138:. A similar construction works for any cubic alternative separable algebra over a field containing a primitive cube root of unity. An Okubo algebra is an algebra constructed in this way from the trace-zero elements of a degree-3
375:
1363:
733:
1269:
251:
1274:
Conversely, an algebra with a non-degenerate symmetric bilinear form satisfying this equation is either a para-Hurwitz algebra or an eight-dimensional
1288:
1539:
Susumu Okubo & J. Marshall Osborn (1981) "Algebras with nondegenerate associative symmetric bilinear forms permitting composition",
633:
1183:
1575:
1460:
1424:
172:
1570:
1510:
1417:
1505:
1452:
388:
and preserves the quadratic form. The overline notation stresses the fact that complex and quaternion
1540:
1368:
is either a
Hurwitz algebra, a para-Hurwitz algebra or an eight-dimensional pseudo-octonion algebra.
180:
1500:
385:
242:
1275:
132:
139:
76:
61:
1554:
1547:
1533:
1470:
29:
1478:
8:
1401:
593:
389:
69:
37:
1409:
1456:
1420:
65:
1451:. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge:
370:{\displaystyle \displaystyle {{\bar {a}}=-a+2(a,1)1,\,\,\,L(a)b=ab,\,\,\,R(a)b=ba.}}
1520:
1474:
1279:
486:
135:
128:
41:
17:
1551:
1544:
1529:
1466:
1446:
150:
1358:{\displaystyle \langle xy|xy\rangle =\langle x|x\rangle \langle y|y\rangle \ }
1564:
196:
599:
These properties are proved starting from polarized version of the identity
33:
1405:
164:
1518:
Okubo, Susumu (1978), "Pseudo-quaternion and pseudo-octonion algebras",
392:
are partial cases of it. These operators have the following properties:
1491:
The term "para-quaternions" is sometimes applied to unrelated algebras.
1448:
Introduction to octonion and other non-associative algebras in physics
728:{\displaystyle \displaystyle {2(a,b)(c,d)=(ac,bd)+(ad,bc).}}
1162:
is a composition algebra not generally unital, known as a
1264:{\displaystyle (x*y)*x=x*(y*x)=\langle x|x\rangle y\ .}
1056:
gives the other identity. Substituting the formula for
145:
1291:
1186:
637:
636:
255:
254:
1357:
1263:
727:
369:
241:has a multiplicative unity, define involution and
232:
1562:
1412:(1998) "Composition and Triality", chapter 8 in
1416:, pp 451–511, Colloquium Publications v 44,
1349:
1335:
1332:
1318:
1312:
1292:
1249:
1235:
396:The involution is an antiautomorphism, i.e.
79:-zero complex matrices, with the product of
195:has characteristic not equal to 2, then a
75:Okubo's example was the algebra of 3-by-3
337:
336:
335:
307:
306:
305:
1127:may be defined in a Hurwitz algebra as
1118:
72:, and do not have an identity element.
1563:
1440:
1438:
1436:
1434:
1432:
1396:
1394:
225:is associated with the quadratic form
1517:
1444:
1385:
146:Construction of Para-Hurwitz algebra
1429:
1391:
13:
1166:. In dimensions 4 and 8 these are
14:
1587:
1177:A para-Hurwitz algebra satisfies
186:
153:composition algebras are called
68:, but are not associative, not
1485:
1378:
1342:
1325:
1302:
1242:
1229:
1217:
1199:
1187:
717:
699:
693:
675:
669:
657:
654:
642:
347:
341:
317:
311:
296:
284:
263:
233:Involution in Hurwitz algebras
32:similar to the one studied by
1:
1418:American Mathematical Society
1371:
963:. By the polarized identity
243:right and left multiplication
7:
1506:Encyclopedia of Mathematics
131:form an 8-dimensional real
107:is the identity matrix and
10:
1592:
1453:Cambridge University Press
1026:. Applied to 1 this gives
1541:Communications in Algebra
489:with respect to the form
181:Euclidean Hurwitz algebra
1576:Non-associative algebras
1414:The Book of Involutions
1384:Susumu Okubo (
1276:pseudo-octonion algebra
62:Lie admissible algebras
30:non-associative algebra
26:pseudo-octonion algebra
1445:Okubo, Susumu (1995).
1359:
1265:
729:
371:
157:. If the ground field
140:central simple algebra
1360:
1266:
730:
372:
36:. Okubo algebras are
1571:Composition algebras
1289:
1184:
1164:para-Hurwitz algebra
1119:Para-Hurwitz algebra
634:
252:
127: = 1. The
70:alternative algebras
38:composition algebras
28:is an 8-dimensional
1550:and 9(20): 2015–73
1402:Alexander Merkurjev
594:alternative algebra
1410:Jean-Pierre Tignol
1355:
1261:
1123:Another operation
725:
724:
367:
366:
129:Hermitian elements
1400:Max-Albert Knus,
1354:
1257:
266:
173:positive-definite
66:power associative
42:flexible algebras
1583:
1543:9(12): 1233–61,
1536:
1528:(4): 1250–1278,
1521:Hadronic Journal
1514:
1492:
1489:
1483:
1482:
1442:
1427:
1398:
1389:
1382:
1364:
1362:
1361:
1356:
1352:
1345:
1328:
1305:
1280:flexible algebra
1270:
1268:
1267:
1262:
1255:
1245:
1161:
1150:
1149:
1144:
1126:
1114:
1095:
1091:
1074:
1063:
1062:
1055:
1054:
1047:
1043:
1031:
1025:
1011:
1000:
992:
962:
943:
940:
931:
903:
896:
893:
888:
881:
878:
872:
869:
864:
857:
854:
845:
831:
823:
813:
793:
783:
772:
762:
751:
744:
734:
732:
731:
726:
723:
626:
591:
587:
568:
548:
527:
521:
505:
492:
487:adjoint operator
484:
480:
470:
459:
449:
437:
434:
420:
412:
410:
407:
402:
383:
376:
374:
373:
368:
365:
268:
267:
259:
240:
228:
224:
223:
221:
220:
217:
214:
194:
178:
170:
163:is the field of
162:
155:Hurwitz algebras
136:division algebra
95: – Tr(
1591:
1590:
1586:
1585:
1584:
1582:
1581:
1580:
1561:
1560:
1501:"Okubo_algebra"
1499:
1496:
1495:
1490:
1486:
1463:
1443:
1430:
1399:
1392:
1383:
1379:
1374:
1341:
1324:
1301:
1290:
1287:
1286:
1278:. Similarly, a
1241:
1185:
1182:
1181:
1168:para-quaternion
1155:
1145:
1140:
1131:
1124:
1121:
1097:
1089:
1070:
1065:
1058:
1057:
1050:
1049:
1045:
1029:
1027:
1007:
1002:
990:
964:
941:
938:
927:
905:
894:
891:
884:
879:
876:
870:
867:
860:
855:
852:
841:
827:
825:
809:
795:
779:
774:
758:
753:
746:
739:
638:
635:
632:
631:
600:
589:
570:
551:
530:
517:
507:
495:
490:
482:
466:
461:
445:
440:
432:
418:
415:
408:
405:
398:
397:
381:
258:
257:
256:
253:
250:
249:
238:
235:
226:
218:
215:
212:
211:
209:
199:
192:
189:
176:
168:
158:
148:
133:non-associative
52:) = (
12:
11:
5:
1589:
1579:
1578:
1573:
1559:
1558:
1537:
1515:
1494:
1493:
1484:
1461:
1428:
1390:
1376:
1375:
1373:
1370:
1366:
1365:
1351:
1348:
1344:
1340:
1337:
1334:
1331:
1327:
1323:
1320:
1317:
1314:
1311:
1308:
1304:
1300:
1297:
1294:
1272:
1271:
1260:
1254:
1251:
1248:
1244:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1152:
1151:
1120:
1117:
736:
735:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
597:
596:
549:
528:
493:
438:
413:
378:
377:
364:
361:
358:
355:
352:
349:
346:
343:
340:
334:
331:
328:
325:
322:
319:
316:
313:
310:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
265:
262:
234:
231:
188:
187:Scalar product
185:
147:
144:
142:over a field.
123: = 3
9:
6:
4:
3:
2:
1588:
1577:
1574:
1572:
1569:
1568:
1566:
1556:
1553:
1549:
1546:
1542:
1538:
1535:
1531:
1527:
1523:
1522:
1516:
1512:
1508:
1507:
1502:
1498:
1497:
1488:
1480:
1476:
1472:
1468:
1464:
1462:0-521-47215-6
1458:
1454:
1450:
1449:
1441:
1439:
1437:
1435:
1433:
1426:
1425:0-8218-0904-0
1422:
1419:
1415:
1411:
1407:
1403:
1397:
1395:
1387:
1381:
1377:
1369:
1346:
1338:
1329:
1321:
1315:
1309:
1306:
1298:
1295:
1285:
1284:
1283:
1281:
1277:
1258:
1252:
1246:
1238:
1232:
1226:
1223:
1220:
1214:
1211:
1208:
1205:
1202:
1196:
1193:
1190:
1180:
1179:
1178:
1175:
1173:
1172:para-octonion
1169:
1165:
1159:
1148:
1143:
1138:
1134:
1130:
1129:
1128:
1116:
1112:
1108:
1104:
1100:
1093:
1086:
1082:
1078:
1073:
1068:
1061:
1053:
1041:
1037:
1033:
1023:
1019:
1015:
1010:
1005:
998:
994:
987:
983:
979:
975:
971:
967:
960:
956:
952:
948:
944:
935:
930:
925:
921:
917:
913:
909:
901:
897:
887:
882:
873:
863:
858:
849:
844:
839:
835:
830:
821:
817:
812:
807:
803:
799:
791:
787:
782:
777:
770:
766:
761:
756:
749:
742:
720:
714:
711:
708:
705:
702:
696:
690:
687:
684:
681:
678:
672:
666:
663:
660:
651:
648:
645:
639:
630:
629:
628:
624:
620:
616:
612:
608:
604:
595:
585:
581:
577:
573:
566:
562:
558:
554:
550:
546:
542:
538:
534:
529:
525:
520:
515:
511:
503:
499:
494:
488:
478:
474:
469:
464:
457:
453:
448:
443:
439:
436:
429:
425:
421:
414:
411:
401:
395:
394:
393:
391:
387:
362:
359:
356:
353:
350:
344:
338:
332:
329:
326:
323:
320:
314:
308:
302:
299:
293:
290:
287:
281:
278:
275:
272:
269:
260:
248:
247:
246:
245:operators by
244:
230:
207:
203:
198:
197:bilinear form
184:
182:
174:
166:
161:
156:
152:
143:
141:
137:
134:
130:
126:
122:
119: +
118:
114:
110:
106:
102:
98:
94:
91: +
90:
86:
82:
78:
73:
71:
67:
63:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
22:Okubo algebra
19:
1525:
1519:
1504:
1487:
1447:
1413:
1380:
1367:
1273:
1176:
1171:
1167:
1163:
1157:
1154:The algebra
1153:
1146:
1141:
1136:
1132:
1122:
1110:
1106:
1102:
1098:
1088:
1084:
1080:
1076:
1071:
1066:
1059:
1051:
1044:. Replacing
1039:
1035:
1028:
1021:
1017:
1013:
1008:
1003:
996:
989:
985:
981:
977:
973:
969:
965:
958:
954:
950:
946:
937:
933:
928:
923:
919:
915:
911:
907:
899:
890:
885:
875:
866:
861:
851:
847:
842:
837:
833:
828:
824:. Similarly
819:
815:
810:
805:
801:
797:
789:
785:
780:
775:
768:
764:
759:
754:
747:
740:
737:
622:
618:
614:
610:
606:
602:
598:
583:
579:
575:
571:
564:
560:
556:
552:
544:
540:
536:
532:
523:
518:
513:
509:
501:
497:
485:denotes the
476:
472:
467:
462:
455:
451:
446:
441:
431:
427:
423:
416:
404:
399:
379:
236:
205:
201:
190:
179:is called a
165:real numbers
159:
154:
149:
124:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
74:
57:
53:
49:
45:
34:Susumu Okubo
25:
21:
15:
1406:Markus Rost
1282:satisfying
953:), 1) = Re(
390:conjugation
1565:Categories
1479:0841.17001
1372:References
1174:algebras.
874:)) = (1, (
818:, 1) = Re(
588:, so that
386:involution
380:Evidently
1511:EMS Press
1350:⟩
1336:⟨
1333:⟩
1319:⟨
1313:⟩
1293:⟨
1250:⟩
1236:⟨
1224:∗
1215:∗
1203:∗
1194:∗
273:−
264:¯
237:Assuming
103:/3 where
87:given by
922:, 1) = (
904:. Hence
859:) = (1,
804:, 1) = (
794:. Hence
738:Setting
481:, where
115:satisfy
1555:0640611
1548:0618901
1534:0510100
1513:, 2001
1471:1356224
752:yields
539:) = Re(
522:)/2 = (
500:) = Re(
222:
210:
175:, then
18:algebra
1532:
1477:
1469:
1459:
1423:
1353:
1256:
1096:gives
592:is an
506:where
430:) 1 =
384:is an
382:
151:Unital
64:, and
988:) = (
980:) = (
945:) = (
932:) = (
889:) = (
846:) = (
836:) = (
814:) = (
800:) = (
609:) = (
491:( , )
77:trace
20:, an
1457:ISBN
1421:ISBN
1386:1978
1170:and
1160:, ∗)
1105:) =
1083:) =
1016:) =
1012:) L(
914:= ((
784:) =
773:and
763:) =
578:) =
559:) =
531:Re((
526:, 1)
471:) =
450:) =
208:) =
167:and
111:and
83:and
1475:Zbl
1064:in
1048:by
1001:so
993:a c
986:a d
982:a c
972:) (
959:b c
951:b c
924:a b
916:a b
908:a b
906:Re(
838:a b
829:a b
820:b a
816:b a
802:a b
798:a b
796:Re(
750:= 1
745:or
743:= 1
607:a b
603:a b
545:b c
533:a b
512:= (
508:Re
502:b a
498:a b
496:Re(
400:a b
191:If
171:is
93:bYX
89:aXY
60:),
24:or
16:In
1567::
1552:MR
1545:MR
1530:MR
1524:,
1509:,
1503:,
1473:.
1467:MR
1465:.
1455:.
1431:^
1408:,
1404:,
1393:^
1139:=
1135:∗
1115:.
1075:)
1034:=
995:,
984:,
976:,
961:))
936:,
926:,
898:,
883:)
850:,
840:,
832:,
808:,
792:)*
771:)*
627::
621:,
617:)(
613:,
605:,
569:,
547:))
535:)
516:+
479:)*
460:,
458:)*
422:=
417:a
403:=
229:.
204:,
183:.
125:ab
97:XY
54:AB
50:BA
40:,
1557:.
1526:1
1481:.
1388:)
1347:y
1343:|
1339:y
1330:x
1326:|
1322:x
1316:=
1310:y
1307:x
1303:|
1299:y
1296:x
1259:.
1253:y
1247:x
1243:|
1239:x
1233:=
1230:)
1227:x
1221:y
1218:(
1212:x
1209:=
1206:x
1200:)
1197:y
1191:x
1188:(
1158:A
1156:(
1147:y
1142:x
1137:y
1133:x
1125:∗
1113:)
1111:a
1109:(
1107:L
1103:a
1101:(
1099:L
1094:)
1092:a
1090:a
1087:(
1085:L
1081:a
1079:(
1077:L
1072:a
1069:(
1067:L
1060:a
1052:a
1046:a
1042:)
1040:a
1038:(
1036:N
1032:a
1030:a
1024:)
1022:a
1020:(
1018:N
1014:a
1009:a
1006:(
1004:L
999:)
997:d
991:a
978:d
974:c
970:a
968:(
966:N
957:(
955:a
949:(
947:a
942:b
939:c
934:a
929:c
920:c
918:)
912:c
910:)
902:)
900:c
895:a
892:b
886:c
880:a
877:b
871:c
868:a
865:(
862:b
856:c
853:a
848:b
843:c
834:c
826:(
822:)
811:b
806:a
790:c
788:(
786:R
781:c
778:(
776:R
769:a
767:(
765:L
760:a
757:(
755:L
748:d
741:b
721:.
718:)
715:c
712:b
709:,
706:d
703:a
700:(
697:+
694:)
691:d
688:b
685:,
682:c
679:a
676:(
673:=
670:)
667:d
664:,
661:c
658:(
655:)
652:b
649:,
646:a
643:(
640:2
625:)
623:b
619:b
615:a
611:a
601:(
590:A
586:)
584:a
582:(
580:R
576:a
574:(
572:R
567:)
565:a
563:(
561:L
557:a
555:(
553:L
543:(
541:a
537:c
524:x
519:x
514:x
510:x
504:)
483:*
477:a
475:(
473:R
468:a
465:(
463:R
456:a
454:(
452:L
447:a
444:(
442:L
435:a
433:a
428:a
426:(
424:N
419:a
409:a
406:b
363:.
360:a
357:b
354:=
351:b
348:)
345:a
342:(
339:R
333:,
330:b
327:a
324:=
321:b
318:)
315:a
312:(
309:L
303:,
300:1
297:)
294:1
291:,
288:a
285:(
282:2
279:+
276:a
270:=
261:a
239:A
227:N
219:2
216:/
213:1
206:b
202:a
200:(
193:K
177:A
169:N
160:K
121:b
117:a
113:b
109:a
105:I
101:I
99:)
85:Y
81:X
58:A
56:)
48:(
46:A
44:(
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