2166:
1136:. Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elements
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1024:
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1991:
1747:
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1223:, that is, the subalgebra generated by a single element is associative. The converse need not hold: the sedenions are power-associative but not alternative.
17:
2013:
where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra.
710:, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
1768:
Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its
2288:
2228:
2197:
2085:
2262:
289:
whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to
883:
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1684:
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206:
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107:
56:
390:
1035:
2238:
2022:
1769:
1139:
286:
39:
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2207:
2095:
1749:. The set of all invertible elements is therefore closed under multiplication and forms a
8:
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1427:
1118:"Artin's theorem" redirects here. For Artin's theorem on primitive elements, see
282:
2220:
1803:
is an alternative algebra, as shown by Guy Roos in 2008: A composition algebra
1758:
188:
Alternative algebras are so named because they are the algebras for which the
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2170:
Associative
Composition Algebra/Transcendental paradigm#Categorical treatment
2165:
1793:
1789:
197:
47:
1816:
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2215:
Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) .
2144:
Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in
1081:
1030:
43:
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189:
2078:
On
Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry
2214:
2032:
1772:. The structure theory of alternative rings is presented in the book
1430:
are unique whenever they exist. Moreover, for any invertible element
1216:
877:
An alternating associator is always totally skew-symmetric. That is,
1102:
1073:
706:
The associator of an alternative algebra is therefore alternating.
177:
1212:), the subalgebra generated by those elements is associative.
874:
is alternative and therefore satisfies all three identities.
172:
is obviously alternative, but so too are some strictly
2126:
Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982) p. 151
1757:
in an alternative ring or algebra is analogous to the
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of Artin's theorem is that alternative algebras are
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2148:by Bruce Gilligan & Guy Roos, volume 468 of
1776:by Zhevlakov, Slin'ko, Shestakov, and Shirshov.
1426:In a unital alternative algebra, multiplicative
2135:Zhevlakov, Slin'ko, Shestakov, Shirshov (1982)
1076:form a non-associative alternative algebra, a
1019:{\displaystyle =\operatorname {sgn}(\sigma )}
384:Both of these identities together imply that
1877:{\displaystyle n(a\times b)=n(a)\times n(b)}
1527:This is equivalent to saying the associator
2072:
1168:in an alternative algebra associate (i.e.,
2188:An Introduction to Nonassociative Algebras
1128:states that in an alternative algebra the
2260:
1069:Every associative algebra is alternative.
27:Algebra where x(xy)=(xx)y and (yx)x=y(xx)
2180:
2059:
2057:
1986:{\displaystyle (a:b)=n(a+b)-n(a)-n(b).}
14:
2281:
2120:
1742:{\displaystyle (xy)^{-1}=y^{-1}x^{-1}}
2111:
2102:
2054:
2045:
1052:. The converse holds so long as the
42:in which multiplication need not be
2129:
24:
25:
2300:
2254:
2217:Rings That Are Nearly Associative
1774:Rings That Are Nearly Associative
1423:hold in any alternative algebra.
1132:generated by any two elements is
183:
2263:"Alternative rings and algebras"
2192:. New York: Dover Publications.
2164:
1895:Define the form ( _ : _ ):
1681:is also invertible with inverse
1413:{\displaystyle (ax)(ya)=a(xy)a}
1350:{\displaystyle ((xa)y)a=x(aya)}
1287:{\displaystyle a(x(ay))=(axa)y}
1095:
2158:
2146:Symmetries in Complex Analysis
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2154:American Mathematical Society
2038:
1779:
1517:{\displaystyle y=x^{-1}(xy).}
1113:
864:{\displaystyle (xy)x=x(yx).}
766:right alternative identity:
696:{\displaystyle (xy)x=x(yx).}
642:. This is equivalent to the
270:{\displaystyle =(xy)z-x(yz)}
7:
2268:Encyclopedia of Mathematics
2016:
1063:
809:{\displaystyle (yx)x=y(xx)}
757:{\displaystyle x(xy)=(xx)y}
714:left alternative identity:
147:{\displaystyle (yx)x=y(xx)}
96:{\displaystyle x(xy)=(xx)y}
10:
2305:
2076:; Smith, Derek A. (2003).
1117:
2261:Zhevlakov, K.A. (2001) ,
2001::1) and the conjugate by
1815:that is a multiplicative
1120:Primitive element theorem
50:. That is, one must have
18:Alternative division ring
2289:Non-associative algebras
2150:Contemporary Mathematics
1080:of dimension 8 over the
592:{\displaystyle =+-==-=0}
174:non-associative algebras
1107:Cayley–Dickson algebras
1078:normed division algebra
1045:{\displaystyle \sigma }
1987:
1878:
1743:
1675:
1652:
1632:
1609:
1589:
1575:vanishes for all such
1569:
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271:
196:. The associator is a
148:
97:
1988:
1879:
1788:over any alternative
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1163:
1161:{\displaystyle x,y,z}
1047:
1021:
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811:
759:
698:
637:
617:
594:
376:
331:
285:is alternating if it
272:
149:
98:
2117:Schafer (1995) p. 30
2108:Schafer (1995) p. 29
2063:Schafer (1995) p. 28
2051:Schafer (1995) p. 27
2023:Algebra over a field
1911:
1823:
1685:
1662:
1658:are invertible then
1642:
1622:
1599:
1579:
1531:
1477:
1454:
1434:
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1236:
1172:
1140:
1087:More generally, any
1036:
884:
822:
770:
718:
654:
626:
606:
391:
341:
296:
207:
108:
57:
2182:Schafer, Richard D.
2074:Conway, John Horton
1801:composition algebra
1109:lose alternativity.
818:flexible identity:
374:{\displaystyle =0.}
170:associative algebra
36:alternative algebra
1993:Then the trace of
1983:
1874:
1739:
1674:{\displaystyle xy}
1671:
1648:
1628:
1605:
1585:
1565:
1514:
1460:
1440:
1410:
1347:
1284:
1228:Moufang identities
1205:{\displaystyle =0}
1202:
1158:
1042:
1016:
861:
806:
754:
693:
632:
612:
589:
371:
329:{\displaystyle =0}
326:
267:
144:
93:
1651:{\displaystyle y}
1631:{\displaystyle x}
1608:{\displaystyle y}
1588:{\displaystyle x}
1463:{\displaystyle y}
1443:{\displaystyle x}
1221:power-associative
645:flexible identity
635:{\displaystyle y}
615:{\displaystyle x}
281:By definition, a
16:(Redirected from
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2080:. A. K. Peters.
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1992:
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1984:
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1786:projective plane
1763:associative ring
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1568:{\displaystyle }
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1089:octonion algebra
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165:in the algebra.
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32:abstract algebra
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2028:Maltsev algebra
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1126:Artin's theorem
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1105:and all higher
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184:The associator
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2230:0-12-779850-1
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2222:
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2199:0-486-68813-5
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2087:1-56881-134-9
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2034:
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2029:
2026:
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2014:
2012:
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1997:is given by (
1996:
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1807:over a field
1806:
1802:
1797:
1795:
1794:Moufang plane
1791:
1790:division ring
1787:
1777:
1775:
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1755:loop of units
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1817:homomorphism
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1765:or algebra.
1754:
1751:Moufang loop
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1096:Non-examples
1082:real numbers
1056:of the base
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2096:1098.17001
2039:References
1888:, Ă—) and (
1780:Occurrence
1130:subalgebra
1114:Properties
1060:is not 2.
708:Conversely
190:associator
2273:EMS Press
2033:Zorn ring
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1951:−
1860:×
1836:×
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1103:sedenions
1074:octonions
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250:−
200:given by
178:octonions
2283:Category
2184:(1995).
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1470:one has
1450:and all
1428:inverses
1064:Examples
1029:for any
602:for all
287:vanishes
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