2072:
of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic. There is a uniform description of finite-dimensional simple quadratic Jordan algebras, independent of characteristic: in characteristic not equal to 2 the theory of quadratic
1567:
is played by JBW algebras. These turn out to be JB algebras which, as Banach spaces, are the dual spaces of Banach spaces. Much of the structure theory of von
Neumann algebras can be carried over to JBW algebras. In particular the JBW factors—those with center reduced to
1538:
1543:
These axioms guarantee that the Jordan algebra is formally real, so that, if a sum of squares of terms is zero, those terms must be zero. The complexifications of JB algebras are called Jordan C*-algebras or JB*-algebras. They have been used extensively in
1035:
squares can only vanish if each one vanishes individually. In 1932, Jordan attempted to axiomatize quantum theory by saying that the algebra of observables of any quantum system should be a formally real algebra that is commutative
375:. It was soon shown that the algebras were not useful in this context, however they have since found many applications in mathematics. The algebras were originally called "r-number systems", but were renamed "Jordan algebras" by
553:
is special. Related to this, Macdonald's theorem states that any polynomial in three variables, having degree one in one of the variables, and which vanishes in every special Jordan algebra, vanishes in every Jordan algebra.
1337:
classified infinite-dimensional simple (and prime non-degenerate) Jordan algebras. They are either of
Hermitian or Clifford type. In particular, the only exceptional simple Jordan algebras are finite-dimensional
1953:
1385:
1067:
ones, which are not themselves direct sums in a nontrivial way. In finite dimensions, the simple formally real Jordan algebras come in four infinite families, together with one exceptional case:
1592:
A Jordan ring is a generalization of Jordan algebras, requiring only that the Jordan ring be over a general ring rather than a field. Alternatively one can define a Jordan ring as a commutative
471:
299:
1153:
661:
1830:
1676:
145:
1556:
to infinite dimensions. Not all JB algebras can be realized as Jordan algebras of self-adjoint operators on a
Hilbert space, exactly as in finite dimensions. The exceptional
221:
1790:
1636:
2280:
811:
749:
2014:
540:
1987:
1757:
1730:
1703:
354:
71:
327:
2097:
1580:. Of these the spin factors can be constructed very simply from real Hilbert spaces. All other JWB factors are either the self-adjoint part of a
2894:
2846:
2748:
2722:
2684:
838:
this is the only simple exceptional Jordan algebra up to isomorphism, it is often referred to as "the" exceptional Jordan algebra. Over the
1838:
3024:
2979:
2778:
2481:
1533:{\displaystyle \displaystyle {\|a\circ b\|\leq \|a\|\cdot \|b\|,\,\,\,\|a^{2}\|=\|a\|^{2},\,\,\,\|a^{2}\|\leq \|a^{2}+b^{2}\|.}}
309:. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element
2938:
2920:
2456:
2436:
2971:
2815:
2473:
851:
2810:
2064:
Quadratic Jordan algebras are a generalization of (linear) Jordan algebras introduced by Kevin McCrimmon (
420:
2047:
226:
1576:, all JWB factors can be realised as Jordan algebras of self-adjoint operators on a Hilbert space closed in the
2069:
2059:
1553:
1372:
1178:
The Jordan algebra of 3×3 self-adjoint octonionic matrices, as above (an exceptional Jordan algebra called the
1113:
1016:
571:
403:
1261:
is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces
1361:
1351:
372:
2039:
1577:
28:
2328:
2275:
1376:
376:
35:
2424:
1795:
1641:
604:
82:
2376:
1958:
Jordan simple superalgebras over an algebraically closed field of characteristic 0 were classified by
2598:
2087:
1308:
187:
31:
2805:
2498:
Jordan, Pascual (1933), "Über
Verallgemeinerungsmöglichkeiten des Formalismus der Quantenmechanik",
1765:
1611:
2955:
2082:
1593:
1581:
1564:
1199:
392:
174:
2963:
2616:
2536:
2406:
2353:
2307:
1195:
767:
2933:, CBMS Regional Conference Series in Mathematics, vol. 67, American Mathematical Society,
2550:(1977), "Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras",
708:
2975:
2934:
2916:
2890:
2842:
2774:
2744:
2718:
2680:
2642:
2567:
2477:
2452:
2432:
2345:
2299:
2050:
not equal to 2 the theory of J-structures is essentially the same as that of Jordan algebras.
2043:
1357:
181:
1584:
or its fixed point subalgebra under a period 2 *-antiautomorphism of the von
Neumann factor.
2985:
2882:
2860:
2834:
2792:
2754:
2698:
2672:
2650:
2632:
2606:
2559:
2528:
2512:
2388:
2337:
2289:
2150:
1992:
1545:
828:
546:
495:
20:
2904:
2856:
2788:
2694:
2628:
2579:
2491:
2402:
2365:
2319:
2138:
1965:
1735:
1708:
1681:
332:
44:
3008:
2989:
2900:
2878:
2864:
2852:
2830:
2796:
2784:
2758:
2702:
2690:
2668:
2654:
2624:
2586:
2575:
2487:
2465:
2398:
2361:
2315:
835:
2472:, American Mathematical Society Colloquium Publications, vol. 39, Providence, R.I.:
2602:
1572:—are completely understood in terms of von Neumann algebras. Apart from the exceptional
2822:
2770:
2737:
2102:
1573:
1557:
1339:
1194:
complex matrices as algebras of observables. However, the spin factors play a role in
1179:
821:
695:
550:
360:
312:
2637:
2294:
3018:
2516:
2416:
1334:
2970:, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI:
2372:
1605:
1012:
839:
699:
688:
2729:
2706:
2410:
2393:
1052:
are unambiguously defined). He proved that any such algebra is a Jordan algebra.
2872:
2662:
1063:. Every formally real Jordan algebra can be written as a direct sum of so-called
2959:
2092:
2031:
2025:
1549:
895:
691:
485:
74:
1044:) and power-associative (the associative law holds for products involving only
816:
is a 27 dimensional, exceptional Jordan algebra (it is exceptional because the
2886:
2838:
2563:
2547:
2519:(1934), "On an algebraic generalization of the quantum mechanical formalism",
1368:
368:
16:
Not-necessarily-associative commutative algebra satisfying (𝑥𝑦)𝑥²=𝑥(𝑦𝑥²)
2571:
2349:
2303:
1302:
for totally real Jordan algebras. It was later studied in full generality by
1059:
classified the finite-dimensional formally real Jordan algebras, also called
996:
defines a derivation. In many important examples, the structure algebra of H(
223:
is independent of how we parenthesize this expression. They also imply that
1186:
Of these possibilities, so far it appears that nature makes use only of the
825:
2646:
2611:
842:
there are three isomorphism classes of simple exceptional Jordan algebras.
663:
Thus the set of all elements fixed by the involution (sometimes called the
2765:
Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) .
817:
758:
1962:. They include several families and some exceptional algebras, notably
1027:
A (possibly nonassociative) algebra over the real numbers is said to be
3002:
2540:
2446:
2357:
2311:
1298:) of the three eigenspaces. This decomposition was first considered by
2155:
1832:
becomes a Jordan superalgebra with respect to the graded Jordan brace
1161:
where the right-hand side is defined using the usual inner product on
410:
using the with same underlying addition and a new multiplication, the
2620:
2532:
2341:
2326:
Albert, A. Adrian (1947), "A structure theory for Jordan algebras",
2931:
Jordan algebras in analysis, operator theory, and quantum mechanics
2676:
1948:{\displaystyle \{x_{i},y_{j}\}=x_{i}y_{j}+(-1)^{ij}y_{j}x_{i}\ .}
968:
A simple example is provided by the
Hermitian Jordan algebras H(
173:, particularly to avoid confusion with the product of a related
2451:, Monographs and Studies in Mathematics, vol. 21, Pitman,
2127:, pp. 35–36, specifically remark before (56) and theorem 8
2042:
and axioms taking the Jordan inversion as basic operation and
383:), who began the systematic study of general Jordan algebras.
2431:, Oxford Mathematical Monographs, Oxford University Press,
2073:
Jordan algebras reduces to that of linear Jordan algebras.
1198:, and all the formally real Jordan algebras are related to
492:, whose product (Lie bracket) is defined by the commutator
2877:, Springer Monographs in Mathematics, Berlin, New York:
2278:(1946), "On Jordan algebras of linear transformations",
1608:
were introduced by Kac, Kantor and
Kaplansky; these are
367:) in an effort to formalize the notion of an algebra of
1371:
are JB algebras, which in finite dimensions are called
1299:
1056:
820:
are not associative). This was the first example of an
476:
These Jordan algebras and their subalgebras are called
395:
is a Jordan algebra if and only if it is commutative.
607:
549:–Cohn theorem states that any Jordan algebra with two
1995:
1968:
1841:
1798:
1768:
1738:
1711:
1684:
1644:
1614:
1389:
1388:
1116:
770:
711:
498:
423:
335:
315:
229:
190:
85:
47:
2915:, North-Holland Mathematics Studies, vol. 104,
2871:
Springer, Tonny A.; Veldkamp, Ferdinand D. (2000) ,
1011:
Derivation and structure algebras also form part of
2230:
2736:
2377:"The Octonions, 3: Projective Octonionic Geometry"
2008:
1981:
1947:
1824:
1784:
1751:
1724:
1697:
1670:
1630:
1532:
1147:
805:
757:2. The set of 3×3 self-adjoint matrices over the
743:
655:
534:
465:
348:
321:
293:
215:
139:
65:
2913:Symmetric Banach manifolds and Jordan C∗-algebras
2874:Octonions, Jordan algebras and exceptional groups
2281:Transactions of the American Mathematical Society
2470:Structure and representations of Jordan algebras
2196:
2500:Nachr. Akad. Wiss. Göttingen. Math. Phys. Kl. I
2120:
2118:
1055:Not every Jordan algebra is formally real, but
1375:. The norm on the real Jordan algebra must be
2381:Bulletin of the American Mathematical Society
2245:
2038:to develop a theory of Jordan algebras using
1101:self-adjoint quaternionic matrices. as above.
949:) can be made into a Lie algebra, called the
8:
2589:(1966), "A general theory of Jordan rings",
2192:
2190:
1868:
1842:
1522:
1496:
1490:
1477:
1462:
1455:
1449:
1436:
1427:
1421:
1415:
1409:
1403:
1391:
1142:
1130:
824:. Its automorphism group is the exceptional
466:{\displaystyle x\circ y={\frac {xy+yx}{2}}.}
1031:if it satisfies the property that a sum of
294:{\displaystyle x^{m}(x^{n}y)=x^{n}(x^{m}y)}
2739:An introduction to nonassociative algebras
2139:"Nazis, émigrés, and abstract mathematics"
937:) is a derivation. Thus the direct sum of
329:, the operations of multiplying by powers
180:The axioms imply that a Jordan algebra is
2715:Algebraic Structures of Symmetric Domains
2636:
2610:
2392:
2293:
2258:
2208:
2181:
2169:
2154:
2065:
2000:
1994:
1973:
1967:
1933:
1923:
1910:
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1878:
1862:
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1816:
1803:
1797:
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1710:
1689:
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1503:
1484:
1476:
1475:
1474:
1465:
1443:
1435:
1434:
1433:
1390:
1387:
1148:{\displaystyle x^{2}=\langle x,x\rangle }
1121:
1115:
792:
769:
733:
710:
606:
497:
436:
422:
340:
334:
314:
279:
266:
247:
234:
228:
195:
189:
84:
46:
2124:
2035:
1732:has a "Lie-like" product with values in
1090:self-adjoint complex matrices, as above.
484:. This construction is analogous to the
2240:
2235:
2114:
1300:Jordan, von Neumann & Wigner (1934)
1104:The Jordan algebra freely generated by
1057:Jordan, von Neumann & Wigner (1934)
905:). The Jordan identity implies that if
406:2), one can construct a Jordan algebra
2445:Hanche-Olsen, H.; Størmer, E. (1984),
1303:
1230:is the operation of multiplication by
380:
364:
3009:Jordan-Banach and Jordan-Lie algebras
2068:). The fundamental identities of the
1257:are 0, 1/2, 1. If the Jordan algebra
1214:is an idempotent in a Jordan algebra
1079:self-adjoint real matrices, as above.
7:
2827:Jordan algebras and algebraic groups
1329:Infinite-dimensional Jordan algebras
570:) is an associative algebra with an
165:in a Jordan algebra is also denoted
1959:
1596:that respects the Jordan identity.
1249: − 1) = 0
359:Jordan algebras were introduced by
2667:, Universitext, Berlin, New York:
656:{\textstyle \sigma (xy+yx)=xy+yx.}
14:
2767:Rings that are nearly associative
2295:10.1090/S0002-9947-1946-0016759-3
1825:{\displaystyle A_{0}\oplus A_{1}}
1671:{\displaystyle J_{0}\oplus J_{1}}
1324:Special kinds and generalizations
846:Derivations and structure algebra
140:{\displaystyle (xy)(xx)=x(y(xx))}
2098:Kantor–Koecher–Tits construction
1316:relative to the idempotent
1048:, so that powers of any element
917:, then the endomorphism sending
38:satisfies the following axioms:
2231:Hanche-Olsen & Størmer 1984
1563:The Jordan algebra analogue of
754:form a special Jordan algebra.
671:, which is sometimes denoted H(
667:elements) form a subalgebra of
216:{\displaystyle x^{n}=x\cdots x}
2743:, Courier Dover Publications,
2717:, Princeton University Press,
1907:
1897:
1785:{\displaystyle \mathbb {Z} /2}
1631:{\displaystyle \mathbb {Z} /2}
1552:Jordan algebraic treatment of
1165:. This is sometimes called a
789:
771:
730:
712:
629:
611:
511:
499:
398:Given any associative algebra
288:
272:
256:
240:
134:
131:
122:
116:
107:
98:
95:
86:
1:
2972:American Mathematical Society
2591:Proc. Natl. Acad. Sci. U.S.A.
2474:American Mathematical Society
2394:10.1090/S0273-0979-01-00934-X
1023:Formally real Jordan algebras
702:matrices with multiplication
2735:Schafer, Richard D. (1996),
2197:Springer & Veldkamp 2000
1792:-graded associative algebra
976:). In this case any element
761:, again with multiplication
157:The product of two elements
2829:, Classics in Mathematics,
2811:Encyclopedia of Mathematics
2429:Analysis on symmetric cones
1560:is the common obstruction.
1360:has been extended to cover
1342:, which have dimension 27.
1253:so the only eigenvalues of
482:exceptional Jordan algebras
3041:
2664:A taste of Jordan algebras
2057:
2023:
1349:
894:). The derivations form a
806:{\displaystyle (xy+yx)/2,}
301:for all positive integers
2887:10.1007/978-3-662-12622-6
2839:10.1007/978-3-642-61970-0
2661:McCrimmon, Kevin (2004),
2564:10.1080/00927877708822224
2552:Communications in Algebra
2383:. Bull. Amer. Math. Soc.
2246:Faraut & Koranyi 1994
2137:Dahn, Ryan (2023-01-01).
2054:Quadratic Jordan algebras
2046:as a basic relation. In
1554:bounded symmetric domains
1373:Euclidean Jordan algebras
1061:Euclidean Jordan algebras
744:{\displaystyle (xy+yx)/2}
558:Hermitian Jordan algebras
377:Abraham Adrian Albert
3025:Non-associative algebras
2448:Jordan operator algebras
2070:quadratic representation
2060:Quadratic Jordan algebra
1705:is a Jordan algebra and
1379:and satisfy the axioms:
1362:Jordan operator algebras
1346:Jordan operator algebras
1017:Freudenthal magic square
2968:The book of involutions
2804:Slin'ko, A.M. (2001) ,
2040:linear algebraic groups
1352:Jordan operator algebra
1169:or a Jordan algebra of
480:, while all others are
478:special Jordan algebras
387:Special Jordan algebras
373:quantum electrodynamics
2713:Ichiro Satake (1980),
2612:10.1073/pnas.56.4.1072
2010:
2009:{\displaystyle K_{10}}
1983:
1949:
1826:
1786:
1753:
1726:
1699:
1672:
1632:
1578:weak operator topology
1534:
1245: − 1)(
1149:
1093:The Jordan algebra of
1082:The Jordan algebra of
1071:The Jordan algebra of
1015:' construction of the
807:
745:
657:
536:
535:{\displaystyle =xy-yx}
467:
350:
323:
295:
217:
141:
67:
29:nonassociative algebra
2521:Annals of Mathematics
2329:Annals of Mathematics
2011:
1984:
1982:{\displaystyle K_{3}}
1950:
1827:
1787:
1754:
1752:{\displaystyle J_{0}}
1727:
1725:{\displaystyle J_{1}}
1700:
1698:{\displaystyle J_{0}}
1673:
1633:
1535:
1150:
808:
746:
658:
537:
468:
391:Notice first that an
351:
349:{\displaystyle x^{n}}
324:
296:
218:
142:
68:
66:{\displaystyle xy=yx}
2956:Merkurjev, Alexander
2929:Upmeier, H. (1987),
2911:Upmeier, H. (1985),
2088:Jordan triple system
1993:
1966:
1839:
1796:
1766:
1736:
1709:
1682:
1642:
1612:
1600:Jordan superalgebras
1565:von Neumann algebras
1386:
1367:The counterparts of
1309:Peirce decomposition
1206:Peirce decomposition
1114:
854:of a Jordan algebra
768:
709:
605:
496:
421:
333:
313:
227:
188:
83:
45:
2964:Tignol, Jean-Pierre
2603:1966PNAS...56.1072M
2417:Online HTML version
2199:, §5.8, p. 153
2083:Freudenthal algebra
1594:nonassociative ring
1200:projective geometry
1108:with the relations
858:is an endomorphism
393:associative algebra
175:associative algebra
2954:Knus, Max-Albert;
2823:Springer, Tonny A.
2034:was introduced by
2006:
1979:
1945:
1822:
1782:
1749:
1722:
1695:
1668:
1628:
1582:von Neumann factor
1530:
1529:
1196:special relativity
1145:
803:
741:
653:
532:
463:
361:Pascual Jordan
346:
319:
291:
213:
137:
63:
2896:978-3-540-66337-9
2848:978-3-540-63632-8
2750:978-0-486-68813-8
2724:978-0-691-08271-4
2686:978-0-387-95447-9
2558:(13): 1375–1400,
2332:, Second Series,
2276:Albert, A. Adrian
2156:10.1063/PT.3.5158
1941:
1638:-graded algebras
1358:operator algebras
951:structure algebra
834:. Since over the
458:
322:{\displaystyle x}
182:power-associative
3032:
2992:
2943:
2925:
2907:
2867:
2818:
2806:"Jordan algebra"
2800:
2761:
2742:
2727:
2709:
2657:
2640:
2614:
2597:(4): 1072–1079,
2587:McCrimmon, Kevin
2582:
2543:
2507:
2494:
2466:Jacobson, Nathan
2461:
2441:
2414:
2396:
2368:
2322:
2297:
2262:
2256:
2250:
2225:
2219:
2206:
2200:
2194:
2185:
2179:
2173:
2167:
2161:
2160:
2158:
2134:
2128:
2122:
2015:
2013:
2012:
2007:
2005:
2004:
1988:
1986:
1985:
1980:
1978:
1977:
1954:
1952:
1951:
1946:
1939:
1938:
1937:
1928:
1927:
1918:
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1669:
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1653:
1637:
1635:
1634:
1629:
1624:
1619:
1546:complex geometry
1539:
1537:
1536:
1531:
1528:
1521:
1520:
1508:
1507:
1489:
1488:
1470:
1469:
1448:
1447:
1154:
1152:
1151:
1146:
1126:
1125:
913:are elements of
812:
810:
809:
804:
796:
750:
748:
747:
742:
737:
662:
660:
659:
654:
601:it follows that
541:
539:
538:
533:
472:
470:
469:
464:
459:
454:
437:
355:
353:
352:
347:
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214:
200:
199:
152:
151:
146:
144:
143:
138:
72:
70:
69:
64:
21:abstract algebra
3040:
3039:
3035:
3034:
3033:
3031:
3030:
3029:
3015:
3014:
2999:
2982:
2953:
2950:
2948:Further reading
2941:
2928:
2923:
2910:
2897:
2879:Springer-Verlag
2870:
2849:
2831:Springer-Verlag
2821:
2803:
2781:
2764:
2751:
2734:
2725:
2712:
2687:
2669:Springer-Verlag
2660:
2585:
2546:
2533:10.2307/1968117
2513:von Neumann, J.
2510:
2497:
2484:
2464:
2459:
2444:
2439:
2422:
2371:
2342:10.2307/1969128
2325:
2274:
2271:
2266:
2265:
2261:, pp. 9–10
2257:
2253:
2226:
2222:
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2203:
2195:
2188:
2180:
2176:
2168:
2164:
2136:
2135:
2131:
2123:
2116:
2111:
2079:
2062:
2056:
2036:Springer (1998)
2030:The concept of
2028:
2022:
1996:
1991:
1990:
1969:
1964:
1963:
1929:
1919:
1906:
1884:
1874:
1858:
1845:
1837:
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1812:
1799:
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1739:
1734:
1733:
1712:
1707:
1706:
1685:
1680:
1679:
1658:
1645:
1640:
1639:
1610:
1609:
1602:
1590:
1512:
1499:
1480:
1461:
1439:
1384:
1383:
1354:
1348:
1340:Albert algebras
1331:
1326:
1306:and called the
1293:
1282:
1271:
1208:
1117:
1112:
1111:
1025:
848:
836:complex numbers
832:
766:
765:
707:
706:
685:
603:
602:
560:
494:
493:
438:
419:
418:
389:
336:
331:
330:
311:
310:
275:
262:
243:
230:
225:
224:
191:
186:
185:
184:, meaning that
150:Jordan identity
149:
148:
81:
80:
43:
42:
17:
12:
11:
5:
3038:
3036:
3028:
3027:
3017:
3016:
3013:
3012:
3006:
3003:Jordan algebra
2998:
2997:External links
2995:
2994:
2993:
2980:
2949:
2946:
2945:
2944:
2939:
2926:
2921:
2908:
2895:
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2771:Academic Press
2762:
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2710:
2685:
2677:10.1007/b97489
2658:
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2508:
2495:
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2420:
2387:(2): 145–205.
2369:
2336:(3): 546–567,
2323:
2288:(3): 524–555,
2270:
2267:
2264:
2263:
2259:McCrimmon 2004
2251:
2249:
2248:
2243:
2238:
2233:
2220:
2211:, pp. 99
2209:McCrimmon 2004
2201:
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2182:McCrimmon 2004
2174:
2170:McCrimmon 2004
2162:
2129:
2113:
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2110:
2107:
2106:
2105:
2103:Scorza variety
2100:
2095:
2090:
2085:
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2058:Main article:
2055:
2052:
2048:characteristic
2044:Hua's identity
2024:Main article:
2021:
2018:
2003:
1999:
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1574:Albert algebra
1558:Albert algebra
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1356:The theory of
1350:Main article:
1347:
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1327:
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1322:
1291:
1287:) ⊕
1280:
1276:) ⊕
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1180:Albert algebra
1175:
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822:Albert algebra
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488:associated to
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404:characteristic
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36:multiplication
25:Jordan algebra
15:
13:
10:
9:
6:
4:
3:
2:
3037:
3026:
3023:
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3020:
3011:at PlanetMath
3010:
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3005:at PlanetMath
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2548:Kac, Victor G
2545:
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1311:
1310:
1305:
1304:Albert (1947)
1301:
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1264:
1260:
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1233:
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845:
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841:
837:
833:
827:
823:
819:
800:
797:
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774:
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738:
734:
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623:
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569:
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460:
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451:
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430:
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417:
416:
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413:
409:
405:
401:
396:
394:
386:
384:
382:
378:
374:
370:
366:
362:
357:
356:all commute.
341:
337:
316:
308:
304:
285:
280:
276:
267:
263:
259:
253:
248:
244:
235:
231:
210:
207:
204:
201:
196:
192:
183:
178:
176:
172:
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164:
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128:
125:
119:
113:
110:
104:
101:
92:
89:
79:
76:
60:
57:
54:
51:
48:
41:
40:
39:
37:
33:
30:
26:
22:
2967:
2960:Rost, Markus
2930:
2912:
2873:
2826:
2809:
2766:
2738:
2714:
2663:
2594:
2590:
2555:
2551:
2527:(1): 29–64,
2524:
2520:
2511:Jordan, P.;
2503:
2499:
2469:
2447:
2428:
2423:Faraut, J.;
2384:
2380:
2333:
2327:
2285:
2279:
2254:
2241:Upmeier 1987
2236:Upmeier 1985
2223:
2216:
2212:
2204:
2184:, p. 99
2177:
2165:
2149:(1): 44–50.
2146:
2142:
2132:
2063:
2029:
2020:J-structures
1957:
1761:
1603:
1591:
1588:Jordan rings
1569:
1562:
1542:
1366:
1355:
1332:
1317:
1313:
1307:
1295:
1288:
1284:
1277:
1273:
1266:
1262:
1258:
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1252:
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1242:
1238:
1231:
1227:
1223:
1219:
1215:
1211:
1209:
1191:
1187:
1185:
1170:
1166:
1162:
1105:
1098:
1094:
1087:
1083:
1076:
1072:
1064:
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1054:
1049:
1045:
1041:
1037:
1032:
1028:
1026:
1010:
1005:
1001:
997:
993:
989:
985:
981:
977:
973:
969:
967:
962:
958:
954:
950:
946:
942:
938:
934:
930:
926:
922:
918:
914:
910:
906:
902:
898:
891:
887:
883:
879:
875:
871:
867:
863:
859:
855:
849:
840:real numbers
815:
756:
753:
700:quaternionic
689:self-adjoint
686:
676:
672:
668:
664:
598:
594:
590:
586:
582:
578:
574:
567:
563:
561:
544:
489:
481:
477:
475:
414:defined by:
411:
407:
399:
397:
390:
358:
306:
302:
179:
170:
166:
162:
158:
156:
32:over a field
24:
18:
2425:Koranyi, A.
2093:Jordan pair
2032:J-structure
2026:J-structure
1369:C*-algebras
1167:spin factor
896:Lie algebra
486:Lie algebra
369:observables
75:commutative
2990:0955.16001
2940:082180717X
2922:0444876510
2865:1024.17018
2797:0487.17001
2759:0145.25601
2703:1044.17001
2655:0139.25502
2517:Wigner, E.
2458:0273086197
2438:0198534779
2269:References
1960:Kac (1977)
1548:to extend
866:such that
852:derivation
577:, then if
572:involution
551:generators
2825:(1998) ,
2816:EMS Press
2572:0092-7872
2506:: 209–217
2468:(2008) ,
2350:0003-486X
2304:0002-9947
1901:−
1810:⊕
1656:⊕
1550:Koecher's
1523:‖
1497:‖
1494:≤
1491:‖
1478:‖
1463:‖
1456:‖
1450:‖
1437:‖
1428:‖
1422:‖
1419:⋅
1416:‖
1410:‖
1407:≤
1404:‖
1398:∘
1392:‖
1333:In 1979,
1143:⟩
1131:⟨
992:)=−
826:Lie group
818:octonions
759:octonions
665:hermitian
609:σ
524:−
428:∘
208:⋯
3019:Category
2966:(1998),
2647:16591377
2427:(1994),
2375:(2002).
2077:See also
1377:complete
929:)−
683:Examples
547:Shirshov
402:(not of
2905:1763974
2857:1490836
2789:0518614
2695:2014924
2629:0202783
2599:Bibcode
2580:0498755
2541:1968117
2492:0251099
2403:1886087
2366:0021546
2358:1969128
2320:0016759
2312:1990270
1604:Jordan
1234:, then
696:complex
379: (
363: (
2988:
2978:
2937:
2919:
2903:
2893:
2863:
2855:
2845:
2795:
2787:
2777:
2757:
2747:
2730:Review
2721:
2707:Errata
2701:
2693:
2683:
2653:
2645:
2638:220000
2635:
2627:
2619:
2578:
2570:
2539:
2490:
2480:
2455:
2435:
2411:586512
2409:
2401:
2364:
2356:
2348:
2318:
2310:
2302:
2217:et seq
2215:, 235
2213:et seq
1940:
1678:where
1226:) and
1065:simple
1002:σ
986:σ
974:σ
677:σ
591:σ
579:σ
575:σ
568:σ
34:whose
2621:57792
2617:JSTOR
2537:JSTOR
2407:S2CID
2354:JSTOR
2308:JSTOR
2227:See:
2109:Notes
1004:) is
984:with
698:, or
27:is a
2976:ISBN
2935:ISBN
2917:ISBN
2891:ISBN
2843:ISBN
2775:ISBN
2745:ISBN
2719:ISBN
2681:ISBN
2643:PMID
2568:ISSN
2478:ISBN
2453:ISBN
2433:ISBN
2346:ISSN
2300:ISSN
2066:1966
1989:and
1762:Any
1013:Tits
941:and
909:and
874:) =
692:real
597:) =
589:and
585:) =
562:If (
545:The
381:1946
365:1933
305:and
161:and
77:law)
23:, a
2986:Zbl
2883:doi
2861:Zbl
2835:doi
2793:Zbl
2755:Zbl
2699:Zbl
2673:doi
2651:Zbl
2633:PMC
2607:doi
2560:doi
2529:doi
2389:doi
2338:doi
2290:doi
2151:doi
1312:of
1281:1/2
1210:If
980:of
965:).
959:str
953:of
943:der
921:to
899:der
862:of
679:).
371:in
19:In
3021::
2984:,
2974:,
2962:;
2958:;
2901:MR
2899:,
2889:,
2881:,
2859:,
2853:MR
2851:,
2841:,
2833:,
2814:,
2808:,
2791:.
2785:MR
2783:.
2773:.
2769:.
2753:,
2728:.
2705:,
2697:,
2691:MR
2689:,
2679:,
2671:,
2649:,
2641:,
2631:,
2625:MR
2623:,
2615:,
2605:,
2595:56
2593:,
2576:MR
2574:,
2566:,
2554:,
2535:,
2525:35
2523:,
2515:;
2504:41
2502:,
2488:MR
2486:,
2476:,
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