933:
3924:
3921:
3915:
3891:{\displaystyle \alpha (a)^{t}}
3879:
3872:
3794:
3788:
3762:
3756:
3747:
3744:
3738:
3664:
3658:
3587:
3577:
3554:
3548:
3398:
3392:
3383:
3380:
3374:
3278:
3272:
3203:
3197:
3191:
3150:
3138:
3132:
3126:
3077:
3051:
3045:
3013:
3007:
2984:
2981:
2975:
2962:
2953:
2950:
2944:
2931:
2922:
2916:
2893:
2887:
2835:
2829:
2807:
2801:
2768:
2756:
2750:
2734:
2728:
2658:
2652:
2614:
2608:
2579:
2553:
2541:
2508:
2502:
2493:
2490:
2484:
2449:
2443:
2417:
2411:
2044:
2038:
2026:
2020:
1983:
1970:
1855:
1849:
1823:
1817:
1781:
1775:
1723:
1717:
1695:
1689:
1432:
1426:
1367:
1361:
1327:
1321:
1282:
1276:
1231:
1219:
972:
966:
930:
927:
921:
912:
909:
903:
894:
879:
719:Infinite dimensional Lie group
1:
9581:American Mathematical Society
9077:Eilenberg–MacLane space
8082:, by considering a fibration
7516:, the exact sequence becomes
6652:First consider the action of
5729:entirely; for instance SL(2,
5009:. Corresponds to the abelian
4893:In low dimensions, there are
4456:{\displaystyle 1\leq k\leq m}
3166:. Then define the product as
2559:{\displaystyle p^{-1}(\{e\})}
2363:{\displaystyle v=e_{i}+e_{j}}
2298:which follows by considering
1754:; it has a natural action on
1050:reflection through the origin
8952:The spin group appears in a
8730:splitting by parity yields:
6428:. In particular we can find
6255:, this implies that the map
5744:The definite signature Spin(
4475:
4303:
4253:
3719:Now define the automorphism
3025:{\displaystyle \gamma (0)=e}
2588:{\displaystyle {\tilde {e}}}
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
9635:Varadarajan, V. S. (2004).
6907:one obtains an isomorphism
6372:Fundamental groups of SO(n)
5741:, but are not isomorphic).
5377:indefinite orthogonal group
4943:) is a short-hand for Spin(
1995:{\displaystyle q(v_{i})=1.}
855:, such that there exists a
217:List of group theory topics
9789:
9575:Friedrich, Thomas (2000),
9551:Princeton University Press
9133:between subgroups of Spin(
6831:, while the stabilizer is
6237:the fundamental group is Z
5554:Topological considerations
5304:. dim = 15
5245:. dim = 10
4559:
4341:is spanned by the spinors
4241:of spinors and a subspace
3303:, a double covering of SO(
2317:{\displaystyle v\otimes v}
1953:{\displaystyle v_{i}\in V}
1121:on a spinor bundle is the
1089:The spin group is used in
1065:) can be constructed as a
1029:and so coincides with the
9612:, (2002) Springer Verlag
9370:binary tetrahedral groups
6631:axis-angle representation
5186:. dim = 6
5115:. dim = 3
5036:. dim = 1
4996:by double phase rotation
4958:Pin(1) = {+i, −i, +1, −1}
4935:) is a short-hand for Cl(
4923:for the complex numbers,
4726:) and the unit circle in
3709:{\displaystyle aa^{t}=1.}
3337:{\displaystyle \{e_{i}\}}
2372:Pauli exclusion principle
2189:{\displaystyle \{e_{i}\}}
1161:over a real vector space
9543:Michelsohn, Marie-Louise
9472:Orientation entanglement
9404:binary octahedral groups
9159:binary polyhedral groups
6905:orbit-stabilizer theorem
5789:maximal compact subgroup
5581:, there is an inclusion
4900:exceptional isomorphisms
4889:Exceptional isomorphisms
4090:{\displaystyle \rho (a)}
2288:{\displaystyle i\neq j,}
1796:special orthogonal group
1253:The Clifford algebra Cl(
1109:: the spin group is the
838:special orthogonal group
335:Elementary abelian group
212:Glossary of group theory
9668:Hatcher, Allen (2002).
9345:maps isomorphically to
9113:Given the double cover
9086:homotopy group in Spin(
6534:is the point manifold,
6376:The fundamental groups
6312:, this map is given by
5347:is constructed through
3106:{\displaystyle \gamma }
2867:{\displaystyle \gamma }
2430:is the double cover of
1167:definite quadratic form
992:) therefore shares its
9768:Topology of Lie groups
9509:) – two-fold cover of
9491:) – two-fold cover of
9323:
9239:
9201:
9168:, for the point group
9137:) and subgroups of SO(
9062:
8817:
8663:
8232:
8180:
8153:
8076:
7990:
7928:
7846:
7766:
7765:{\displaystyle n>3}
7739:
7738:{\displaystyle n>3}
7711:
7658:
7611:
7510:
7509:{\displaystyle n>3}
7484:
7483:{\displaystyle k<n}
7458:
7407:
7246:
7060:
6979:
6897:
6825:
6757:
6707:
6678:
6623:
6573:
6528:
6496:
6495:{\displaystyle n>3}
6470:
6418:
6217:
5711:
5679:
5640:
5298:
5239:
5180:
5109:
5030:
4840:
4826:where the equivalence
4817:
4716:
4670:
4545:
4521:
4484:
4457:
4425:
4335:
4312:
4262:
4235:
4215:
4157:
4134:
4091:
4062:
4061:{\displaystyle a\in V}
4032:
4031:{\displaystyle v\in V}
4003:
3946:
3892:
3856:
3826:
3769:
3710:
3671:
3626:
3561:
3514:
3405:
3338:
3299:For a quadratic space
3285:
3253:
3210:
3160:
3107:
3087:
3058:
3026:
2991:
2900:
2868:
2854:. These define a path
2848:
2778:
2705:
2665:
2621:
2589:
2560:
2515:
2456:
2424:
2394:Geometric construction
2364:
2318:
2289:
2260:
2190:
2142:
2067:
1996:
1954:
1918:
1862:
1830:
1788:
1733:
1644:
1608:
1575:
1540:
1520:
1497:
1400:
1399:{\displaystyle v\in V}
1374:
1342:
1244:
979:
953:the multiplication on
940:
751:Linear algebraic group
493:
468:
431:
9734:Karoubi, Max (2008).
9408:hyperoctahedral group
9324:
9240:
9202:
9153:on subgroups of Spin(
9073:short exact sequences
9063:
8818:
8757:+1) = PSO(2
8664:
8233:
8181:
8179:{\displaystyle H^{n}}
8154:
8077:
7991:
7929:
7847:
7767:
7740:
7712:
7659:
7612:
7511:
7485:
7459:
7408:
7247:
7061:
6980:
6898:
6826:
6758:
6708:
6679:
6624:
6574:
6529:
6527:{\displaystyle SO(1)}
6497:
6471:
6419:
6218:
5835:)/{(1, 1), (−1, −1)}.
5712:
5680:
5641:
5299:
5240:
5181:
5110:
5031:
5029:{\displaystyle D_{1}}
4875:Seiberg–Witten theory
4841:
4839:{\displaystyle \sim }
4818:
4717:
4671:
4546:
4522:
4485:
4458:
4426:
4336:
4313:
4263:
4236:
4216:
4158:
4135:
4092:
4063:
4033:
4004:
3947:
3893:
3857:
3855:{\displaystyle a^{*}}
3827:
3770:
3711:
3672:
3627:
3562:
3515:
3406:
3339:
3286:
3254:
3211:
3161:
3108:
3088:
3059:
3027:
2992:
2901:
2869:
2849:
2779:
2706:
2666:
2622:
2590:
2561:
2516:
2457:
2425:
2365:
2319:
2290:
2261:
2191:
2143:
2068:
1997:
1955:
1919:
1863:
1831:
1789:
1734:
1645:
1609:
1576:
1541:
1521:
1498:
1401:
1375:
1343:
1245:
1052:, generally denoted −
988:As a Lie group, Spin(
980:
941:
494:
469:
432:
9253:
9211:
9176:
8967:
8780:
8765:which are the three
8753:+1) → SO(2
8706:), then Spin is the
8264:
8198:
8163:
8086:
8007:
7938:
7859:
7776:
7750:
7723:
7668:
7621:
7520:
7494:
7468:
7419:
7258:
7078:
6994:
6911:
6835:
6767:
6717:
6688:
6656:
6583:
6538:
6506:
6480:
6432:
6380:
5866:
5697:
5665:
5588:
5333:indefinite signature
5327:Indefinite signature
5268:
5209:
5137:
5066:
5013:
4830:
4737:
4686:
4576:
4535:
4504:
4467:
4435:
4345:
4325:
4275:
4245:
4225:
4197:
4189:an even number, its
4144:
4124:
4072:
4046:
4016:
3959:
3906:
3866:
3839:
3782:
3723:
3684:
3643:
3574:
3527:
3418:
3359:
3315:
3263:
3223:
3170:
3117:
3097:
3068:
3036:
3001:
2910:
2878:
2858:
2788:
2715:
2675:
2631:
2599:
2570:
2525:
2469:
2434:
2402:
2328:
2302:
2270:
2207:
2167:
2080:
2011:
1964:
1931:
1875:
1840:
1808:
1762:
1657:
1621:
1585:
1550:
1530:
1526:is the dimension of
1510:
1417:
1384:
1373:{\displaystyle q(v)}
1355:
1267:
1191:
1137:(effectively in the
1107:Riemannian manifolds
1079:spin representations
957:
873:
859:of Lie groups (when
857:short exact sequence
481:
456:
419:
9710:essential dimension
9541:Lawson, H. Blaine;
9462:Table of Lie groups
9410:(symmetries of the
8881:is trivial), so SO(
8775:compact Lie algebra
8742:) → PSO(2
7072:of homotopy groups
7070:long exact sequence
6327:. And finally, for
5264:, corresponding to
5205:, corresponding to
5062:, corresponding to
4977:the complex numbers
4913:simple Lie algebras
4042:has degree 1 (i.e.
2384:Minkowski spacetime
1960:is of unit length:
1135:in curved spacetime
125:Group homomorphisms
35:Algebraic structure
9712:of spin groups is
9671:Algebraic topology
9319:
9235:
9197:
9102:Discrete subgroups
9058:
8813:
8767:compact real forms
8738:) → SO(2
8722:) → PSO(
8659:
8657:
8650:
8436:
8228:
8176:
8149:
8072:
7986:
7924:
7842:
7762:
7735:
7707:
7654:
7607:
7506:
7480:
7454:
7403:
7242:
7056:
6975:
6893:
6821:
6753:
6703:
6674:
6638:algebraic topology
6619:
6569:
6524:
6492:
6466:
6414:
6213:
6208:
6072:
5961:
5887:
5841:fundamental groups
5773:identity component
5707:
5675:
5636:
5413:is connected; for
5294:
5235:
5176:
5105:
5026:
4836:
4813:
4712:
4666:
4541:
4517:
4516:
4497:is defined as the
4480:
4453:
4421:
4331:
4308:
4258:
4231:
4211:
4156:{\displaystyle -a}
4153:
4130:
4087:
4058:
4028:
3999:
3942:
3888:
3852:
3822:
3765:
3706:
3667:
3622:
3557:
3510:
3401:
3334:
3281:
3249:
3206:
3156:
3103:
3083:
3054:
3022:
2987:
2896:
2864:
2844:
2774:
2701:
2661:
2617:
2585:
2556:
2511:
2452:
2420:
2360:
2314:
2285:
2256:
2186:
2138:
2063:
1992:
1950:
1914:
1858:
1826:
1784:
1729:
1640:
1604:
1571:
1536:
1516:
1493:
1396:
1370:
1338:
1240:
1127:general relativity
975:
936:
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
9745:978-3-540-79889-7
9590:978-0-8218-2055-1
9560:978-0-691-08542-5
9503:Metaplectic group
9442:Clifford analysis
9329:and the subgroup
9131:Galois connection
9075:starting with an
9047:
9030:
9013:
8996:
8979:
8718:) → SO(
8646:
8632:
8579:
8565:
8521:
8513:
8204:
8109:
8092:
8058:
8019:
8001:
8000:
7957:
7865:
7828:
7795:
7687:
7640:
7584:
7545:
7351:
7312:
7216:
7142:
7103:
7023:
7000:
6936:
6917:
6876:
6848:
6842:
6780:
6774:
6629:(shown using the
6071:
5960:
5886:
5803:) × SO(
5349:Clifford algebras
5335:, the spin group
4544:{\displaystyle W}
4478:
4419:
4334:{\displaystyle W}
4306:
4268:of anti-spinors:
4256:
4234:{\displaystyle W}
4133:{\displaystyle a}
3635:Observe that Pin(
3346:orthonormal basis
3194:
3153:
3129:
3080:
2771:
2582:
2090:
2057:
1836:is a subgroup of
1539:{\displaystyle V}
1519:{\displaystyle n}
1147:Hawking radiation
1119:affine connection
828:whose underlying
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
9780:
9749:
9696:
9695:
9693:
9691:
9676:
9665:
9659:
9658:
9632:
9623:
9621:(See Chapter 1.)
9606:
9595:
9593:
9572:
9566:
9564:
9538:
9511:symplectic group
9493:orthogonal group
9437:Clifford algebra
9394:
9360:
9344:
9328:
9326:
9325:
9320:
9315:
9314:
9309:
9300:
9299:
9285:
9276:
9275:
9261:
9244:
9242:
9241:
9236:
9234:
9233:
9219:
9206:
9204:
9203:
9198:
9190:
9189:
9184:
9151:closure operator
9124:
9082:
9067:
9065:
9064:
9059:
9048:
9045:
9031:
9028:
9014:
9011:
8997:
8994:
8980:
8977:
8958:orthogonal group
8956:anchored by the
8943:
8927:
8880:
8864:simply connected
8857:
8840:are equal, but π
8839:
8822:
8820:
8819:
8814:
8806:
8792:
8791:
8772:
8705:
8668:
8666:
8665:
8660:
8658:
8654:
8653:
8647:
8644:
8633:
8630:
8611:
8610:
8605:
8596:
8595:
8590:
8580:
8577:
8566:
8563:
8538:
8537:
8532:
8522:
8519:
8514:
8511:
8504:
8503:
8498:
8440:
8439:
8419:
8418:
8413:
8404:
8403:
8398:
8368:
8367:
8362:
8332:
8331:
8326:
8299:
8256:
8237:
8235:
8234:
8229:
8227:
8226:
8205:
8202:
8185:
8183:
8182:
8177:
8175:
8174:
8158:
8156:
8155:
8150:
8145:
8144:
8132:
8131:
8110:
8107:
8093:
8090:
8081:
8079:
8078:
8073:
8059:
8056:
8042:
8041:
8020:
8017:
7995:
7993:
7992:
7987:
7985:
7984:
7979:
7958:
7955:
7950:
7949:
7933:
7931:
7930:
7925:
7911:
7906:
7905:
7893:
7892:
7887:
7866:
7863:
7851:
7849:
7848:
7843:
7829:
7826:
7821:
7820:
7796:
7793:
7788:
7787:
7771:
7769:
7768:
7763:
7744:
7742:
7741:
7736:
7716:
7714:
7713:
7708:
7688:
7685:
7680:
7679:
7663:
7661:
7660:
7655:
7641:
7638:
7633:
7632:
7616:
7614:
7613:
7608:
7585:
7582:
7577:
7576:
7546:
7543:
7538:
7537:
7515:
7513:
7512:
7507:
7489:
7487:
7486:
7481:
7463:
7461:
7460:
7455:
7444:
7443:
7431:
7430:
7412:
7410:
7409:
7404:
7396:
7395:
7377:
7376:
7352:
7349:
7344:
7343:
7313:
7310:
7305:
7304:
7289:
7288:
7270:
7269:
7251:
7249:
7248:
7243:
7217:
7214:
7209:
7208:
7187:
7186:
7168:
7167:
7143:
7140:
7135:
7134:
7104:
7101:
7096:
7095:
7065:
7063:
7062:
7057:
7052:
7051:
7024:
7021:
7001:
6998:
6984:
6982:
6981:
6976:
6971:
6970:
6937:
6934:
6932:
6918:
6915:
6903:. Thus from the
6902:
6900:
6899:
6894:
6877:
6874:
6860:
6859:
6849:
6846:
6843:
6840:
6830:
6828:
6827:
6822:
6820:
6819:
6792:
6791:
6781:
6778:
6775:
6772:
6762:
6760:
6759:
6754:
6712:
6710:
6709:
6704:
6702:
6701:
6696:
6683:
6681:
6680:
6675:
6643:
6628:
6626:
6625:
6620:
6618:
6617:
6612:
6578:
6576:
6575:
6570:
6568:
6567:
6533:
6531:
6530:
6525:
6501:
6499:
6498:
6493:
6475:
6473:
6472:
6467:
6444:
6443:
6423:
6421:
6420:
6415:
6392:
6391:
6367:
6363:
6359:
6348:
6337:
6326:
6311:
6300:
6289:
6282:
6254:
6236:
6222:
6220:
6219:
6214:
6212:
6211:
6188:
6187:
6182:
6149:
6105:
6097:
6073:
6069:
6031:
5993:
5992:
5987:
5962:
5958:
5888:
5884:
5878:
5877:
5750:simply connected
5716:
5714:
5713:
5708:
5706:
5705:
5684:
5682:
5681:
5676:
5674:
5673:
5645:
5643:
5642:
5637:
5629:
5600:
5599:
5563:simply connected
5549:
5424:
5412:
5400:
5389:
5370:
5346:
5314:
5303:
5301:
5300:
5295:
5293:
5292:
5280:
5279:
5244:
5242:
5241:
5236:
5234:
5233:
5221:
5220:
5185:
5183:
5182:
5177:
5175:
5174:
5162:
5161:
5149:
5148:
5114:
5112:
5111:
5106:
5104:
5103:
5091:
5090:
5078:
5077:
5035:
5033:
5032:
5027:
5025:
5024:
5008:
4988:, which acts on
4955:the real numbers
4939:) and that Spin(
4883:electromagnetism
4869:
4857:
4845:
4843:
4842:
4837:
4822:
4820:
4819:
4814:
4809:
4804:
4800:
4799:
4798:
4751:
4750:
4749:
4721:
4719:
4718:
4713:
4711:
4681:complexification
4675:
4673:
4672:
4667:
4611:
4610:
4609:
4596:
4595:
4590:
4550:
4548:
4547:
4542:
4526:
4524:
4523:
4518:
4512:
4499:exterior algebra
4489:
4487:
4486:
4481:
4479:
4471:
4462:
4460:
4459:
4454:
4430:
4428:
4427:
4422:
4420:
4415:
4413:
4408:
4404:
4403:
4402:
4384:
4383:
4357:
4356:
4340:
4338:
4337:
4332:
4317:
4315:
4314:
4309:
4307:
4299:
4288:
4267:
4265:
4264:
4259:
4257:
4249:
4240:
4238:
4237:
4232:
4220:
4218:
4217:
4212:
4210:
4191:complexification
4188:
4162:
4160:
4159:
4154:
4139:
4137:
4136:
4131:
4096:
4094:
4093:
4088:
4067:
4065:
4064:
4059:
4037:
4035:
4034:
4029:
4008:
4006:
4005:
4000:
3995:
3994:
3951:
3949:
3948:
3943:
3897:
3895:
3894:
3889:
3887:
3886:
3861:
3859:
3858:
3853:
3851:
3850:
3831:
3829:
3828:
3823:
3774:
3772:
3771:
3766:
3715:
3713:
3712:
3707:
3699:
3698:
3676:
3674:
3673:
3668:
3631:
3629:
3628:
3623:
3618:
3617:
3608:
3607:
3595:
3594:
3566:
3564:
3563:
3558:
3519:
3517:
3516:
3511:
3506:
3505:
3496:
3495:
3483:
3482:
3470:
3469:
3464:
3460:
3459:
3458:
3446:
3445:
3436:
3435:
3410:
3408:
3407:
3402:
3354:antiautomorphism
3343:
3341:
3340:
3335:
3330:
3329:
3290:
3288:
3287:
3282:
3258:
3256:
3255:
3250:
3248:
3247:
3235:
3234:
3215:
3213:
3212:
3207:
3196:
3195:
3187:
3165:
3163:
3162:
3157:
3155:
3154:
3146:
3131:
3130:
3122:
3112:
3110:
3109:
3104:
3092:
3090:
3089:
3084:
3082:
3081:
3073:
3063:
3061:
3060:
3055:
3031:
3029:
3028:
3023:
2996:
2994:
2993:
2988:
2974:
2973:
2943:
2942:
2905:
2903:
2902:
2897:
2873:
2871:
2870:
2865:
2853:
2851:
2850:
2845:
2828:
2827:
2800:
2799:
2783:
2781:
2780:
2775:
2773:
2772:
2764:
2749:
2748:
2727:
2726:
2710:
2708:
2707:
2702:
2700:
2699:
2687:
2686:
2670:
2668:
2667:
2662:
2626:
2624:
2623:
2618:
2594:
2592:
2591:
2586:
2584:
2583:
2575:
2565:
2563:
2562:
2557:
2540:
2539:
2520:
2518:
2517:
2512:
2461:
2459:
2458:
2453:
2429:
2427:
2426:
2421:
2369:
2367:
2366:
2361:
2359:
2358:
2346:
2345:
2323:
2321:
2320:
2315:
2294:
2292:
2291:
2286:
2265:
2263:
2262:
2257:
2255:
2254:
2245:
2244:
2229:
2228:
2219:
2218:
2195:
2193:
2192:
2187:
2182:
2181:
2147:
2145:
2144:
2139:
2131:
2130:
2118:
2117:
2105:
2104:
2092:
2091:
2088:
2072:
2070:
2069:
2064:
2059:
2058:
2055:
2001:
1999:
1998:
1993:
1982:
1981:
1959:
1957:
1956:
1951:
1943:
1942:
1923:
1921:
1920:
1915:
1910:
1909:
1897:
1896:
1887:
1886:
1867:
1865:
1864:
1859:
1835:
1833:
1832:
1827:
1793:
1791:
1790:
1785:
1774:
1773:
1738:
1736:
1735:
1730:
1716:
1715:
1688:
1687:
1669:
1668:
1649:
1647:
1646:
1641:
1639:
1638:
1613:
1611:
1610:
1605:
1597:
1596:
1580:
1578:
1577:
1572:
1570:
1562:
1561:
1545:
1543:
1542:
1537:
1525:
1523:
1522:
1517:
1502:
1500:
1499:
1494:
1492:
1491:
1473:
1472:
1460:
1459:
1447:
1446:
1405:
1403:
1402:
1397:
1379:
1377:
1376:
1371:
1347:
1345:
1344:
1339:
1334:
1330:
1300:
1292:
1259:quotient algebra
1249:
1247:
1246:
1241:
1209:
1198:
1159:Clifford algebra
1071:Clifford algebra
1057:
1027:simply connected
1020:
1006:
984:
982:
981:
976:
945:
943:
942:
937:
893:
892:
887:
865:
854:
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
65:
54:
41:
30:
21:
9788:
9787:
9783:
9782:
9781:
9779:
9778:
9777:
9753:
9752:
9746:
9733:
9730:
9728:Further reading
9705:
9700:
9699:
9689:
9687:
9685:
9674:
9667:
9666:
9662:
9647:
9634:
9633:
9626:
9607:
9598:
9591:
9574:
9573:
9569:
9561:
9540:
9539:
9535:
9530:
9481:
9476:
9432:
9393:
9387:
9381:
9354:
9346:
9338:
9330:
9304:
9280:
9256:
9251:
9250:
9214:
9209:
9208:
9179:
9174:
9173:
9127:lattice theorem
9114:
9104:
9085:
9080:
8965:
8964:
8954:Whitehead tower
8950:
8948:Whitehead tower
8933:
8930:component group
8909:
8888:
8879:
8875:
8871:
8867:
8852:
8847:
8843:
8834:
8827:homotopy groups
8778:
8777:
8770:
8700:
8678:Quotient groups
8675:
8673:Quotient groups
8656:
8655:
8649:
8648:
8631: and
8612:
8600:
8585:
8582:
8581:
8564: and
8539:
8527:
8524:
8523:
8505:
8493:
8486:
8478:
8442:
8441:
8435:
8434:
8420:
8408:
8393:
8390:
8389:
8369:
8357:
8354:
8353:
8333:
8321:
8314:
8306:
8262:
8261:
8251:
8248:
8218:
8196:
8195:
8166:
8161:
8160:
8136:
8123:
8084:
8083:
8033:
8005:
8004:
7974:
7941:
7936:
7935:
7897:
7879:
7857:
7856:
7812:
7779:
7774:
7773:
7748:
7747:
7721:
7720:
7717:are isomorphic
7671:
7666:
7665:
7624:
7619:
7618:
7568:
7529:
7518:
7517:
7492:
7491:
7466:
7465:
7435:
7422:
7417:
7416:
7381:
7368:
7335:
7296:
7274:
7261:
7256:
7255:
7194:
7172:
7159:
7126:
7087:
7076:
7075:
7037:
6992:
6991:
6956:
6909:
6908:
6838:
6833:
6832:
6805:
6770:
6765:
6764:
6715:
6714:
6691:
6686:
6685:
6654:
6653:
6604:
6581:
6580:
6559:
6536:
6535:
6504:
6503:
6478:
6477:
6435:
6430:
6429:
6426:homotopy theory
6383:
6378:
6377:
6374:
6365:
6361:
6350:
6339:
6328:
6325:
6313:
6302:
6299:
6295:
6291:
6288:
6284:
6272:
6260:
6256:
6245:
6240:
6227:
6207:
6206:
6189:
6177:
6174:
6173:
6150:
6142:
6141:
6106:
6090:
6089:
6032:
6024:
6023:
5994:
5982:
5979:
5978:
5921:
5911:
5869:
5864:
5863:
5799:), which is SO(
5778:
5720:
5695:
5694:
5663:
5662:
5622:
5591:
5586:
5585:
5575:universal cover
5556:
5531:
5414:
5402:
5391:
5379:
5360:
5356:
5336:
5329:
5309:
5284:
5271:
5266:
5265:
5225:
5212:
5207:
5206:
5166:
5153:
5140:
5135:
5134:
5095:
5082:
5069:
5064:
5063:
5016:
5011:
5010:
4997:
4919:for the reals,
4909:Dynkin diagrams
4891:
4859:
4847:
4828:
4827:
4790:
4771:
4767:
4740:
4735:
4734:
4684:
4683:
4600:
4585:
4574:
4573:
4564:
4558:
4533:
4532:
4502:
4501:
4465:
4464:
4433:
4432:
4391:
4366:
4365:
4361:
4348:
4343:
4342:
4323:
4322:
4273:
4272:
4243:
4242:
4223:
4222:
4195:
4194:
4180:
4169:
4142:
4141:
4122:
4121:
4070:
4069:
4044:
4043:
4014:
4013:
3986:
3957:
3956:
3904:
3903:
3878:
3864:
3863:
3842:
3837:
3836:
3780:
3779:
3721:
3720:
3690:
3682:
3681:
3641:
3640:
3609:
3599:
3586:
3572:
3571:
3525:
3524:
3497:
3487:
3474:
3450:
3437:
3427:
3426:
3422:
3421:
3416:
3415:
3357:
3356:
3321:
3313:
3312:
3297:
3295:Double covering
3261:
3260:
3239:
3226:
3221:
3220:
3168:
3167:
3115:
3114:
3095:
3094:
3066:
3065:
3034:
3033:
2999:
2998:
2965:
2934:
2908:
2907:
2876:
2875:
2856:
2855:
2819:
2791:
2786:
2785:
2740:
2718:
2713:
2712:
2691:
2678:
2673:
2672:
2629:
2628:
2597:
2596:
2568:
2567:
2528:
2523:
2522:
2467:
2466:
2432:
2431:
2400:
2399:
2396:
2350:
2337:
2326:
2325:
2300:
2299:
2268:
2267:
2246:
2236:
2220:
2210:
2205:
2204:
2173:
2165:
2164:
2122:
2109:
2096:
2083:
2078:
2077:
2050:
2009:
2008:
1973:
1962:
1961:
1934:
1929:
1928:
1901:
1888:
1878:
1873:
1872:
1838:
1837:
1806:
1805:
1760:
1759:
1660:
1655:
1654:
1619:
1618:
1588:
1583:
1582:
1553:
1548:
1547:
1528:
1527:
1508:
1507:
1483:
1464:
1451:
1438:
1415:
1414:
1382:
1381:
1353:
1352:
1305:
1301:
1265:
1264:
1189:
1188:
1155:
1143:quantum gravity
1123:spin connection
1111:structure group
1103:spin structures
1087:
1053:
1031:universal cover
1015:
997:
955:
954:
882:
871:
870:
860:
840:
820:, denoted Spin(
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
479:
478:
454:
453:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
15:
12:
11:
5:
9786:
9784:
9776:
9775:
9770:
9765:
9755:
9754:
9751:
9750:
9744:
9729:
9726:
9725:
9724:
9717:
9704:
9703:External links
9701:
9698:
9697:
9683:
9660:
9645:
9624:
9596:
9589:
9567:
9559:
9532:
9531:
9529:
9526:
9525:
9524:
9518:
9500:
9480:
9479:Related groups
9477:
9475:
9474:
9469:
9464:
9459:
9457:Spin structure
9454:
9449:
9444:
9439:
9433:
9431:
9428:
9420:
9419:
9416:cross-polytope
9400:
9389:
9383:
9348:
9332:
9318:
9313:
9308:
9303:
9298:
9295:
9292:
9289:
9284:
9279:
9274:
9271:
9268:
9265:
9260:
9232:
9229:
9226:
9223:
9218:
9196:
9193:
9188:
9183:
9103:
9100:
9083:
9069:
9068:
9057:
9054:
9051:
9043:
9040:
9037:
9034:
9026:
9023:
9020:
9017:
9009:
9006:
9003:
9000:
8992:
8989:
8986:
8983:
8975:
8972:
8949:
8946:
8886:
8877:
8873:
8869:
8845:
8841:
8812:
8809:
8805:
8801:
8798:
8795:
8790:
8787:
8763:
8762:
8747:
8728:
8727:
8682:covering group
8674:
8671:
8670:
8669:
8652:
8642:
8639:
8636:
8628:
8625:
8622:
8619:
8616:
8613:
8609:
8604:
8599:
8594:
8589:
8584:
8583:
8575:
8572:
8569:
8561:
8558:
8555:
8552:
8549:
8546:
8543:
8540:
8536:
8531:
8526:
8525:
8517:
8512: or
8509:
8506:
8502:
8497:
8492:
8491:
8489:
8484:
8481:
8479:
8477:
8474:
8471:
8468:
8465:
8462:
8459:
8456:
8453:
8450:
8447:
8444:
8443:
8438:
8433:
8430:
8427:
8424:
8421:
8417:
8412:
8407:
8402:
8397:
8392:
8391:
8388:
8385:
8382:
8379:
8376:
8373:
8370:
8366:
8361:
8356:
8355:
8352:
8349:
8346:
8343:
8340:
8337:
8334:
8330:
8325:
8320:
8319:
8317:
8312:
8309:
8307:
8305:
8302:
8298:
8294:
8291:
8288:
8285:
8282:
8279:
8276:
8273:
8270:
8269:
8247:
8244:
8225:
8221:
8217:
8214:
8211:
8208:
8173:
8169:
8148:
8143:
8139:
8135:
8130:
8126:
8122:
8119:
8116:
8113:
8105:
8102:
8099:
8096:
8071:
8068:
8065:
8062:
8054:
8051:
8048:
8045:
8040:
8036:
8032:
8029:
8026:
8023:
8015:
8012:
7999:
7998:
7983:
7978:
7973:
7970:
7967:
7964:
7961:
7953:
7948:
7944:
7923:
7920:
7917:
7914:
7910:
7904:
7900:
7896:
7891:
7886:
7883:
7878:
7875:
7872:
7869:
7841:
7838:
7835:
7832:
7824:
7819:
7815:
7811:
7808:
7805:
7802:
7799:
7791:
7786:
7782:
7761:
7758:
7755:
7734:
7731:
7728:
7706:
7703:
7700:
7697:
7694:
7691:
7683:
7678:
7674:
7653:
7650:
7647:
7644:
7636:
7631:
7627:
7606:
7603:
7600:
7597:
7594:
7591:
7588:
7580:
7575:
7571:
7567:
7564:
7561:
7558:
7555:
7552:
7549:
7541:
7536:
7532:
7528:
7525:
7505:
7502:
7499:
7479:
7476:
7473:
7453:
7450:
7447:
7442:
7438:
7434:
7429:
7425:
7402:
7399:
7394:
7391:
7388:
7384:
7380:
7375:
7371:
7367:
7364:
7361:
7358:
7355:
7347:
7342:
7338:
7334:
7331:
7328:
7325:
7322:
7319:
7316:
7308:
7303:
7299:
7295:
7292:
7287:
7284:
7281:
7277:
7273:
7268:
7264:
7241:
7238:
7235:
7232:
7229:
7226:
7223:
7220:
7212:
7207:
7204:
7201:
7197:
7193:
7190:
7185:
7182:
7179:
7175:
7171:
7166:
7162:
7158:
7155:
7152:
7149:
7146:
7138:
7133:
7129:
7125:
7122:
7119:
7116:
7113:
7110:
7107:
7099:
7094:
7090:
7086:
7083:
7055:
7050:
7047:
7044:
7040:
7036:
7033:
7030:
7027:
7019:
7016:
7013:
7010:
7007:
7004:
6974:
6969:
6966:
6963:
6959:
6955:
6952:
6949:
6946:
6943:
6940:
6931:
6927:
6924:
6921:
6892:
6889:
6886:
6883:
6880:
6872:
6869:
6866:
6863:
6858:
6855:
6852:
6818:
6815:
6812:
6808:
6804:
6801:
6798:
6795:
6790:
6787:
6784:
6752:
6749:
6746:
6743:
6740:
6737:
6734:
6731:
6728:
6725:
6722:
6700:
6695:
6673:
6670:
6667:
6664:
6661:
6648:
6647:
6616:
6611:
6608:
6603:
6600:
6597:
6594:
6591:
6588:
6566:
6562:
6558:
6555:
6552:
6549:
6546:
6543:
6523:
6520:
6517:
6514:
6511:
6491:
6488:
6485:
6465:
6462:
6459:
6456:
6453:
6450:
6447:
6442:
6438:
6413:
6410:
6407:
6404:
6401:
6398:
6395:
6390:
6386:
6373:
6370:
6323:
6297:
6293:
6286:
6270:
6258:
6238:
6224:
6223:
6210:
6205:
6202:
6199:
6196:
6193:
6190:
6186:
6181:
6176:
6175:
6172:
6169:
6166:
6163:
6160:
6157:
6154:
6151:
6148:
6144:
6143:
6140:
6137:
6134:
6131:
6128:
6125:
6122:
6119:
6116:
6113:
6110:
6107:
6104:
6100:
6096:
6092:
6091:
6088:
6085:
6082:
6079:
6076:
6070: or
6066:
6063:
6060:
6057:
6054:
6051:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6026:
6025:
6022:
6019:
6016:
6013:
6010:
6007:
6004:
6001:
5998:
5995:
5991:
5986:
5981:
5980:
5977:
5974:
5971:
5968:
5965:
5959: or
5955:
5952:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5920:
5917:
5916:
5914:
5909:
5906:
5903:
5900:
5897:
5894:
5891:
5881:
5876:
5872:
5837:
5836:
5776:
5718:
5704:
5672:
5647:
5646:
5635:
5632:
5628:
5625:
5621:
5618:
5615:
5612:
5609:
5606:
5603:
5598:
5594:
5555:
5552:
5528:
5527:
5517:
5507:
5501:
5491:
5481:
5475:
5457:
5447:
5437:
5358:
5328:
5325:
5306:
5305:
5291:
5287:
5283:
5278:
5274:
5258:
5247:
5246:
5232:
5228:
5224:
5219:
5215:
5199:
5188:
5187:
5173:
5169:
5165:
5160:
5156:
5152:
5147:
5143:
5131:
5117:
5116:
5102:
5098:
5094:
5089:
5085:
5081:
5076:
5072:
5052:
5038:
5037:
5023:
5019:
4978:
4967:
4966:
4959:
4956:
4890:
4887:
4835:
4824:
4823:
4812:
4808:
4803:
4797:
4793:
4789:
4786:
4783:
4780:
4777:
4774:
4770:
4766:
4763:
4760:
4757:
4754:
4748:
4743:
4710:
4706:
4703:
4700:
4697:
4694:
4691:
4677:
4676:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4608:
4603:
4599:
4594:
4589:
4584:
4581:
4568:exact sequence
4560:Main article:
4557:
4554:
4540:
4515:
4511:
4477:
4474:
4452:
4449:
4446:
4443:
4440:
4418:
4412:
4407:
4401:
4398:
4394:
4390:
4387:
4382:
4379:
4376:
4373:
4369:
4364:
4360:
4355:
4351:
4330:
4319:
4318:
4305:
4302:
4297:
4294:
4291:
4287:
4283:
4280:
4255:
4252:
4230:
4209:
4205:
4202:
4168:
4165:
4152:
4149:
4129:
4086:
4083:
4080:
4077:
4057:
4054:
4051:
4027:
4024:
4021:
4010:
4009:
3998:
3993:
3989:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3885:
3881:
3877:
3874:
3871:
3849:
3845:
3833:
3832:
3821:
3818:
3815:
3812:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3764:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3717:
3716:
3705:
3702:
3697:
3693:
3689:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3633:
3632:
3621:
3616:
3612:
3606:
3602:
3598:
3593:
3589:
3585:
3582:
3579:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3521:
3520:
3509:
3504:
3500:
3494:
3490:
3486:
3481:
3477:
3473:
3468:
3463:
3457:
3453:
3449:
3444:
3440:
3434:
3430:
3425:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3333:
3328:
3324:
3320:
3296:
3293:
3280:
3277:
3274:
3271:
3268:
3246:
3242:
3238:
3233:
3229:
3205:
3202:
3199:
3193:
3190:
3184:
3181:
3178:
3175:
3152:
3149:
3143:
3140:
3137:
3134:
3128:
3125:
3102:
3079:
3076:
3053:
3050:
3047:
3044:
3041:
3021:
3018:
3015:
3012:
3009:
3006:
2986:
2983:
2980:
2977:
2972:
2968:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2941:
2937:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2895:
2892:
2889:
2886:
2883:
2863:
2843:
2840:
2837:
2834:
2831:
2826:
2822:
2818:
2815:
2812:
2809:
2806:
2803:
2798:
2794:
2770:
2767:
2761:
2758:
2755:
2752:
2747:
2743:
2739:
2736:
2733:
2730:
2725:
2721:
2698:
2694:
2690:
2685:
2681:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2616:
2613:
2610:
2607:
2604:
2581:
2578:
2555:
2552:
2549:
2546:
2543:
2538:
2535:
2531:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2451:
2448:
2445:
2442:
2439:
2419:
2416:
2413:
2410:
2407:
2395:
2392:
2357:
2353:
2349:
2344:
2340:
2336:
2333:
2313:
2310:
2307:
2296:
2295:
2284:
2281:
2278:
2275:
2253:
2249:
2243:
2239:
2235:
2232:
2227:
2223:
2217:
2213:
2185:
2180:
2176:
2172:
2137:
2134:
2129:
2125:
2121:
2116:
2112:
2108:
2103:
2099:
2095:
2086:
2074:
2073:
2062:
2053:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
1991:
1988:
1985:
1980:
1976:
1972:
1969:
1949:
1946:
1941:
1937:
1925:
1924:
1913:
1908:
1904:
1900:
1895:
1891:
1885:
1881:
1857:
1854:
1851:
1848:
1845:
1825:
1822:
1819:
1816:
1813:
1783:
1780:
1777:
1772:
1769:
1740:
1739:
1728:
1725:
1722:
1719:
1714:
1711:
1708:
1705:
1700:
1697:
1694:
1691:
1686:
1683:
1680:
1677:
1672:
1667:
1663:
1650:is defined as
1637:
1634:
1631:
1628:
1603:
1600:
1595:
1591:
1569:
1565:
1560:
1556:
1535:
1515:
1504:
1503:
1490:
1486:
1482:
1479:
1476:
1471:
1467:
1463:
1458:
1454:
1450:
1445:
1441:
1437:
1434:
1431:
1428:
1425:
1422:
1395:
1392:
1389:
1369:
1366:
1363:
1360:
1349:
1348:
1337:
1333:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1304:
1299:
1295:
1291:
1287:
1284:
1281:
1278:
1275:
1272:
1257:) is then the
1251:
1250:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1208:
1204:
1201:
1197:
1174:tensor algebra
1154:
1151:
1133:to be written
1131:Dirac equation
1086:
1083:
974:
971:
968:
965:
962:
947:
946:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
891:
886:
881:
878:
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9785:
9774:
9771:
9769:
9766:
9764:
9761:
9760:
9758:
9747:
9741:
9737:
9732:
9731:
9727:
9722:
9718:
9715:
9711:
9707:
9706:
9702:
9686:
9684:9780521795401
9680:
9673:
9672:
9664:
9661:
9656:
9652:
9648:
9642:
9638:
9631:
9629:
9625:
9622:
9619:
9618:3-540-42627-2
9615:
9611:
9608:Jürgen Jost,
9605:
9603:
9601:
9597:
9592:
9586:
9582:
9578:
9571:
9568:
9562:
9556:
9552:
9548:
9547:Spin Geometry
9544:
9537:
9534:
9527:
9522:
9519:
9516:
9512:
9508:
9504:
9501:
9498:
9494:
9490:
9486:
9483:
9482:
9478:
9473:
9470:
9468:
9465:
9463:
9460:
9458:
9455:
9453:
9452:Spinor bundle
9450:
9448:
9445:
9443:
9440:
9438:
9435:
9434:
9429:
9427:
9425:
9417:
9413:
9409:
9405:
9401:
9398:
9392:
9386:
9379:
9375:
9371:
9367:
9366:
9365:
9362:
9358:
9352:
9342:
9336:
9316:
9311:
9301:
9296:
9293:
9290:
9287:
9277:
9272:
9269:
9266:
9263:
9248:
9230:
9227:
9224:
9221:
9194:
9191:
9186:
9171:
9167:
9162:
9160:
9156:
9152:
9148:
9144:
9140:
9136:
9132:
9129:, there is a
9128:
9122:
9118:
9111:
9109:
9101:
9099:
9097:
9093:
9089:
9078:
9074:
9052:
9035:
9018:
9001:
8984:
8970:
8963:
8962:
8961:
8959:
8955:
8947:
8945:
8941:
8937:
8931:
8925:
8921:
8917:
8913:
8907:
8903:
8898:
8896:
8892:
8884:
8865:
8861:
8855:
8849:
8837:
8832:
8828:
8823:
8810:
8799:
8796:
8776:
8768:
8760:
8756:
8752:
8748:
8745:
8741:
8737:
8733:
8732:
8731:
8725:
8721:
8717:
8713:
8712:
8711:
8709:
8703:
8698:
8694:
8690:
8685:
8683:
8679:
8672:
8640:
8637:
8634:
8626:
8623:
8620:
8617:
8614:
8607:
8597:
8592:
8573:
8570:
8567:
8559:
8556:
8553:
8550:
8547:
8544:
8541:
8534:
8515:
8507:
8500:
8487:
8482:
8480:
8469:
8466:
8463:
8457:
8454:
8448:
8431:
8428:
8425:
8422:
8415:
8405:
8400:
8386:
8383:
8380:
8377:
8374:
8371:
8364:
8350:
8347:
8344:
8341:
8338:
8335:
8328:
8315:
8310:
8308:
8292:
8289:
8283:
8280:
8274:
8260:
8259:
8258:
8254:
8245:
8243:
8241:
8240:Lorentz group
8215:
8212:
8209:
8193:
8189:
8171:
8167:
8146:
8141:
8137:
8120:
8117:
8114:
8097:
8063:
8049:
8046:
8030:
8027:
8024:
8010:
7997:
7981:
7971:
7962:
7946:
7942:
7918:
7915:
7908:
7902:
7898:
7894:
7889:
7876:
7870:
7853:
7833:
7817:
7813:
7809:
7800:
7784:
7780:
7759:
7756:
7753:
7745:
7732:
7729:
7726:
7698:
7695:
7692:
7676:
7672:
7645:
7629:
7625:
7604:
7601:
7589:
7573:
7569:
7556:
7553:
7550:
7534:
7530:
7523:
7503:
7500:
7497:
7477:
7474:
7471:
7451:
7448:
7440:
7436:
7427:
7423:
7413:
7400:
7392:
7389:
7386:
7382:
7373:
7369:
7356:
7340:
7336:
7323:
7320:
7317:
7301:
7297:
7285:
7282:
7279:
7275:
7266:
7262:
7252:
7239:
7227:
7224:
7221:
7205:
7202:
7199:
7195:
7183:
7180:
7177:
7173:
7164:
7160:
7147:
7131:
7127:
7114:
7111:
7108:
7092:
7088:
7081:
7073:
7071:
7066:
7053:
7048:
7045:
7042:
7038:
7028:
7011:
7008:
7005:
6990:
6985:
6972:
6967:
6964:
6961:
6957:
6953:
6947:
6944:
6941:
6929:
6922:
6906:
6887:
6884:
6881:
6870:
6864:
6853:
6816:
6813:
6810:
6806:
6802:
6796:
6785:
6747:
6744:
6741:
6738:
6735:
6732:
6729:
6723:
6720:
6698:
6668:
6662:
6659:
6650:
6649:
6645:
6644:
6641:
6639:
6634:
6632:
6614:
6601:
6595:
6589:
6586:
6564:
6560:
6556:
6550:
6544:
6541:
6518:
6512:
6509:
6489:
6486:
6483:
6457:
6451:
6448:
6440:
6436:
6427:
6405:
6399:
6396:
6388:
6384:
6371:
6369:
6358:
6354:
6347:
6343:
6335:
6331:
6321:
6317:
6309:
6305:
6280:
6276:
6268:
6264:
6252:
6248:
6242:
6234:
6230:
6203:
6200:
6197:
6194:
6191:
6184:
6170:
6167:
6164:
6161:
6158:
6155:
6152:
6135:
6132:
6129:
6123:
6117:
6114:
6111:
6098:
6083:
6080:
6077:
6061:
6058:
6055:
6049:
6043:
6040:
6037:
6020:
6017:
6014:
6011:
6008:
6005:
6002:
5999:
5996:
5989:
5972:
5969:
5966:
5950:
5947:
5944:
5938:
5932:
5929:
5926:
5918:
5912:
5907:
5898:
5895:
5892:
5874:
5870:
5862:
5861:
5860:
5858:
5854:
5850:
5846:
5842:
5834:
5830:
5826:
5825:
5824:
5822:
5818:
5814:
5810:
5806:
5802:
5798:
5794:
5790:
5786:
5782:
5774:
5770:
5766:
5761:
5759:
5755:
5751:
5747:
5742:
5740:
5736:
5733:) and PSL(2,
5732:
5728:
5724:
5692:
5688:
5660:
5656:
5652:
5633:
5626:
5623:
5616:
5610:
5604:
5596:
5592:
5584:
5583:
5582:
5580:
5576:
5572:
5568:
5564:
5560:
5553:
5551:
5547:
5543:
5539:
5535:
5526:
5524:
5519:Spin(6, 2) =
5518:
5516:
5514:
5509:Spin(3, 3) =
5508:
5506:
5503:Spin(4, 2) =
5502:
5500:
5498:
5493:Spin(5, 1) =
5492:
5490:
5488:
5483:Spin(3, 2) =
5482:
5480:
5477:Spin(4, 1) =
5476:
5474:
5472:
5466:
5464:
5459:Spin(2, 2) =
5458:
5456:
5454:
5449:Spin(3, 1) =
5448:
5446:
5444:
5439:Spin(2, 1) =
5438:
5436:
5434:
5429:Spin(1, 1) =
5428:
5427:
5426:
5422:
5418:
5410:
5406:
5398:
5394:
5387:
5383:
5378:
5374:
5368:
5364:
5354:
5350:
5344:
5340:
5334:
5326:
5324:
5322:
5318:
5312:
5289:
5285:
5281:
5276:
5272:
5263:
5259:
5256:
5252:
5251:
5250:
5230:
5226:
5222:
5217:
5213:
5204:
5200:
5197:
5193:
5192:
5191:
5171:
5167:
5163:
5158:
5154:
5150:
5145:
5141:
5132:
5130:
5126:
5122:
5121:
5120:
5100:
5096:
5092:
5087:
5083:
5079:
5074:
5070:
5061:
5057:
5053:
5051:
5047:
5043:
5042:
5041:
5021:
5017:
5007:
5004:
5000:
4995:
4991:
4987:
4983:
4979:
4976:
4972:
4971:
4970:
4964:
4960:
4957:
4954:
4950:
4949:
4948:
4946:
4942:
4938:
4934:
4930:
4926:
4922:
4918:
4914:
4910:
4906:
4902:
4901:
4896:
4888:
4886:
4884:
4880:
4876:
4871:
4867:
4863:
4855:
4851:
4833:
4810:
4806:
4801:
4795:
4791:
4787:
4781:
4775:
4772:
4768:
4764:
4758:
4752:
4741:
4733:
4732:
4731:
4729:
4725:
4704:
4698:
4692:
4689:
4682:
4663:
4654:
4648:
4642:
4636:
4630:
4627:
4618:
4612:
4601:
4592:
4579:
4572:
4571:
4570:
4569:
4563:
4555:
4553:
4538:
4530:
4529:endomorphisms
4513:
4509:
4500:
4496:
4491:
4472:
4450:
4447:
4444:
4441:
4438:
4416:
4410:
4405:
4399:
4396:
4392:
4388:
4385:
4380:
4377:
4374:
4371:
4367:
4362:
4358:
4353:
4349:
4328:
4300:
4295:
4292:
4289:
4281:
4278:
4271:
4270:
4269:
4250:
4228:
4203:
4200:
4192:
4187:
4183:
4179:of dimension
4178:
4174:
4166:
4164:
4150:
4147:
4127:
4119:
4115:
4111:
4107:
4102:
4100:
4081:
4075:
4055:
4052:
4049:
4041:
4025:
4022:
4019:
3996:
3991:
3987:
3983:
3980:
3977:
3974:
3968:
3962:
3955:
3954:
3953:
3936:
3930:
3918:
3912:
3909:
3901:
3883:
3875:
3869:
3847:
3843:
3819:
3816:
3813:
3810:
3806:
3803:
3800:
3797:
3791:
3785:
3778:
3777:
3776:
3759:
3753:
3750:
3741:
3735:
3732:
3729:
3726:
3703:
3700:
3695:
3691:
3687:
3680:
3679:
3678:
3661:
3655:
3652:
3649:
3646:
3638:
3619:
3614:
3610:
3604:
3600:
3596:
3591:
3583:
3580:
3570:
3569:
3568:
3551:
3545:
3542:
3539:
3536:
3533:
3530:
3507:
3502:
3498:
3492:
3488:
3484:
3479:
3475:
3471:
3466:
3461:
3455:
3451:
3447:
3442:
3438:
3432:
3428:
3423:
3414:
3413:
3412:
3395:
3389:
3386:
3377:
3371:
3368:
3365:
3362:
3355:
3352:. Define an
3351:
3347:
3326:
3322:
3310:
3306:
3302:
3294:
3292:
3291:a Lie group.
3275:
3269:
3266:
3244:
3240:
3236:
3231:
3227:
3217:
3200:
3188:
3182:
3179:
3176:
3173:
3147:
3141:
3135:
3123:
3100:
3074:
3048:
3042:
3039:
3019:
3016:
3010:
3004:
2978:
2970:
2966:
2959:
2956:
2947:
2939:
2935:
2928:
2925:
2919:
2913:
2890:
2884:
2881:
2861:
2841:
2838:
2832:
2824:
2820:
2816:
2813:
2810:
2804:
2796:
2792:
2765:
2759:
2753:
2745:
2741:
2737:
2731:
2723:
2719:
2696:
2692:
2688:
2683:
2679:
2671:choose paths
2655:
2649:
2646:
2643:
2640:
2637:
2634:
2611:
2605:
2602:
2576:
2547:
2536:
2533:
2529:
2505:
2499:
2496:
2487:
2481:
2478:
2475:
2472:
2465:
2446:
2440:
2437:
2414:
2408:
2405:
2393:
2391:
2389:
2388:supersymmetry
2385:
2381:
2380:spinor bundle
2377:
2373:
2355:
2351:
2347:
2342:
2338:
2334:
2331:
2311:
2308:
2305:
2282:
2279:
2276:
2273:
2251:
2247:
2241:
2237:
2233:
2230:
2225:
2221:
2215:
2211:
2203:
2202:
2201:
2199:
2178:
2174:
2161:
2159:
2155:
2151:
2135:
2132:
2127:
2123:
2119:
2114:
2110:
2106:
2101:
2097:
2093:
2084:
2060:
2051:
2047:
2041:
2035:
2032:
2029:
2023:
2017:
2014:
2007:
2006:
2005:
2002:
1989:
1986:
1978:
1974:
1967:
1947:
1944:
1939:
1935:
1911:
1906:
1902:
1898:
1893:
1889:
1883:
1879:
1871:
1870:
1869:
1852:
1846:
1843:
1820:
1814:
1811:
1804:
1799:
1797:
1778:
1757:
1753:
1749:
1745:
1726:
1720:
1698:
1692:
1670:
1665:
1661:
1653:
1652:
1651:
1617:
1601:
1598:
1593:
1589:
1563:
1558:
1554:
1533:
1513:
1488:
1484:
1480:
1477:
1474:
1469:
1465:
1461:
1456:
1452:
1448:
1443:
1439:
1435:
1429:
1423:
1420:
1413:
1412:
1411:
1409:
1393:
1390:
1387:
1364:
1358:
1335:
1331:
1324:
1318:
1315:
1312:
1309:
1306:
1302:
1297:
1293:
1285:
1279:
1273:
1270:
1263:
1262:
1261:
1260:
1256:
1237:
1234:
1228:
1225:
1222:
1216:
1213:
1210:
1202:
1199:
1187:
1186:
1185:
1183:
1179:
1175:
1171:
1168:
1164:
1160:
1152:
1150:
1148:
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1115:spinor bundle
1112:
1108:
1104:
1100:
1096:
1092:
1084:
1082:
1080:
1076:
1072:
1068:
1064:
1059:
1056:
1051:
1047:
1042:
1040:
1038:
1032:
1028:
1024:
1018:
1012:
1010:
1004:
1000:
995:
991:
986:
969:
963:
960:
952:
933:
924:
918:
915:
906:
900:
897:
889:
876:
869:
868:
867:
863:
858:
852:
848:
844:
839:
835:
831:
827:
823:
819:
815:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
31:
19:
9735:
9721:OEIS:A096336
9714:OEIS:A280191
9688:. Retrieved
9670:
9663:
9636:
9620:
9609:
9576:
9570:
9546:
9536:
9521:String group
9514:
9506:
9496:
9488:
9421:
9396:
9390:
9384:
9373:
9363:
9356:
9350:
9340:
9334:
9246:
9169:
9165:
9163:
9154:
9146:
9142:
9138:
9134:
9120:
9116:
9112:
9108:point groups
9105:
9095:
9092:string group
9087:
9070:
8951:
8939:
8935:
8923:
8919:
8915:
8911:
8905:
8901:
8899:
8894:
8890:
8882:
8859:
8853:
8850:
8848:may differ.
8835:
8824:
8769:(or two, if
8764:
8758:
8754:
8750:
8743:
8739:
8735:
8729:
8723:
8719:
8715:
8707:
8701:
8696:
8686:
8676:
8252:
8249:
8192:contractible
8002:
7854:
7718:
7414:
7253:
7074:
7067:
6986:
6651:
6635:
6375:
6356:
6352:
6345:
6341:
6333:
6329:
6319:
6315:
6307:
6303:
6283:is given by
6278:
6274:
6266:
6262:
6250:
6246:
6243:
6232:
6228:
6225:
5856:
5852:
5848:
5844:
5838:
5832:
5828:
5820:
5816:
5812:
5808:
5804:
5800:
5796:
5792:
5784:
5780:
5768:
5764:
5762:
5757:
5753:
5745:
5743:
5738:
5734:
5730:
5726:
5725:) determine
5722:
5690:
5686:
5658:
5650:
5648:
5578:
5570:
5566:
5557:
5545:
5541:
5537:
5533:
5529:
5522:
5512:
5496:
5486:
5470:
5462:
5452:
5442:
5432:
5420:
5416:
5408:
5404:
5396:
5392:
5385:
5381:
5366:
5362:
5353:double cover
5342:
5338:
5330:
5320:
5310:
5307:
5254:
5253:Cl(6)= M(4,
5248:
5195:
5194:Cl(5)= M(2,
5189:
5128:
5124:
5118:
5045:
5039:
5005:
5002:
4998:
4993:
4989:
4974:
4968:
4952:
4944:
4940:
4936:
4932:
4924:
4920:
4916:
4905:root systems
4898:
4895:isomorphisms
4892:
4872:
4865:
4861:
4853:
4849:
4825:
4727:
4723:
4678:
4565:
4556:Complex case
4495:spinor space
4494:
4492:
4320:
4185:
4181:
4176:
4173:Weyl spinors
4170:
4167:Spinor space
4117:
4113:
4112:) and of SO(
4109:
4105:
4103:
4098:
4039:
4011:
3899:
3834:
3718:
3636:
3634:
3522:
3349:
3308:
3304:
3300:
3298:
3218:
2464:covering map
2397:
2297:
2197:
2162:
2157:
2153:
2149:
2075:
2003:
1926:
1800:
1755:
1747:
1743:
1741:
1616:spin algebra
1615:
1505:
1350:
1254:
1252:
1181:
1177:
1169:
1162:
1156:
1153:Construction
1105:on (pseudo-)
1088:
1074:
1062:
1060:
1054:
1043:
1036:
1022:
1016:
1013:
1002:
998:
989:
987:
948:
861:
850:
846:
842:
834:double cover
821:
817:
811:
640:
628:
616:
604:
592:
580:
568:
556:
327:
284:
271:
260:
249:
245:Cyclic group
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
18:Spin algebra
9690:24 February
9149:), and the
8691:, which is
8190:, which is
8188:hyperboloid
7719:as long as
6364:is sent to
6349:is sent to
5050:quaternions
4929:quaternions
4879:gauge group
4846:identifies
2997:satisfying
2711:satisfying
2163:If the set
1927:where each
1752:Lie algebra
1750:. It is a
1009:Lie algebra
814:mathematics
530:Topological
369:alternating
9763:Lie groups
9757:Categories
9646:0821835742
9528:References
9424:pin groups
9339:< Spin(
8889:while PSO(
8693:centerless
8645: even
8578: even
7855:And since
7772:, we have
6318:→ (1,1) ∈
6292:(1, 1) ∈ Z
6226:Thus once
5851:), taking
5748:) are all
5689:determine
5530:Note that
5423:) = (1, 1)
5260:Spin(6) =
5201:Spin(5) =
5054:Spin(3) =
4980:Spin(2) =
4961:Spin(1) =
4915:. Writing
4321:The space
4120:) because
4116:) by Spin(
3677:for which
3307:) by Spin(
1007:, and its
818:spin group
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
9485:Pin group
9412:hypercube
9399:-simplex.
9302:×
9278:≅
9192:×
9125:, by the
9042:→
9025:→
9008:→
8991:→
8978:Fivebrane
8974:→
8971:…
8773:) of the
8699:) is for
8598:⊕
8520: odd
8458:
8449:
8406:⊕
8284:
8275:
8224:↑
8134:→
8129:↑
8104:→
8050:π
8047:≅
8039:↑
8011:π
7972:≅
7943:π
7934:, we get
7916:±
7895:≅
7877:≅
7814:π
7810:≅
7781:π
7746:, so for
7696:−
7673:π
7626:π
7599:→
7570:π
7566:→
7554:−
7531:π
7527:→
7490:. So for
7424:π
7390:−
7370:π
7366:→
7337:π
7333:→
7321:−
7298:π
7294:→
7283:−
7263:π
7240:⋯
7237:→
7225:−
7203:−
7196:π
7192:→
7181:−
7161:π
7157:→
7128:π
7124:→
7112:−
7089:π
7085:→
7082:⋯
7046:−
7035:→
7018:→
7009:−
6989:fibration
6965:−
6954:≅
6945:−
6885:−
6814:−
6742:⋯
6663:
6602:≅
6557:≅
6452:
6437:π
6400:
6385:π
6340:(1, 0) ∈
6290:going to
6099:×
5871:π
5831:) × Spin(
5617:
5611:⊂
5593:π
5559:Connected
5540:) = Spin(
5521:SU(2, 2,
5282:≅
5223:≅
5164:×
5151:≅
5093:≅
5080:≅
4834:∼
4811:∼
4788:×
4776:
4753:
4705:⊗
4693:
4661:→
4649:
4643:×
4631:
4625:→
4613:
4598:→
4583:→
4510:⋀
4476:¯
4448:≤
4442:≤
4386:−
4378:−
4350:η
4304:¯
4296:⊕
4282:⊗
4254:¯
4204:⊗
4148:−
4108:) by Pin(
4076:ρ
4053:∈
4023:∈
3992:∗
3963:ρ
3952:given by
3931:
3925:→
3913:
3870:α
3848:∗
3814:∈
3801:−
3786:α
3754:
3748:→
3736:
3730::
3727:α
3656:
3650:∈
3546:
3540:∈
3485:⋯
3448:⋯
3390:
3384:→
3372:
3270:
3241:γ
3228:γ
3192:~
3189:γ
3177:⋅
3151:~
3127:~
3124:γ
3101:γ
3078:~
3075:γ
3043:
3005:γ
2967:γ
2957:⋅
2936:γ
2914:γ
2885:
2862:γ
2821:γ
2793:γ
2769:~
2742:γ
2720:γ
2693:γ
2680:γ
2650:
2644:∈
2606:
2580:~
2534:−
2500:
2494:→
2482:
2441:
2409:
2309:⊗
2277:≠
2234:−
2136:⋯
2133:⊕
2120:⊕
2107:⊕
2048:∩
2036:
2018:
1945:∈
1899:⋯
1847:
1815:
1803:pin group
1481:⊕
1478:⋯
1475:⊕
1462:⊕
1449:⊕
1424:
1391:∈
1316:−
1310:⊗
1274:
1238:⋯
1235:⊕
1226:⊗
1217:⊕
1211:⊕
994:dimension
964:
931:→
919:
913:→
901:
895:→
880:→
826:Lie group
701:Conformal
589:Euclidean
196:nilpotent
9736:K-Theory
9655:55487352
9545:(1989).
9430:See also
9355:< SO(
8928:and the
8771:SO = PSO
6351:(1,1) ∈
5627:′
5505:SU(2, 2)
5479:Sp(1, 1)
5123:Cl(4) =
5044:Cl(3) =
4973:Cl(2) =
4951:Cl(1) =
4927:for the
3835:and let
3032:. Since
2906:defined
2376:fermions
1099:electron
1095:fermions
1067:subgroup
830:manifold
824:), is a
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
9773:Spinors
9594:page 15
9565:page 14
9402:higher
9368:higher
9119:) → SO(
9094:String(
8918:) ⊂ SO(
8914:) × SO(
8858:, Spin(
8708:maximal
6366:(1, −1)
5653:′) the
5649:with Z(
5375:of the
5317:Spin(8)
4038:. When
3862:denote
2521:. Then
2076:where
1794:of the
1165:with a
1091:physics
1021:, Spin(
951:lifting
845:) = SO(
836:of the
832:is the
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
9742:
9681:
9653:
9643:
9616:
9587:
9557:
9513:, Sp(2
9447:Spinor
9245:in SO(
8995:String
8856:> 2
8838:> 1
8749:Spin(2
8734:Spin(2
8704:> 2
8246:Center
8194:, and
8159:where
7617:hence
6646:Proof
6579:, and
6362:(0, 1)
6310:> 2
6301:. For
6269:)) → π
6261:(Spin(
6253:> 2
6235:> 2
5843:of SO(
5791:of SO(
5775:, Spin
5655:center
5573:′ the
5511:SL(4,
5495:SL(2,
5485:Sp(4,
5469:SL(2,
5461:SL(2,
5451:SL(2,
5441:SL(2,
5431:GL(1,
5399:> 2
5390:. For
5371:, the
5313:= 7, 8
4012:where
3344:be an
2784:, and
2627:, for
1614:. The
1506:where
1408:graded
1351:where
1139:tetrad
1117:. The
1046:kernel
1019:> 2
1005:− 1)/2
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
9675:(PDF)
9467:Anyon
9115:Spin(
8934:Spin(
8862:) is
8844:and π
8714:Spin(
6773:Orbit
6306:= 2,
6285:1 ∈ Z
5827:Spin(
5823:) is
5717:and π
5532:Spin(
5403:Spin(
5337:Spin(
5315:(see
5262:SU(4)
5203:Sp(2)
5060:SU(2)
5056:Sp(1)
4986:SO(2)
4858:with
3113:with
1113:of a
1061:Spin(
1025:) is
730:Sp(∞)
727:SU(∞)
140:image
9740:ISBN
9708:The
9692:2023
9679:ISBN
9651:OCLC
9641:ISBN
9614:ISBN
9585:ISBN
9555:ISBN
9505:Mp(2
9495:, O(
9487:Pin(
9012:Spin
8851:For
8825:The
8761:+1),
8455:Spin
8281:Spin
7757:>
7730:>
7664:and
7501:>
7475:<
7464:for
6841:Stab
6487:>
6476:for
6360:and
6314:1 ∈
6273:(SO(
6201:>
6156:>
6000:>
5752:for
5561:and
5048:the
4982:U(1)
4963:O(1)
4773:Spin
4742:Spin
4602:Spin
4493:The
4431:for
3348:for
3267:Spin
3040:Spin
2647:Spin
2603:Spin
2479:Spin
2406:Spin
2374:for
2324:for
2266:for
2089:even
2056:even
2015:Spin
1801:The
1581:and
1014:For
898:Spin
816:the
724:O(∞)
713:Loop
532:and
9388:→ A
9382:2⋅A
9110:).
9098:).
8932:of
8910:SO(
8897:).
8876:= Z
8872:= π
8255:≥ 3
7852:.
6684:on
6633:).
6336:= 2
6322:× Z
6296:× Z
5819:,
5811:,
5795:,
5783:,
5760:).
5685:of
5657:of
5577:of
5380:SO(
5355:of
5331:In
5249:--
5190:--
5119:--
5040:--
4992:in
4969:--
4881:of
4864:, −
4193:is
4184:= 2
4068:),
3910:Pin
3411:by
3093:of
2874:in
2382:on
2033:Pin
1812:Pin
1180:of
1073:Cl(
1035:SO(
1033:of
864:≠ 2
841:SO(
812:In
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
9759::
9649:.
9627:^
9599:^
9583:,
9579:,
9553:.
9549:.
9418:).
9380:,
9361:.
9353:+1
9337:+1
9161:.
9029:SO
8960::
8944:.
8938:,
8922:,
8904:,
8746:),
8726:),
8203:SO
8108:SO
8091:SO
8057:SO
8018:SO
7996:.
7956:SO
7864:SO
7827:SO
7794:SO
7686:SO
7639:SO
7583:SO
7544:SO
7350:SO
7311:SO
7215:SO
7141:SO
7102:SO
7022:SO
6999:SO
6935:SO
6916:SO
6875:SO
6847:SO
6779:SO
6660:SO
6640:.
6449:SO
6397:SO
6368:.
6355:×
6344:×
6338:,
6332:=
6281:))
6277:,
6265:,
6249:,
6231:,
5885:SO
5859::
5855:≥
5847:,
5767:,
5550:.
5544:,
5536:,
5467:×
5419:,
5407:,
5401:,
5395:+
5384:,
5365:,
5357:SO
5341:,
5127:⊕
5058:=
5001:↦
4984:=
4885:.
4870:.
4860:(−
4852:,
4690:Cl
4664:1.
4628:SO
4163:.
3751:Cl
3733:Cl
3704:1.
3653:Cl
3543:Cl
3387:Cl
3369:Cl
3216:.
2882:SO
2497:SO
2438:SO
2124:Cl
2111:Cl
2098:Cl
2085:Cl
2052:Cl
1990:1.
1844:Cl
1798:.
1662:Cl
1590:Cl
1555:Cl
1546:,
1485:Cl
1466:Cl
1453:Cl
1440:Cl
1421:Cl
1271:Cl
1081:.
1058:.
1041:.
996:,
985:.
961:SO
934:1.
916:SO
866:)
849:,
615:U(
591:E(
579:O(
37:→
9748:.
9723:.
9716:.
9694:.
9657:.
9563:.
9517:)
9515:n
9507:n
9499:)
9497:n
9489:n
9397:n
9391:n
9385:n
9374:n
9359:)
9357:n
9351:k
9349:2
9347:Z
9343:)
9341:n
9335:k
9333:2
9331:Z
9317:,
9312:2
9307:Z
9297:1
9294:+
9291:k
9288:2
9283:Z
9273:2
9270:+
9267:k
9264:4
9259:C
9247:n
9231:1
9228:+
9225:k
9222:2
9217:Z
9195:G
9187:2
9182:C
9170:G
9166:G
9155:n
9147:n
9143:n
9139:n
9135:n
9123:)
9121:n
9117:n
9096:n
9088:n
9084:3
9081:π
9056:)
9053:n
9050:(
9046:O
9039:)
9036:n
9033:(
9022:)
9019:n
9016:(
9005:)
9002:n
8999:(
8988:)
8985:n
8982:(
8942:)
8940:q
8936:p
8926:)
8924:q
8920:p
8916:q
8912:p
8906:q
8902:p
8895:n
8891:n
8887:2
8883:n
8878:1
8874:1
8870:0
8868:π
8866:(
8860:n
8854:n
8846:1
8842:0
8836:k
8811:.
8808:)
8804:R
8800:,
8797:n
8794:(
8789:o
8786:s
8759:n
8755:n
8751:n
8744:n
8740:n
8736:n
8724:n
8720:n
8716:n
8702:n
8697:n
8641:q
8638:,
8635:p
8627:,
8624:k
8621:4
8618:=
8615:n
8608:2
8603:Z
8593:2
8588:Z
8574:q
8571:,
8568:p
8560:,
8557:2
8554:+
8551:k
8548:4
8545:=
8542:n
8535:4
8530:Z
8516:q
8508:p
8501:2
8496:Z
8488:{
8483:=
8476:)
8473:)
8470:q
8467:,
8464:p
8461:(
8452:(
8446:Z
8432:k
8429:4
8426:=
8423:n
8416:2
8411:Z
8401:2
8396:Z
8387:2
8384:+
8381:k
8378:4
8375:=
8372:n
8365:4
8360:Z
8351:1
8348:+
8345:k
8342:2
8339:=
8336:n
8329:2
8324:Z
8316:{
8311:=
8304:)
8301:)
8297:C
8293:,
8290:n
8287:(
8278:(
8272:Z
8253:n
8220:)
8216:n
8213:,
8210:1
8207:(
8172:n
8168:H
8147:,
8142:n
8138:H
8125:)
8121:n
8118:,
8115:1
8112:(
8101:)
8098:n
8095:(
8070:)
8067:)
8064:n
8061:(
8053:(
8044:)
8035:)
8031:n
8028:,
8025:1
8022:(
8014:(
7982:2
7977:Z
7969:)
7966:)
7963:3
7960:(
7952:(
7947:1
7922:}
7919:1
7913:{
7909:/
7903:3
7899:S
7890:3
7885:P
7882:R
7874:)
7871:3
7868:(
7840:)
7837:)
7834:3
7831:(
7823:(
7818:1
7807:)
7804:)
7801:n
7798:(
7790:(
7785:1
7760:3
7754:n
7733:3
7727:n
7705:)
7702:)
7699:1
7693:n
7690:(
7682:(
7677:1
7652:)
7649:)
7646:n
7643:(
7635:(
7630:1
7605:,
7602:0
7596:)
7593:)
7590:n
7587:(
7579:(
7574:1
7563:)
7560:)
7557:1
7551:n
7548:(
7540:(
7535:1
7524:0
7504:3
7498:n
7478:n
7472:k
7452:0
7449:=
7446:)
7441:n
7437:S
7433:(
7428:k
7401:.
7398:)
7393:1
7387:n
7383:S
7379:(
7374:1
7363:)
7360:)
7357:n
7354:(
7346:(
7341:1
7330:)
7327:)
7324:1
7318:n
7315:(
7307:(
7302:1
7291:)
7286:1
7280:n
7276:S
7272:(
7267:2
7234:)
7231:)
7228:1
7222:n
7219:(
7211:(
7206:1
7200:k
7189:)
7184:1
7178:n
7174:S
7170:(
7165:k
7154:)
7151:)
7148:n
7145:(
7137:(
7132:k
7121:)
7118:)
7115:1
7109:n
7106:(
7098:(
7093:k
7054:.
7049:1
7043:n
7039:S
7032:)
7029:n
7026:(
7015:)
7012:1
7006:n
7003:(
6973:.
6968:1
6962:n
6958:S
6951:)
6948:1
6942:n
6939:(
6930:/
6926:)
6923:n
6920:(
6891:)
6888:1
6882:n
6879:(
6871:=
6868:)
6865:v
6862:(
6857:)
6854:n
6851:(
6817:1
6811:n
6807:S
6803:=
6800:)
6797:v
6794:(
6789:)
6786:n
6783:(
6751:)
6748:0
6745:,
6739:,
6736:0
6733:,
6730:1
6727:(
6724:=
6721:v
6699:n
6694:R
6672:)
6669:n
6666:(
6615:3
6610:P
6607:R
6599:)
6596:3
6593:(
6590:O
6587:S
6565:1
6561:S
6554:)
6551:2
6548:(
6545:O
6542:S
6522:)
6519:1
6516:(
6513:O
6510:S
6490:3
6484:n
6464:)
6461:)
6458:n
6455:(
6446:(
6441:1
6412:)
6409:)
6406:n
6403:(
6394:(
6389:1
6357:Z
6353:Z
6346:Z
6342:Z
6334:q
6330:p
6324:2
6320:Z
6316:Z
6308:q
6304:p
6298:2
6294:2
6287:2
6279:q
6275:p
6271:1
6267:q
6263:p
6259:1
6257:π
6251:q
6247:p
6239:2
6233:q
6229:p
6204:2
6198:q
6195:,
6192:p
6185:2
6180:Z
6171:2
6168:=
6165:q
6162:,
6159:2
6153:p
6147:Z
6139:)
6136:2
6133:,
6130:2
6127:(
6124:=
6121:)
6118:q
6115:,
6112:p
6109:(
6103:Z
6095:Z
6087:)
6084:1
6081:,
6078:2
6075:(
6065:)
6062:0
6059:,
6056:2
6053:(
6050:=
6047:)
6044:q
6041:,
6038:p
6035:(
6029:Z
6021:1
6018:,
6015:0
6012:=
6009:q
6006:,
6003:2
5997:p
5990:2
5985:Z
5976:)
5973:0
5970:,
5967:1
5964:(
5954:)
5951:1
5948:,
5945:1
5942:(
5939:=
5936:)
5933:q
5930:,
5927:p
5924:(
5919:0
5913:{
5908:=
5905:)
5902:)
5899:q
5896:,
5893:p
5890:(
5880:(
5875:1
5857:q
5853:p
5849:q
5845:p
5833:q
5829:p
5821:q
5817:p
5813:q
5809:p
5805:q
5801:p
5797:q
5793:p
5785:q
5781:p
5779:(
5777:0
5769:q
5765:p
5758:n
5754:n
5746:n
5739:Z
5735:R
5731:R
5727:G
5723:G
5721:(
5719:1
5703:g
5691:G
5687:G
5671:g
5659:G
5651:G
5634:,
5631:)
5624:G
5620:(
5614:Z
5608:)
5605:G
5602:(
5597:1
5579:G
5571:G
5567:G
5548:)
5546:p
5542:q
5538:q
5534:p
5525:)
5523:H
5515:)
5513:R
5499:)
5497:H
5489:)
5487:R
5473:)
5471:R
5465:)
5463:R
5455:)
5453:C
5445:)
5443:R
5435:)
5433:R
5421:q
5417:p
5415:(
5411:)
5409:q
5405:p
5397:q
5393:p
5388:)
5386:q
5382:p
5369:)
5367:q
5363:p
5361:(
5359:0
5345:)
5343:q
5339:p
5321:n
5311:n
5290:3
5286:A
5277:3
5273:D
5255:C
5231:2
5227:C
5218:2
5214:B
5196:H
5172:1
5168:A
5159:1
5155:A
5146:2
5142:D
5129:H
5125:H
5101:1
5097:A
5088:1
5084:C
5075:1
5071:B
5046:H
5022:1
5018:D
5006:z
5003:u
4999:z
4994:R
4990:z
4975:C
4953:R
4945:R
4941:n
4937:R
4933:n
4925:H
4921:C
4917:R
4868:)
4866:u
4862:a
4856:)
4854:u
4850:a
4848:(
4807:/
4802:)
4796:1
4792:S
4785:)
4782:V
4779:(
4769:(
4765:=
4762:)
4759:V
4756:(
4747:C
4728:C
4724:V
4709:C
4702:)
4699:V
4696:(
4658:)
4655:1
4652:(
4646:U
4640:)
4637:n
4634:(
4622:)
4619:n
4616:(
4607:C
4593:2
4588:Z
4580:1
4539:W
4514:W
4473:W
4451:m
4445:k
4439:1
4417:2
4411:/
4406:)
4400:k
4397:2
4393:e
4389:i
4381:1
4375:k
4372:2
4368:e
4363:(
4359:=
4354:k
4329:W
4301:W
4293:W
4290:=
4286:C
4279:V
4251:W
4229:W
4208:C
4201:V
4186:m
4182:n
4177:V
4151:a
4128:a
4118:V
4114:V
4110:V
4106:V
4099:a
4085:)
4082:a
4079:(
4056:V
4050:a
4040:a
4026:V
4020:v
3997:,
3988:a
3984:v
3981:a
3978:=
3975:v
3972:)
3969:a
3966:(
3940:)
3937:V
3934:(
3928:O
3922:)
3919:V
3916:(
3900:V
3884:t
3880:)
3876:a
3873:(
3844:a
3820:,
3817:V
3811:v
3807:,
3804:v
3798:=
3795:)
3792:v
3789:(
3763:)
3760:V
3757:(
3745:)
3742:V
3739:(
3701:=
3696:t
3692:a
3688:a
3665:)
3662:V
3659:(
3647:a
3637:V
3620:.
3615:t
3611:a
3605:t
3601:b
3597:=
3592:t
3588:)
3584:b
3581:a
3578:(
3555:)
3552:V
3549:(
3537:b
3534:,
3531:a
3508:.
3503:i
3499:e
3493:j
3489:e
3480:k
3476:e
3472:=
3467:t
3462:)
3456:k
3452:e
3443:j
3439:e
3433:i
3429:e
3424:(
3399:)
3396:V
3393:(
3381:)
3378:V
3375:(
3366::
3363:t
3350:V
3332:}
3327:i
3323:e
3319:{
3309:V
3305:V
3301:V
3279:)
3276:n
3273:(
3245:b
3237:,
3232:a
3204:)
3201:1
3198:(
3183:=
3180:b
3174:a
3148:e
3142:=
3139:)
3136:0
3133:(
3052:)
3049:n
3046:(
3020:e
3017:=
3014:)
3011:0
3008:(
2985:)
2982:)
2979:t
2976:(
2971:b
2963:(
2960:p
2954:)
2951:)
2948:t
2945:(
2940:a
2932:(
2929:p
2926:=
2923:)
2920:t
2917:(
2894:)
2891:n
2888:(
2842:b
2839:=
2836:)
2833:1
2830:(
2825:b
2817:,
2814:a
2811:=
2808:)
2805:1
2802:(
2797:a
2766:e
2760:=
2757:)
2754:0
2751:(
2746:b
2738:=
2735:)
2732:0
2729:(
2724:a
2697:b
2689:,
2684:a
2659:)
2656:n
2653:(
2641:b
2638:,
2635:a
2615:)
2612:n
2609:(
2577:e
2554:)
2551:}
2548:e
2545:{
2542:(
2537:1
2530:p
2509:)
2506:n
2503:(
2491:)
2488:n
2485:(
2476::
2473:p
2450:)
2447:n
2444:(
2418:)
2415:n
2412:(
2356:j
2352:e
2348:+
2343:i
2339:e
2335:=
2332:v
2312:v
2306:v
2283:,
2280:j
2274:i
2252:i
2248:e
2242:j
2238:e
2231:=
2226:j
2222:e
2216:i
2212:e
2198:V
2184:}
2179:i
2175:e
2171:{
2158:k
2154:V
2150:V
2128:4
2115:2
2102:0
2094:=
2061:,
2045:)
2042:V
2039:(
2030:=
2027:)
2024:V
2021:(
1987:=
1984:)
1979:i
1975:v
1971:(
1968:q
1948:V
1940:i
1936:v
1912:,
1907:k
1903:v
1894:2
1890:v
1884:1
1880:v
1856:)
1853:V
1850:(
1824:)
1821:V
1818:(
1782:)
1779:n
1776:(
1771:o
1768:s
1756:V
1748:n
1744:V
1727:,
1724:)
1721:n
1718:(
1713:n
1710:i
1707:p
1704:s
1699:=
1696:)
1693:V
1690:(
1685:n
1682:i
1679:p
1676:s
1671:=
1666:2
1636:n
1633:i
1630:p
1627:s
1602:V
1599:=
1594:1
1568:R
1564:=
1559:0
1534:V
1514:n
1489:n
1470:2
1457:1
1444:0
1436:=
1433:)
1430:V
1427:(
1394:V
1388:v
1368:)
1365:v
1362:(
1359:q
1336:,
1332:)
1328:)
1325:v
1322:(
1319:q
1313:v
1307:v
1303:(
1298:/
1294:V
1290:T
1286:=
1283:)
1280:V
1277:(
1255:V
1232:)
1229:V
1223:V
1220:(
1214:V
1207:R
1203:=
1200:V
1196:T
1182:V
1178:V
1176:T
1170:q
1163:V
1075:n
1063:n
1055:I
1039:)
1037:n
1023:n
1017:n
1003:n
1001:(
999:n
990:n
973:)
970:n
967:(
928:)
925:n
922:(
910:)
907:n
904:(
890:2
885:Z
877:1
862:n
853:)
851:R
847:n
843:n
822:n
801:e
794:t
787:v
683:8
681:E
675:7
673:E
667:6
665:E
659:4
657:F
651:2
649:G
643:)
641:n
631:)
629:n
619:)
617:n
607:)
605:n
595:)
593:n
583:)
581:n
571:)
569:n
559:)
557:n
499:)
486:Z
474:)
461:Z
437:)
424:Z
415:(
328:p
293:Q
285:n
282:D
272:n
269:A
261:n
258:S
250:n
247:Z
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
