513:, must be matrix algebras over the real numbers or the quaternions. Thus a real or pseudoreal representation is a direct sum of irreducible real representations and irreducible quaternionic representations. It is real if no quaternionic representations occur in the decomposition.
321:
605:
440:
97:
482:
139:
533:
of quaternions themselves under left multiplication. By restricting this to the unit quaternions, we obtain a quaternionic representation of the
801:
761:
244:
621:
of Spin(5). An example of a non-unitary quaternionic representation would be the two dimensional irreducible representation of Spin(5,1).
663:
Among the compact real forms of the simple Lie groups, irreducible quaternionic representations only exist for the Lie groups of type
394:
554:
832:
733:
487:
A representation which is isomorphic to its complex conjugate, but which is not a real representation, is sometimes called a
378:) and in this case quaternionic representations are also known as symplectic representations. Such representations, amongst
745:
737:
379:
383:
363:
165:
510:
656:, 1). These representations have the same type of real or quaternionic structure as the spinors of Spin(
817:
347:
55:
35:
173:
43:
413:
70:
618:
390:
327:
185:
192:
451:
108:
797:
775:
757:
749:
177:
771:
767:
405:
62:
17:
790:
398:
397:. Here a real representation is taken to be a complex representation with an invariant
47:
27:
Representation of a group or algebra in terms of an algebra with quaternionic structure
826:
371:
219:
534:
526:
375:
529:. There is an obvious one-dimensional quaternionic vector space, namely the space
525:
in three dimensions. Each (proper) rotation is represented by a quaternion with
331:
31:
753:
499:
402:
181:
59:
779:
522:
748:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
649:
316:{\displaystyle \rho (gh)=\rho (g)\rho (h){\text{ for all }}g,h\in G.}
548:) also happens to be a unitary quaternionic representation because
498:
can be understood by viewing them as representations of the real
210:), the group of invertible quaternion-linear transformations of
505:. Such a representation will be a direct sum of central simple
521:
A common example involves the quaternionic representation of
184:). From this point of view, quaternionic representation of a
214:. In particular, a quaternionic matrix representation of
600:{\displaystyle \rho (g)^{\dagger }\rho (g)=\mathbf {1} }
648:
is an integer. In physics, one often encounters the
557:
454:
416:
247:
111:
73:
789:
599:
476:
434:
315:
133:
91:
624:More generally, the spin representations of Spin(
494:Real and pseudoreal representations of a group
8:
389:Quaternionic representations are similar to
358:admits an invariant complex symplectic form
592:
571:
556:
459:
453:
415:
290:
246:
116:
110:
72:
792:Linear Representations of Finite Groups
742:Representation theory. A first course
393:in that they are isomorphic to their
7:
25:
334:can be defined in a similar way.
144:Together with the imaginary unit
593:
395:complex conjugate representation
326:Quaternionic representations of
617:Another unitary example is the
350:and the quaternionic structure
338:Properties and related concepts
238:(e) is the identity matrix and
586:
580:
568:
561:
435:{\displaystyle j\colon V\to V}
426:
287:
281:
275:
269:
260:
251:
92:{\displaystyle j\colon V\to V}
83:
1:
746:Graduate Texts in Mathematics
354:is a unitary operator, then
788:Serre, Jean-Pierre (1977),
382:, can be picked out by the
380:irreducible representations
40:quaternionic representation
849:
754:10.1007/978-1-4612-0979-9
509:-algebras, which, by the
489:pseudoreal representation
477:{\displaystyle j^{2}=+1.}
384:Frobenius-Schur indicator
370:is a representation of a
364:symplectic representation
166:quaternionic vector space
134:{\displaystyle j^{2}=-1.}
18:Pseudoreal representation
628:) are quaternionic when
511:Artin-Wedderburn theorem
164:with the structure of a
818:Symplectic vector space
660: − 1).
544:: Spin(3) → GL(1,
366:. This always holds if
148:and the antilinear map
636:, 4 + 8
632:equals 3 + 8
601:
478:
436:
348:unitary representation
317:
135:
93:
56:quaternionic structure
833:Representation theory
640:, and 5 + 8
602:
479:
437:
318:
136:
94:
36:representation theory
555:
540:This representation
452:
414:
391:real representations
245:
226:(g) to each element
109:
71:
796:, Springer-Verlag,
619:spin representation
292: for all
644:dimensions, where
597:
474:
432:
313:
193:group homomorphism
131:
89:
54:with an invariant
803:978-0-387-90190-9
763:978-0-387-97495-8
362:, and hence is a
293:
16:(Redirected from
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806:
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783:
606:
604:
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445:which satisfies
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178:division algebra
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102:which satisfies
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839:
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823:
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787:
764:
734:Fulton, William
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406:equivariant map
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222:of quaternions
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69:
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63:equivariant map
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23:
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12:
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44:representation
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583:
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543:
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536:
532:
528:
524:
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500:group algebra
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372:compact group
369:
365:
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329:
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266:
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241:
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220:square matrix
217:
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163:
159:
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147:
128:
125:
122:
117:
113:
105:
104:
103:
86:
80:
77:
74:
67:
66:
65:
64:
61:
57:
53:
50:vector space
49:
45:
41:
37:
33:
19:
791:
741:
717:
711:
706:
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691:
686:
680:
675:
669:
664:
662:
657:
653:
645:
641:
637:
633:
629:
625:
623:
616:
614:in Spin(3).
611:
609:
545:
541:
539:
535:spinor group
530:
520:
506:
502:
495:
493:
488:
486:
444:
388:
376:finite group
367:
359:
355:
351:
343:
341:
332:Lie algebras
325:
235:
231:
227:
223:
215:
211:
207:
203:
199:
195:
188:
169:
161:
157:
153:
149:
145:
143:
101:
51:
39:
32:mathematical
29:
738:Harris, Joe
401:, i.e., an
328:associative
202:→ GL(
182:quaternions
58:, i.e., an
727:References
403:antilinear
234:such that
218:assigns a
172:becomes a
60:antilinear
780:246650103
578:ρ
573:†
559:ρ
537:Spin(3).
527:unit norm
523:rotations
427:→
421::
305:∈
279:ρ
267:ρ
249:ρ
176:over the
126:−
84:→
78::
34:field of
827:Category
812:See also
740:(1991).
652:of Spin(
610:for all
517:Examples
374:(e.g. a
772:1153249
650:spinors
206:,
168:(i.e.,
160:equips
48:complex
800:
778:
770:
760:
716:, and
542:ρ
360:ω
236:ρ
224:ρ
196:φ
174:module
346:is a
191:is a
186:group
46:on a
42:is a
798:ISBN
776:OCLC
758:ISBN
330:and
38:, a
750:doi
342:If
230:of
180:of
30:In
829::
774:.
768:MR
766:.
756:.
744:.
736:;
723:.
714:+2
705:,
696:,
694:+2
685:,
683:+1
674:,
672:+1
491:.
472:1.
386:.
198::
156:,
154:ij
129:1.
807:.
784:.
782:.
752::
721:7
718:E
712:k
710:4
707:D
702:k
698:C
692:k
690:4
687:B
681:k
679:4
676:B
670:k
668:4
665:A
658:d
654:d
646:k
642:k
638:k
634:k
630:d
626:d
612:g
594:1
590:=
587:)
584:g
581:(
569:)
565:g
562:(
546:H
531:H
507:R
503:R
496:G
469:+
466:=
461:2
457:j
430:V
424:V
418:j
368:V
356:V
352:j
344:V
311:.
308:G
302:h
299:,
296:g
288:)
285:h
282:(
276:)
273:g
270:(
264:=
261:)
258:h
255:g
252:(
232:G
228:g
216:g
212:V
208:H
204:V
200:G
189:G
170:V
162:V
158:j
150:k
146:i
123:=
118:2
114:j
87:V
81:V
75:j
52:V
20:)
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