2083:
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
1578:
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 factorsâthose with center reduced to
1549:
1554:
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
1046:
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
386:. 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
564:
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.
1348:
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
1964:
1396:
1078:
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:
1603:
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
482:
310:
1164:
672:
1841:
1687:
156:
1567:
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
232:
1801:
1647:
2291:
822:
760:
2025:
551:
1998:
1768:
1741:
1714:
365:
82:
338:
2108:
1591:. 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
2905:
2857:
2759:
2733:
2695:
849:
this is the only simple exceptional Jordan algebra up to isomorphism, it is often referred to as "the" exceptional Jordan algebra. Over the
1849:
3035:
2990:
2789:
2492:
1544:{\displaystyle \displaystyle {\|a\circ b\|\leq \|a\|\cdot \|b\|,\,\,\,\|a^{2}\|=\|a\|^{2},\,\,\,\|a^{2}\|\leq \|a^{2}+b^{2}\|.}}
320:. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element
2949:
2931:
2467:
2447:
2982:
2826:
2484:
862:
2821:
2075:
Quadratic Jordan algebras are a generalization of (linear) Jordan algebras introduced by Kevin McCrimmon (
431:
2058:
237:
1587:, all JWB factors can be realised as Jordan algebras of self-adjoint operators on a Hilbert space closed in the
2080:
2070:
1564:
1383:
1189:
The Jordan algebra of 3Ă3 self-adjoint octonionic matrices, as above (an exceptional Jordan algebra called the
1124:
1027:
582:
414:
1272:
is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces
1372:
1362:
383:
2050:
1588:
39:
2339:
2286:
1387:
387:
46:
2435:
1806:
1652:
615:
93:
2387:
1969:
Jordan simple superalgebras over an algebraically closed field of characteristic 0 were classified by
2609:
2098:
1319:
198:
42:
2816:
2509:
Jordan, Pascual (1933), "Ăber
Verallgemeinerungsmöglichkeiten des Formalismus der Quantenmechanik",
1776:
1622:
2966:
2093:
1604:
1592:
1575:
1210:
403:
185:
2974:
2627:
2547:
2417:
2364:
2318:
1206:
778:
2944:, CBMS Regional Conference Series in Mathematics, vol. 67, American Mathematical Society,
2561:(1977), "Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras",
719:
2986:
2945:
2927:
2901:
2853:
2785:
2755:
2729:
2691:
2653:
2578:
2488:
2463:
2443:
2356:
2310:
2061:
not equal to 2 the theory of J-structures is essentially the same as that of Jordan algebras.
2054:
1368:
192:
1595:
or its fixed point subalgebra under a period 2 *-antiautomorphism of the von
Neumann factor.
2996:
2893:
2871:
2845:
2803:
2765:
2709:
2683:
2661:
2643:
2617:
2570:
2539:
2523:
2399:
2348:
2300:
2161:
2003:
1556:
839:
557:
506:
31:
2915:
2867:
2799:
2705:
2639:
2590:
2502:
2413:
2376:
2330:
2149:
1976:
1746:
1719:
1692:
343:
55:
3019:
3000:
2911:
2889:
2875:
2863:
2841:
2807:
2795:
2769:
2713:
2701:
2679:
2665:
2635:
2597:
2586:
2498:
2476:
2409:
2372:
2326:
846:
2483:, American Mathematical Society Colloquium Publications, vol. 39, Providence, R.I.:
2613:
1583:âare completely understood in terms of von Neumann algebras. Apart from the exceptional
2833:
2781:
2748:
2113:
1584:
1568:
1350:
1205:
complex matrices as algebras of observables. However, the spin factors play a role in
1190:
832:
706:
561:
371:
323:
2648:
2305:
3029:
2527:
2427:
1345:
17:
2981:, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI:
2383:
1616:
1023:
850:
710:
699:
2740:
2717:
2421:
2404:
1063:
are unambiguously defined). He proved that any such algebra is a Jordan algebra.
2883:
2673:
1074:. Every formally real Jordan algebra can be written as a direct sum of so-called
2970:
2103:
2042:
2036:
1560:
906:
702:
496:
85:
1055:) and power-associative (the associative law holds for products involving only
827:
is a 27 dimensional, exceptional Jordan algebra (it is exceptional because the
2897:
2849:
2574:
2558:
2530:(1934), "On an algebraic generalization of the quantum mechanical formalism",
1379:
379:
27:
Not-necessarily-associative commutative algebra satisfying (đ„đŠ)đ„ÂČ=đ„(đŠđ„ÂČ)
2582:
2360:
2314:
1313:
for totally real Jordan algebras. It was later studied in full generality by
1070:
classified the finite-dimensional formally real Jordan algebras, also called
1007:
defines a derivation. In many important examples, the structure algebra of H(
234:
is independent of how we parenthesize this expression. They also imply that
1197:
Of these possibilities, so far it appears that nature makes use only of the
836:
2657:
2622:
853:
there are three isomorphism classes of simple exceptional Jordan algebras.
674:
Thus the set of all elements fixed by the involution (sometimes called the
2776:
Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) .
828:
769:
1973:. They include several families and some exceptional algebras, notably
1038:
A (possibly nonassociative) algebra over the real numbers is said to be
3013:
2551:
2457:
2368:
2322:
1309:) of the three eigenspaces. This decomposition was first considered by
2166:
1843:
becomes a Jordan superalgebra with respect to the graded Jordan brace
1172:
where the right-hand side is defined using the usual inner product on
421:
using the with same underlying addition and a new multiplication, the
2631:
2543:
2352:
2337:
Albert, A. Adrian (1947), "A structure theory for Jordan algebras",
2942:
Jordan algebras in analysis, operator theory, and quantum mechanics
2687:
1959:{\displaystyle \{x_{i},y_{j}\}=x_{i}y_{j}+(-1)^{ij}y_{j}x_{i}\ .}
979:
A simple example is provided by the
Hermitian Jordan algebras H(
184:, particularly to avoid confusion with the product of a related
2462:, Monographs and Studies in Mathematics, vol. 21, Pitman,
2138:, pp. 35â36, specifically remark before (56) and theorem 8
2053:
and axioms taking the Jordan inversion as basic operation and
394:), who began the systematic study of general Jordan algebras.
2442:, Oxford Mathematical Monographs, Oxford University Press,
2084:
Jordan algebras reduces to that of linear Jordan algebras.
1209:, and all the formally real Jordan algebras are related to
503:, whose product (Lie bracket) is defined by the commutator
2888:, Springer Monographs in Mathematics, Berlin, New York:
2289:(1946), "On Jordan algebras of linear transformations",
1619:
were introduced by Kac, Kantor and
Kaplansky; these are
378:) in an effort to formalize the notion of an algebra of
1382:
are JB algebras, which in finite dimensions are called
1310:
1067:
831:
are not associative). This was the first example of an
487:
These Jordan algebras and their subalgebras are called
406:
is a Jordan algebra if and only if it is commutative.
618:
560:âCohn theorem states that any Jordan algebra with two
2006:
1979:
1852:
1809:
1779:
1749:
1722:
1695:
1655:
1625:
1400:
1399:
1127:
781:
722:
509:
434:
346:
326:
240:
201:
96:
58:
2926:, North-Holland Mathematics Studies, vol. 104,
2882:
Springer, Tonny A.; Veldkamp, Ferdinand D. (2000) ,
1022:
Derivation and structure algebras also form part of
2241:
2747:
2388:"The Octonions, 3: Projective Octonionic Geometry"
2019:
1992:
1958:
1835:
1795:
1762:
1735:
1708:
1681:
1641:
1543:
1158:
816:
768:2. The set of 3Ă3 self-adjoint matrices over the
754:
666:
545:
476:
359:
332:
304:
226:
150:
76:
2924:Symmetric Banach manifolds and Jordan Câ-algebras
2885:Octonions, Jordan algebras and exceptional groups
2292:Transactions of the American Mathematical Society
2481:Structure and representations of Jordan algebras
2207:
2511:Nachr. Akad. Wiss. Göttingen. Math. Phys. Kl. I
2131:
2129:
1066:Not every Jordan algebra is formally real, but
1386:. The norm on the real Jordan algebra must be
2392:Bulletin of the American Mathematical Society
2256:
2049:to develop a theory of Jordan algebras using
1112:self-adjoint quaternionic matrices. as above.
960:) can be made into a Lie algebra, called the
8:
2600:(1966), "A general theory of Jordan rings",
2203:
2201:
1879:
1853:
1533:
1507:
1501:
1488:
1473:
1466:
1460:
1447:
1438:
1432:
1426:
1420:
1414:
1402:
1153:
1141:
835:. Its automorphism group is the exceptional
477:{\displaystyle x\circ y={\frac {xy+yx}{2}}.}
1042:if it satisfies the property that a sum of
305:{\displaystyle x^{m}(x^{n}y)=x^{n}(x^{m}y)}
2750:An introduction to nonassociative algebras
2150:"Nazis, émigrés, and abstract mathematics"
948:) is a derivation. Thus the direct sum of
340:, the operations of multiplying by powers
191:The axioms imply that a Jordan algebra is
2726:Algebraic Structures of Symmetric Domains
2647:
2621:
2403:
2304:
2269:
2219:
2192:
2180:
2165:
2076:
2011:
2005:
1984:
1978:
1944:
1934:
1921:
1899:
1889:
1873:
1860:
1851:
1827:
1814:
1808:
1785:
1781:
1780:
1778:
1754:
1748:
1727:
1721:
1700:
1694:
1673:
1660:
1654:
1631:
1627:
1626:
1624:
1527:
1514:
1495:
1487:
1486:
1485:
1476:
1454:
1446:
1445:
1444:
1401:
1398:
1159:{\displaystyle x^{2}=\langle x,x\rangle }
1132:
1126:
803:
780:
744:
721:
617:
508:
447:
433:
351:
345:
325:
290:
277:
258:
245:
239:
206:
200:
95:
57:
2135:
2046:
1743:has a "Lie-like" product with values in
1101:self-adjoint complex matrices, as above.
495:. This construction is analogous to the
2251:
2246:
2125:
1311:Jordan, von Neumann & Wigner (1934)
1115:The Jordan algebra freely generated by
1068:Jordan, von Neumann & Wigner (1934)
916:). The Jordan identity implies that if
417:2), one can construct a Jordan algebra
2456:Hanche-Olsen, H.; StĂžrmer, E. (1984),
1314:
1241:is the operation of multiplication by
391:
375:
3020:Jordan-Banach and Jordan-Lie algebras
2079:). The fundamental identities of the
1268:are 0, 1/2, 1. If the Jordan algebra
1225:is an idempotent in a Jordan algebra
1090:self-adjoint real matrices, as above.
7:
2838:Jordan algebras and algebraic groups
1340:Infinite-dimensional Jordan algebras
581:) is an associative algebra with an
176:in a Jordan algebra is also denoted
1970:
1607:that respects the Jordan identity.
1260: − 1) = 0
370:Jordan algebras were introduced by
2678:, Universitext, Berlin, New York:
667:{\textstyle \sigma (xy+yx)=xy+yx.}
25:
2778:Rings that are nearly associative
2306:10.1090/S0002-9947-1946-0016759-3
1836:{\displaystyle A_{0}\oplus A_{1}}
1682:{\displaystyle J_{0}\oplus J_{1}}
1335:Special kinds and generalizations
857:Derivations and structure algebra
151:{\displaystyle (xy)(xx)=x(y(xx))}
2109:KantorâKoecherâTits construction
1327:relative to the idempotent
1059:, so that powers of any element
928:, then the endomorphism sending
49:satisfies the following axioms:
2242:Hanche-Olsen & StĂžrmer 1984
1574:The Jordan algebra analogue of
765:form a special Jordan algebra.
682:, which is sometimes denoted H(
678:elements) form a subalgebra of
227:{\displaystyle x^{n}=x\cdots x}
2754:, Courier Dover Publications,
2728:, Princeton University Press,
1918:
1908:
1796:{\displaystyle \mathbb {Z} /2}
1642:{\displaystyle \mathbb {Z} /2}
1563:Jordan algebraic treatment of
1176:. This is sometimes called a
800:
782:
741:
723:
640:
622:
522:
510:
409:Given any associative algebra
299:
283:
267:
251:
145:
142:
133:
127:
118:
109:
106:
97:
1:
2983:American Mathematical Society
2602:Proc. Natl. Acad. Sci. U.S.A.
2485:American Mathematical Society
2405:10.1090/S0273-0979-01-00934-X
1034:Formally real Jordan algebras
713:matrices with multiplication
2746:Schafer, Richard D. (1996),
2208:Springer & Veldkamp 2000
1803:-graded associative algebra
987:). In this case any element
772:, again with multiplication
168:The product of two elements
2840:, Classics in Mathematics,
2822:Encyclopedia of Mathematics
2440:Analysis on symmetric cones
1571:is the common obstruction.
1371:has been extended to cover
1353:, which have dimension 27.
1264:so the only eigenvalues of
493:exceptional Jordan algebras
3052:
2675:A taste of Jordan algebras
2068:
2034:
1360:
905:). The derivations form a
817:{\displaystyle (xy+yx)/2,}
312:for all positive integers
2898:10.1007/978-3-662-12622-6
2850:10.1007/978-3-642-61970-0
2672:McCrimmon, Kevin (2004),
2575:10.1080/00927877708822224
2563:Communications in Algebra
2394:. Bull. Amer. Math. Soc.
2257:Faraut & Koranyi 1994
2148:Dahn, Ryan (2023-01-01).
2065:Quadratic Jordan algebras
2057:as a basic relation. In
1565:bounded symmetric domains
1384:Euclidean Jordan algebras
1072:Euclidean Jordan algebras
755:{\displaystyle (xy+yx)/2}
569:Hermitian Jordan algebras
388:Abraham Adrian Albert
3036:Non-associative algebras
2459:Jordan operator algebras
2081:quadratic representation
2071:Quadratic Jordan algebra
1716:is a Jordan algebra and
1390:and satisfy the axioms:
1373:Jordan operator algebras
1357:Jordan operator algebras
1028:Freudenthal magic square
2979:The book of involutions
2815:Slin'ko, A.M. (2001) ,
2051:linear algebraic groups
1363:Jordan operator algebra
1180:or a Jordan algebra of
491:, while all others are
489:special Jordan algebras
398:Special Jordan algebras
384:quantum electrodynamics
2724:Ichiro Satake (1980),
2623:10.1073/pnas.56.4.1072
2021:
2020:{\displaystyle K_{10}}
1994:
1960:
1837:
1797:
1764:
1737:
1710:
1683:
1643:
1589:weak operator topology
1545:
1256: − 1)(
1160:
1104:The Jordan algebra of
1093:The Jordan algebra of
1082:The Jordan algebra of
1026:' construction of the
818:
756:
668:
547:
546:{\displaystyle =xy-yx}
478:
361:
334:
306:
228:
152:
78:
40:nonassociative algebra
2532:Annals of Mathematics
2340:Annals of Mathematics
2022:
1995:
1993:{\displaystyle K_{3}}
1961:
1838:
1798:
1765:
1763:{\displaystyle J_{0}}
1738:
1736:{\displaystyle J_{1}}
1711:
1709:{\displaystyle J_{0}}
1684:
1644:
1546:
1161:
819:
757:
669:
548:
479:
402:Notice first that an
362:
360:{\displaystyle x^{n}}
335:
307:
229:
153:
79:
77:{\displaystyle xy=yx}
18:ShirshovâCohn theorem
2967:Merkurjev, Alexander
2940:Upmeier, H. (1987),
2922:Upmeier, H. (1985),
2099:Jordan triple system
2004:
1977:
1850:
1807:
1777:
1747:
1720:
1693:
1653:
1623:
1611:Jordan superalgebras
1576:von Neumann algebras
1397:
1378:The counterparts of
1320:Peirce decomposition
1217:Peirce decomposition
1125:
865:of a Jordan algebra
779:
720:
616:
507:
432:
344:
324:
238:
199:
94:
56:
2975:Tignol, Jean-Pierre
2614:1966PNAS...56.1072M
2428:Online HTML version
2210:, §5.8, p. 153
2094:Freudenthal algebra
1605:nonassociative ring
1211:projective geometry
1119:with the relations
869:is an endomorphism
404:associative algebra
186:associative algebra
2965:Knus, Max-Albert;
2834:Springer, Tonny A.
2045:was introduced by
2017:
1990:
1956:
1833:
1793:
1760:
1733:
1706:
1679:
1639:
1593:von Neumann factor
1541:
1540:
1207:special relativity
1156:
814:
752:
664:
543:
474:
372:Pascual Jordan
357:
330:
302:
224:
148:
74:
2907:978-3-540-66337-9
2859:978-3-540-63632-8
2761:978-0-486-68813-8
2735:978-0-691-08271-4
2697:978-0-387-95447-9
2569:(13): 1375â1400,
2343:, Second Series,
2287:Albert, A. Adrian
2167:10.1063/PT.3.5158
1952:
1649:-graded algebras
1369:operator algebras
962:structure algebra
845:. Since over the
469:
333:{\displaystyle x}
193:power-associative
16:(Redirected from
3043:
3003:
2954:
2936:
2918:
2878:
2829:
2817:"Jordan algebra"
2811:
2772:
2753:
2738:
2720:
2668:
2651:
2625:
2608:(4): 1072â1079,
2598:McCrimmon, Kevin
2593:
2554:
2518:
2505:
2477:Jacobson, Nathan
2472:
2452:
2425:
2407:
2379:
2333:
2308:
2273:
2267:
2261:
2236:
2230:
2217:
2211:
2205:
2196:
2190:
2184:
2178:
2172:
2171:
2169:
2145:
2139:
2133:
2026:
2024:
2023:
2018:
2016:
2015:
1999:
1997:
1996:
1991:
1989:
1988:
1965:
1963:
1962:
1957:
1950:
1949:
1948:
1939:
1938:
1929:
1928:
1904:
1903:
1894:
1893:
1878:
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1864:
1842:
1840:
1839:
1834:
1832:
1831:
1819:
1818:
1802:
1800:
1799:
1794:
1789:
1784:
1769:
1767:
1766:
1761:
1759:
1758:
1742:
1740:
1739:
1734:
1732:
1731:
1715:
1713:
1712:
1707:
1705:
1704:
1688:
1686:
1685:
1680:
1678:
1677:
1665:
1664:
1648:
1646:
1645:
1640:
1635:
1630:
1557:complex geometry
1550:
1548:
1547:
1542:
1539:
1532:
1531:
1519:
1518:
1500:
1499:
1481:
1480:
1459:
1458:
1165:
1163:
1162:
1157:
1137:
1136:
924:are elements of
823:
821:
820:
815:
807:
761:
759:
758:
753:
748:
673:
671:
670:
665:
612:it follows that
552:
550:
549:
544:
483:
481:
480:
475:
470:
465:
448:
366:
364:
363:
358:
356:
355:
339:
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336:
331:
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309:
308:
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295:
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263:
262:
250:
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233:
231:
230:
225:
211:
210:
163:
162:
157:
155:
154:
149:
83:
81:
80:
75:
32:abstract algebra
21:
3051:
3050:
3046:
3045:
3044:
3042:
3041:
3040:
3026:
3025:
3010:
2993:
2964:
2961:
2959:Further reading
2952:
2939:
2934:
2921:
2908:
2890:Springer-Verlag
2881:
2860:
2842:Springer-Verlag
2832:
2814:
2792:
2775:
2762:
2745:
2736:
2723:
2698:
2680:Springer-Verlag
2671:
2596:
2557:
2544:10.2307/1968117
2524:von Neumann, J.
2521:
2508:
2495:
2475:
2470:
2455:
2450:
2433:
2382:
2353:10.2307/1969128
2336:
2285:
2282:
2277:
2276:
2272:, pp. 9â10
2268:
2264:
2237:
2233:
2218:
2214:
2206:
2199:
2191:
2187:
2179:
2175:
2147:
2146:
2142:
2134:
2127:
2122:
2090:
2073:
2067:
2047:Springer (1998)
2041:The concept of
2039:
2033:
2007:
2002:
2001:
1980:
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1940:
1930:
1917:
1895:
1885:
1869:
1856:
1848:
1847:
1823:
1810:
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1775:
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1750:
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1723:
1718:
1717:
1696:
1691:
1690:
1669:
1656:
1651:
1650:
1621:
1620:
1613:
1601:
1523:
1510:
1491:
1472:
1450:
1395:
1394:
1365:
1359:
1351:Albert algebras
1342:
1337:
1317:and called the
1304:
1293:
1282:
1219:
1128:
1123:
1122:
1036:
859:
847:complex numbers
843:
777:
776:
718:
717:
696:
614:
613:
571:
505:
504:
449:
430:
429:
400:
347:
342:
341:
322:
321:
286:
273:
254:
241:
236:
235:
202:
197:
196:
195:, meaning that
161:Jordan identity
160:
159:
92:
91:
54:
53:
28:
23:
22:
15:
12:
11:
5:
3049:
3047:
3039:
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3028:
3027:
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3023:
3017:
3014:Jordan algebra
3009:
3008:External links
3006:
3005:
3004:
2991:
2960:
2957:
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2955:
2950:
2937:
2932:
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2782:Academic Press
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2688:10.1007/b97489
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2398:(2): 145â205.
2380:
2347:(3): 546â567,
2334:
2299:(3): 524â555,
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2274:
2270:McCrimmon 2004
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2249:
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2231:
2222:, pp. 99
2220:McCrimmon 2004
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2193:McCrimmon 2004
2185:
2181:McCrimmon 2004
2173:
2140:
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2121:
2118:
2117:
2116:
2114:Scorza variety
2111:
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2101:
2096:
2089:
2086:
2069:Main article:
2066:
2063:
2059:characteristic
2055:Hua's identity
2035:Main article:
2032:
2029:
2014:
2010:
1987:
1983:
1967:
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1569:Albert algebra
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1367:The theory of
1361:Main article:
1358:
1355:
1341:
1338:
1336:
1333:
1302:
1298:) â
1291:
1287:) â
1280:
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1195:
1194:
1191:Albert algebra
1186:
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833:Albert algebra
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698:1. The set of
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499:associated to
485:
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415:characteristic
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47:multiplication
36:Jordan algebra
26:
24:
14:
13:
10:
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6:
4:
3:
2:
3048:
3037:
3034:
3033:
3031:
3022:at PlanetMath
3021:
3018:
3016:at PlanetMath
3015:
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2559:Kac, Victor G
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1315:Albert (1947)
1312:
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1275:
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838:
834:
830:
811:
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766:
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619:
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566:
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420:
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412:
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405:
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393:
389:
385:
381:
377:
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368:
367:all commute.
352:
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90:
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68:
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44:
41:
37:
33:
19:
2978:
2971:Rost, Markus
2941:
2923:
2884:
2837:
2820:
2777:
2749:
2725:
2674:
2605:
2601:
2566:
2562:
2538:(1): 29â64,
2535:
2531:
2522:Jordan, P.;
2514:
2510:
2480:
2458:
2439:
2434:Faraut, J.;
2395:
2391:
2344:
2338:
2296:
2290:
2265:
2252:Upmeier 1987
2247:Upmeier 1985
2234:
2227:
2223:
2215:
2195:, p. 99
2188:
2176:
2160:(1): 44â50.
2157:
2153:
2143:
2074:
2040:
2031:J-structures
1968:
1772:
1614:
1602:
1599:Jordan rings
1580:
1573:
1553:
1377:
1366:
1343:
1328:
1324:
1318:
1306:
1299:
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949:
945:
941:
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933:
929:
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921:
917:
913:
909:
902:
898:
894:
890:
886:
882:
878:
874:
870:
866:
860:
851:real numbers
826:
767:
764:
711:quaternionic
700:self-adjoint
697:
687:
683:
679:
675:
609:
605:
601:
597:
593:
589:
585:
578:
574:
572:
555:
500:
492:
488:
486:
425:defined by:
422:
418:
410:
408:
401:
369:
317:
313:
190:
181:
177:
173:
169:
167:
43:over a field
35:
29:
2436:Koranyi, A.
2104:Jordan pair
2043:J-structure
2037:J-structure
1380:C*-algebras
1178:spin factor
907:Lie algebra
497:Lie algebra
380:observables
86:commutative
3001:0955.16001
2951:082180717X
2933:0444876510
2876:1024.17018
2808:0487.17001
2770:0145.25601
2714:1044.17001
2666:0139.25502
2528:Wigner, E.
2469:0273086197
2449:0198534779
2280:References
1971:Kac (1977)
1559:to extend
877:such that
863:derivation
588:, then if
583:involution
562:generators
2836:(1998) ,
2827:EMS Press
2583:0092-7872
2517:: 209â217
2479:(2008) ,
2361:0003-486X
2315:0002-9947
1912:−
1821:⊕
1667:⊕
1561:Koecher's
1534:‖
1508:‖
1505:≤
1502:‖
1489:‖
1474:‖
1467:‖
1461:‖
1448:‖
1439:‖
1433:‖
1430:⋅
1427:‖
1421:‖
1418:≤
1415:‖
1409:∘
1403:‖
1344:In 1979,
1154:⟩
1142:⟨
1003:)=−
837:Lie group
829:octonions
770:octonions
676:hermitian
620:σ
535:−
439:∘
219:⋯
3030:Category
2977:(1998),
2658:16591377
2438:(1994),
2386:(2002).
2088:See also
1388:complete
940:)−
694:Examples
558:Shirshov
413:(not of
2916:1763974
2868:1490836
2800:0518614
2706:2014924
2640:0202783
2610:Bibcode
2591:0498755
2552:1968117
2503:0251099
2414:1886087
2377:0021546
2369:1969128
2331:0016759
2323:1990270
1615:Jordan
1245:, then
707:complex
390: (
374: (
2999:
2989:
2948:
2930:
2914:
2904:
2874:
2866:
2856:
2806:
2798:
2788:
2768:
2758:
2741:Review
2732:
2718:Errata
2712:
2704:
2694:
2664:
2656:
2649:220000
2646:
2638:
2630:
2589:
2581:
2550:
2501:
2491:
2466:
2446:
2422:586512
2420:
2412:
2375:
2367:
2359:
2329:
2321:
2313:
2228:et seq
2226:, 235
2224:et seq
1951:
1689:where
1237:) and
1076:simple
1013:σ
997:σ
985:σ
688:σ
602:σ
590:σ
586:σ
579:σ
45:whose
2632:57792
2628:JSTOR
2548:JSTOR
2418:S2CID
2365:JSTOR
2319:JSTOR
2238:See:
2120:Notes
1015:) is
995:with
709:, or
38:is a
2987:ISBN
2946:ISBN
2928:ISBN
2902:ISBN
2854:ISBN
2786:ISBN
2756:ISBN
2730:ISBN
2692:ISBN
2654:PMID
2579:ISSN
2489:ISBN
2464:ISBN
2444:ISBN
2357:ISSN
2311:ISSN
2077:1966
2000:and
1773:Any
1024:Tits
952:and
920:and
885:) =
703:real
608:) =
600:and
596:) =
573:If (
556:The
392:1946
376:1933
316:and
172:and
88:law)
34:, a
2997:Zbl
2894:doi
2872:Zbl
2846:doi
2804:Zbl
2766:Zbl
2710:Zbl
2684:doi
2662:Zbl
2644:PMC
2618:doi
2571:doi
2540:doi
2400:doi
2349:doi
2301:doi
2162:doi
1323:of
1292:1/2
1221:If
991:of
976:).
970:str
964:of
954:der
932:to
910:der
873:of
690:).
382:in
30:In
3032::
2995:,
2985:,
2973:;
2969:;
2912:MR
2910:,
2900:,
2892:,
2870:,
2864:MR
2862:,
2852:,
2844:,
2825:,
2819:,
2802:.
2796:MR
2794:.
2784:.
2780:.
2764:,
2739:.
2716:,
2708:,
2702:MR
2700:,
2690:,
2682:,
2660:,
2652:,
2642:,
2636:MR
2634:,
2626:,
2616:,
2606:56
2604:,
2587:MR
2585:,
2577:,
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