2155:
1125:. 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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1980:
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1212:, that is, the subalgebra generated by a single element is associative. The converse need not hold: the sedenions are power-associative but not alternative.
2277:
2217:
2186:
2074:
2002:
where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra.
699:, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
2261:
2142:
1757:
Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its
2256:
1095:
2251:
278:
whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to
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182:
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1738:. The set of all invertible elements is therefore closed under multiplication and forms a
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1107:"Artin's theorem" redirects here. For Artin's theorem on primitive elements, see
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is an alternative algebra, as shown by Guy Roos in 2008: A composition algebra
1118:
178:
177:
Alternative algebras are so named because they are the algebras for which the
2159:
Associative
Composition Algebra/Transcendental paradigm#Categorical treatment
2021:
1205:
2204:
Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) .
2133:
Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in
1091:
1062:
166:
2067:
On
Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry
1761:. The structure theory of alternative rings is presented in the book
1419:
are unique whenever they exist. Moreover, for any invertible element
866:
An alternating associator is always totally skew-symmetric. That is,
695:
The associator of an alternative algebra is therefore alternating.
1201:), the subalgebra generated by those elements is associative.
863:
is alternative and therefore satisfies all three identities.
161:
is obviously alternative, but so too are some strictly
2115:
Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982) p. 151
1746:
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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2137:by Bruce Gilligan & Guy Roos, volume 468 of
1765:by Zhevlakov, Slin'ko, Shestakov, and Shirshov.
1415:In a unital alternative algebra, multiplicative
2124:Zhevlakov, Slin'ko, Shestakov, Shirshov (1982)
1065:form a non-associative alternative algebra, a
1008:{\displaystyle =\operatorname {sgn}(\sigma )}
373:Both of these identities together imply that
8:
1866:{\displaystyle n(a\times b)=n(a)\times n(b)}
1516:This is equivalent to saying the associator
1157:in an alternative algebra associate (i.e.,
2177:An Introduction to Nonassociative Algebras
1117:states that in an alternative algebra the
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1058:Every associative algebra is alternative.
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16:Algebra where x(xy)=(xx)y and (yx)x=y(xx)
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2046:
2033:
1975:{\displaystyle (a:b)=n(a+b)-n(a)-n(b).}
1731:{\displaystyle (xy)^{-1}=y^{-1}x^{-1}}
7:
1041:. The converse holds so long as the
31:in which multiplication need not be
14:
2206:Rings That Are Nearly Associative
1763:Rings That Are Nearly Associative
1412:hold in any alternative algebra.
1121:generated by any two elements is
2252:"Alternative rings and algebras"
2181:. New York: Dover Publications.
2153:
1884:Define the form ( _ : _ ):
1670:is also invertible with inverse
1402:{\displaystyle (ax)(ya)=a(xy)a}
1339:{\displaystyle ((xa)y)a=x(aya)}
1276:{\displaystyle a(x(ay))=(axa)y}
2135:Symmetries in Complex Analysis
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2143:American Mathematical Society
1506:{\displaystyle y=x^{-1}(xy).}
853:{\displaystyle (xy)x=x(yx).}
755:right alternative identity:
685:{\displaystyle (xy)x=x(yx).}
631:. This is equivalent to the
259:{\displaystyle =(xy)z-x(yz)}
2257:Encyclopedia of Mathematics
798:{\displaystyle (yx)x=y(xx)}
746:{\displaystyle x(xy)=(xx)y}
703:left alternative identity:
136:{\displaystyle (yx)x=y(xx)}
85:{\displaystyle x(xy)=(xx)y}
2294:
2065:; Smith, Derek A. (2003).
1106:
2250:Zhevlakov, K.A. (2001) ,
1990::1) and the conjugate by
1804:that is a multiplicative
1109:Primitive element theorem
39:. That is, one must have
2278:Non-associative algebras
2139:Contemporary Mathematics
1069:of dimension 8 over the
581:{\displaystyle =+-==-=0}
163:non-associative algebras
1096:Cayley–Dickson algebras
1067:normed division algebra
1034:{\displaystyle \sigma }
1976:
1867:
1732:
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1564:vanishes for all such
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185:. The associator is a
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1777:over any alternative
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1150:{\displaystyle x,y,z}
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274:is alternating if it
261:
138:
87:
2106:Schafer (1995) p. 30
2097:Schafer (1995) p. 29
2052:Schafer (1995) p. 28
2040:Schafer (1995) p. 27
2012:Algebra over a field
1900:
1812:
1674:
1651:
1647:are invertible then
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1611:
1588:
1568:
1520:
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1443:
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1076:More generally, any
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873:
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595:
380:
330:
285:
196:
97:
46:
2171:Schafer, Richard D.
2063:Conway, John Horton
1790:composition algebra
1098:lose alternativity.
807:flexible identity:
363:{\displaystyle =0.}
159:associative algebra
25:alternative algebra
1982:Then the trace of
1972:
1863:
1728:
1663:{\displaystyle xy}
1660:
1637:
1617:
1594:
1574:
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1503:
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1217:Moufang identities
1194:{\displaystyle =0}
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1147:
1031:
1005:
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601:
578:
360:
318:{\displaystyle =0}
315:
256:
133:
82:
1640:{\displaystyle y}
1620:{\displaystyle x}
1597:{\displaystyle y}
1577:{\displaystyle x}
1452:{\displaystyle y}
1432:{\displaystyle x}
1210:power-associative
634:flexible identity
624:{\displaystyle y}
604:{\displaystyle x}
270:By definition, a
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2180:
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2069:. A. K. Peters.
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1981:
1979:
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1775:projective plane
1752:associative ring
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1557:{\displaystyle }
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1078:octonion algebra
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154:in the algebra.
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21:abstract algebra
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2017:Maltsev algebra
2008:
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1115:Artin's theorem
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1094:and all higher
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1080:is alternative.
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1986:is given by (
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1796:over a field
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1783:Moufang plane
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1779:division ring
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1744:loop of units
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1806:homomorphism
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1754:or algebra.
1743:
1740:Moufang loop
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1085:Non-examples
1071:real numbers
1045:of the base
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165:such as the
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2028:References
1877:, Ă—) and (
1769:Occurrence
1119:subalgebra
1103:Properties
1049:is not 2.
697:Conversely
179:associator
2262:EMS Press
2022:Zorn ring
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1940:−
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1825:×
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1092:sedenions
1063:octonions
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239:−
189:given by
167:octonions
2272:Category
2173:(1995).
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1459:one has
1439:and all
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1053:Examples
1018:for any
591:for all
276:vanishes
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