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| In [[mathematics]], '''Hurwitz's theorem''' is a theorem of [[Adolf Hurwitz]] (1859–1919), published posthumously in 1923, on finite-dimensional unital [[real numbers|real]] [[non-associative algebra]]s endowed with a [[positive-definite quadratic form]]. The theorem states that if the [[quadratic form]] defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the [[real number]]s, the [[complex number]]s, the [[quaternion]]s or the [[octonion]]s. The non-associative algebras occurring are called '''Hurwitz algebras''' or '''[[composition algebra]]s'''. The problem has an equivalent formulation in terms of quadratic forms {{math|''q''(''x'')}}, composability requiring the existence of a bilinear "composition" {{math|''z''(''x'', ''y'')}} such that {{math|1=''q''(''x'') ''q''(''y'') = ''q''(''z'' (''x'', ''y''))}}. Subsequent proofs have used the [[Cayley–Dickson construction]]. Although neither commutative nor associative, composition algebras have the special property of being [[alternative algebra]]s, i.e. left and right multiplication preserves squares, a weakened version of associativity. Their theory has subsequently been generalized to arbitrary quadratic forms and arbitrary [[field (mathematics)|fields]].<ref>See:
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| *{{harvnb|Lam|2005}}
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| *{{harvnb|Rajwade|1993}}
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| *{{harvnb|Shapiro|2000}}</ref>
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| Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the [[Hurwitz problem]], solved also in {{harvtxt|Radon|1922}}. Subsequent proofs of the restrictions on the dimension have been given by {{harvtxt|Eckmann|1943}} using the [[representation theory of finite groups]] and by {{harvtxt|Lee|1948}} and {{harvtxt|Chevalley|1954}} using [[Clifford algebra]]s. Hurwitz's theorem has been applied in [[algebraic topology]] to problems on [[vector fields on spheres#Radon–Hurwitz numbers|vector fields on spheres]] and the [[homotopy group]]s of the [[classical group]]s<ref>See:
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| *{{harvnb|Eckmann|1989}}
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| *{{harvnb|Eckmann|1999}}</ref> and in [[quantum mechanics]] to the [[Symmetric cone#Simple Euclidean Jordan algebras|classification of simple Jordan algebras]].<ref>{{harvnb|Jordan|von Neumann|Wigner|1934}}</ref>
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| ==Euclidean Hurwitz algebras==
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| ===Definition===
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| A '''Hurwitz algebra''' or '''composition algebra''' is a finite-dimensional nonassociative algebra {{mvar|A}} with identity endowed with a nondegenerate quadratic form {{mvar|q}} such that {{math|1=''q''(''a b'') = ''q''(''a'') ''q''(''b'')}}. If the underlying coefficient field is the [[real number|reals]] and {{mvar|q}} is positive-definite, so that {{math|1=(''a'', ''b'') = {{sfrac|1|2}}[''q''(''a'' + ''b'') − ''q''(''a'') − ''q''(''b'')]}} is an [[inner product space|inner product]], then {{mvar|A}} is called a '''Euclidean Hurwitz algebra'''.<ref>{{harvnb|Faraut|Koranyi|1994|p82}}</ref>
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| If {{mvar|A}} is a Euclidean Hurwitz algebra and {{mvar|a}} is in {{mvar|A}}, define the involution and right and left multiplication operators by
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| :<math>\displaystyle{a^*=-a +2(a,1)1,\,\,\, L(a)b = ab,\,\,\, R(a)b=ba.}</math>
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| Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties:
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| * the involution is an antiautomorphism, i.e. {{math|1=(''a b'')*=''b''* ''a''*}}
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| * {{math|1=''a a''* = ‖ ''a'' ‖<sup>2</sup> 1 = ''a''* ''a''}}
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| * {{math|1=''L''(''a''*) = ''L''(''a'')*}}, {{math|1=''R''(''a''*) = ''R''(''a'')*}}, so that the involution on the algebra corresponds to taking [[adjoint operator|adjoints]]
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| * {{math|1=Re(''a b'') = Re(''b a'')}} if {{math|1=Re ''x'' = (''x'' + ''x''*)/2 = (''x'', 1)1}}
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| * {{math|1=Re(''a b'') ''c'' = Re ''a''(''b c'')}}
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| * {{math|1=''L''(''a''<sup>2</sup>) = ''L''(''a'')<sup>2</sup>}}, {{math|1=''R''(''a''<sup>2</sup>) = ''R''(''a'')<sup>2</sup>}}, so that {{mvar|A}} is an [[alternating algebra]]
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| These properties are proved starting from the polarized version of the identity {{math|1=(''a b'', ''a b'') = (''a'', ''a'')(''b'', ''b'')}}:
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| :<math>\displaystyle{2(a,b)(c,d)=(ac,bd) + (ad,bc).}</math>
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| Setting {{math|1=''b'' = 1}} or {{math|1=''d'' = 1}} yields {{math|1=''L''(''a''*) = ''L''(''a'')*}} and {{math|1=''R''(''c''*) = ''R''(''c'')*}}.
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| Hence {{math|1=Re(''a b'') = (''a b'', 1)1 = (''a'', ''b''*)1 = (''b a'', 1)1 = Re(''b a'')}}.
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| Similarly {{math|1= Re (''a b'')''c'' = ((''a b'')''c'',1)1 = (''a b'', ''c''*)1 = (''b'', ''a''* ''c''*)1 = (''bc'',''a''*)1 = (''a''(''bc''),1)1 = Re ''a''(''b c'')}}.
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| Hence {{math|1= ((''ab'')*,c) = (''ab'',''c''*) = (''b'',''a''*''c''*) = (1,''b''*(''a''*''c''*)) = (1,(''b''*''a''*)''c''*) = (''b''*''a''*,''c'')}}, so that {{math|1=(''ab'')* = ''b''*''a''*}}.
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| By the polarized identity {{math|1=‖ ''a'' ‖<sup>2</sup> (''c'', ''d'') = (''a c'', ''a d'') = (''a''* ''a c'', ''d'')}} so {{math|1=''L''(''a''*) L(''a'') = ‖ ''a'' ‖<sup>2</sup>}}. Applied to 1 this gives {{math|1=''a''* ''a'' = ‖ ''a'' ‖<sup>2</sup>}}. Replacing {{mvar|a}} by {{math|''a''*}} gives the other identity.
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| Substituting the formula for {{math|''a''*}} in {{math|1=''L''(''a''*) ''L''(''a'') = ''L''(''a''* ''a'')}} gives {{math|1=''L''(''a'')<sup>2</sup> = ''L''(''a''<sup>2</sup>)}}.
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| ===Classification===
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| It is routine to check that the real numbers {{math|'''R'''}}, the complex numbers {{math|'''C'''}} and the quaternions {{math|'''H'''}} are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions {{math|'''R''' ⊂ '''C''' ⊂ '''H'''}}.
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| Analysing such an inclusion leads to the [[Cayley–Dickson construction]], formalized by [[Abraham Adrian Albert|A.A. Albert]]. Let {{mvar|A}} be a Euclidean Hurwitz algebra and {{mvar|B}} a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a [[unit vector]] {{mvar|j}} in {{mvar|A}} orthogonal to {{mvar|B}}. Since {{math|1=(''j'', 1) = 0}}, it follows that {{math|1=''j''* = −''j''}} and hence {{math|1=''j''<sup>2</sup> = −1}}. Let {{mvar|C}} be subalgebra generated by {{mvar|B}} and {{mvar|j}}. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following [[Cayley–Dickson construction|'''Cayley–Dickson multiplication laws''']]:
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| :<math>\displaystyle{C=B\oplus Bj, \,\,\, (a+bj)^*=a^* - bj, \,\,\, (a+bj)(c+dj)=(ac -d^*b) +(bc^*+da)j.}</math>
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| To check this note that {{mvar|B}} and {{math|''B j''}} are orthogonal, since {{mvar|j}} is orthogonal to {{mvar|B}}. If {{mvar|a}} is in {{mvar|B}}, then {{math|1=''j a'' = ''a''* ''j''}}, since by orthogonal {{math|1=0 = 2 (''j'', ''a''*) = ''j a'' − ''a''* ''j''}}. The formula for the involution follows. To show that {{math|''B'' ⊕ ''B j''}} is closed under multiplication note that {{math|1=''Bj'' = ''j B''}}. Since {{math|''B j''}} is orthogonal to 1, {{math|1=(''b j'')* = −''b j''}}.
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| * {{math|1=''b''(''c j'') = (''c b'')''j''}} since {{math|1=(''b'', ''j'') = 0}} so that, for {{mvar|x}} in {{mvar|A}}, {{math|1=(''b''(''c j''), ''x'') = (''b''(''j x''), ''j''(''c j'') = −(''b''(''j x''), ''c''*) = −(''c b'', (''j x'')*) = −((''c b'')''j'', ''x''*) = ((''c b'')''j'', ''x'')}}.
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| * {{math|1=(''j c'')''b'' = ''j''(''b c'')}} taking adjoints above.
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| * {{math|1=(''b j'')(''c j'') = −''c''* ''b''}} since {{math|1=(''b'', ''c j'')}} = 0, so that, for {{mvar|x}} in {{mvar|A}}, {{math|1=((''b j'')(''c j''), ''x'') = −((''c j'')''x''*, ''b j'') = (''b x''*, (''c j'')''j'') = −(''c''* ''b'', ''x'')}}.
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| Imposing the multiplicativity of the norm on {{mvar|C}} for {{math|''a'' + ''b j''}} and {{math|''c'' + ''d j''}} gives:
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| :<math>\displaystyle{(\|a\|^2+\|b\|^2)(\|c\|^2+\|d\|^2)=\|ac -d^*b\|^2 + \|bc^*+da\|^2,}</math>
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| which leads to
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| :<math>\displaystyle{(ac,d^*b)=(bc^*,da).}</math>
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| Hence {{math|1=''d''(''a c'') = (''d a'')''c''}}, so that {{mvar|B}} ''must be associative''.
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| This analysis applies to the inclusion of {{math|1='''R'''}} in {{math|1='''C'''}} and {{math|1='''C'''}} in {{math|1='''H'''}}. Taking {{math|1='''O''' = '''H''' ⊕ '''H'''}} with the product and inner product above gives a noncommutative nonassociative algebra generated by {{math|1='''J''' = (0, 1)}}. This recovers the usual definition of the [[octonion]]s or [[Cayley number]]s. If {{mvar|A}} is a Euclidean algebra, it must contain {{math|'''R'''}}. If it is strictly larger than {{math|'''R'''}}, the argument above shows that it contains {{math|'''C'''}}. If it is larger than {{math|'''C'''}}, it contains {{math|'''H'''}}. If it is larger still, it must contain {{math|'''O'''}}. But there the process must stop, because {{math|'''O'''}} is not associative. In fact {{math|'''H'''}} is not commutative and {{math|1=''a''(''b j'') = (''b a'') ''j'' ≠ (''a b'')''j''}} in {{math|1='''O'''}}.<ref>{{harvnb|Faraut|Koranyi|1994|pp=81–86}}</ref>
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| '''{{smallcaps|Theorem.}}''' The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.
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| ==Other proofs==
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| The proofs of {{harvtxt|Lee|1948}} and {{harvtxt|Chevalley|1954}} use [[Clifford algebra]]s to show that the dimension {{mvar|N}} of {{mvar|A}} must be 1, 2, 4 or 8. In fact the operators {{math|''L''(''a'')}} with {{math|1=(''a'', 1) = 0}} satisfy {{math|1=''L''(''a'')<sup>2</sup> = −‖ ''a'' ‖<sup>2</sup>}} and so form a real Clifford algebra. If {{mvar|a}} is a unit vector, then {{math|''L''(''a'')}} is skew-adjoint with square {{math|−''I''}}. So {{mvar|N}} must be [[even number|even]]. The real Clifford algebra and its [[complexification]] act on the complexification of {{mvar|A}}, an {{mvar|N}}-dimensional complex space. Since {{mvar|N}} is even, {{math|''N'' − 1}} is odd, so the Clifford algebra has exactly two complex [[irreducible representation]]s of dimension {{math|2<sup>''N''/2 − 1</sup>}}. So this [[power of 2]] must divide {{mvar|N}}. It is easy to see that this implies {{mvar|N}} can only be 1, 2, 4 or 8.
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| The proof of {{harvtxt|Eckmann|1954}} uses the representation theory of finite groups, or the projective representation theory of elementary Abelian 2-groups, known to be equivalent to the representation theory of real Clifford algebras. Indeed taking an orthonormal basis {{math|''e''<sub>''i''</sub>}} of the orthogonal complement of 1 gives rise to operators {{math|1=''U''<sub>''i''</sub> = ''L''(''e''<sub>''i''</sub>)}}
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| satisfying
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| :<math>\displaystyle{U_i^2=-I,\,\,\, U_iU_j=-U_jU_i \,\, (i\ne j).}</math>
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| This is a [[projective representation]] of a direct product of {{math|''N'' − 1}} groups of order 2. ({{mvar|N}} is assumed to be greater than 1.) The operators {{math|''U''<sub>''i''</sub>}} by construction are skew-symmetric and orthogonal. In fact Eckmann constructed operators of this type in a slightly different but equivalent way. It is in fact the method originally followed in {{harvtxt|Hurwitz|1923}}.<ref>See:
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| *{{harvnb|Hurwitz|1923|p=11}}
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| *{{harvnb|Herstein|1968|pp=141–144}}</ref> Assume that there is a composition law for two forms
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| :<math>\displaystyle{(x_1^2 + \cdots +x_N^2)(y_1^2 + \cdots + y_N^2) =z_1^2 + \cdots + z_N^2,}</math>
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| where {{math|''z''<sub>''i''</sub>}} is bilinear in {{mvar|x}} and {{mvar|y}}. Thus
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| :<math>\displaystyle{z_i=\sum_{j=1}^N a_{ij}(x)y_j}</math>
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| where the matrix {{math|1=''T''(''x'') = (''a''<sub>''ij''</sub>)}} is linear in {{mvar|x}}. The relations above are equivalent to
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| :<math>\displaystyle{T(x)T(x)^t=x_1^2 +\cdots + x_N^2.}</math>
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| Writing
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| :<math>\displaystyle{T(x)=T_1x_1 + \cdots + T_Nx_N,}</math>
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| the relations become
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| :<math>\displaystyle{T_iT^t_j+T_jT_i^t =2\delta_{ij}I.}</math>
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| Now set {{math|1=''V''<sub>''i''</sub> = (''T''<sub>''N''</sub>)<sup>''t''</sup> ''T''<sub>''i''</sub>}}. Thus {{math|1=''V''<sub>''N''</sub> = ''I''}} and the {{math|''V''<sub>1</sub>, … , ''V''<sub>''N'' − 1</sub>}} are skew-adjoint, orthogonal satisfying exactly the same relations as the {{math|''U''<sub>''i''</sub>}}'s:
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| :<math>\displaystyle{V_i^2=-I,\,\,\, V_iV_j=-V_jV_i \,\, (i\ne j).}</math>
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| Since {{math|''V''<sub>''i''</sub>}} is an orthogonal matrix with square {{math|−''I''}} on a real vector space, {{mvar|N}} is even.
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| Let {{mvar|G}} be the finite group generated by elements {{math|''v''<sub>''i''</sub>}} such that
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| :<math>\displaystyle{v_i^2=\varepsilon,\,\,\, v_iv_j=\varepsilon v_jv_i \,\, (i\ne j),}</math>
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| where {{mvar|ε}} is central of order 2. The commutator subgroup {{math|[''G'', ''G'']}} is just formed of 1 and {{mvar|ε}}. If {{mvar|N}} is odd this coincides with the center while if {{mvar|N}} is even the center has order 4 with extra elements {{math|1=γ = ''v''<sub>1</sub>, … , ''v''<sub>''N'' − 1</sub>}} and {{math|''ε'' γ}}. If {{mvar|g}} in {{mvar|G}} is not in the center its conjugacy class is exactly {{mvar|g}} and {{math|''ε g''}}. Thus there are
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| {{math|2<sup>''N'' − 1 </sup> + 1}} conjugacy classes for {{mvar|N}} odd and {{math|2<sup>''N'' − 1 </sup> + 2}} for {{mvar|N}} even. {{mvar|G}} has {{math|1={{!}} {{mvar|G}} / [''G'', ''G''] {{!}} = 2<sup>''N'' − 1</sup>}} 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since {{mvar|N}} is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals {{math|{{!}} ''G'' {{!}}}} and the dimensions divide {{math|{{!}} ''G'' {{!}}}}, the two irreducibles must have dimension {{math|2<sup>(''N'' − 2)/2</sup>}}. When {{mvar|N}} is even, there are two and their dimension must divide the order of the group, so is a power of two, so they must both have dimension {{math|2<sup>(''N'' − 2)/2</sup>}}. The space on which the {{math|''V''<sub>''i''</sub>}}'s act can be complexified. It will have complex dimension {{mvar|N}}. It breaks up into some of complex irreducible representations of {{mvar|G}}, all having dimension {{math|2<sup>(''N'' − 2)/2</sup>}}. In particular this dimension is {{math|≤ ''N''}}, so {{mvar|N}} is less than or equal to 8. If {{math|1=''N'' = 6}}, the dimension is 4, which does not divide 6. So ''N'' can only be 1, 2, 4 or 8.
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| ==Applications to Jordan algebras==
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| Let {{mvar|A}} be a Euclidean Hurwitz algebra and let {{math|''M''<sub>''n''</sub>(''A'')}} be the algebra of {{mvar|n}}-by-{{mvar|n}} matrices over {{mvar|A}}. It is a unital nonassociative algebra with an involution given by
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| :<math>\displaystyle{(x_{ij})^*=(x_{ji}^*).}</math>
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| The trace {{math|Tr(''X'')}} is defined as the sum of the diagonal elements of {{mvar|X}} and the real-valued trace by
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| {{math|1=Tr<sub>'''R'''</sub>(''X'') = Re Tr(''X'')}}. The real-valued trace satisfies:
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| :<math>\displaystyle{\mathrm{Tr}_{\mathbf{R}}\,XY=\mathrm{Tr}_{\mathbf{R}}\,YX,\,\,\, \mathrm{Tr}_{\mathbf{R}}\,(XY)Z=\mathrm{Tr}_{\mathbf{R}}\, X(YZ).}</math>
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| These are immediate consequences of the known identities for {{math|1=''n'' = 1}}.
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| In {{mvar|A}} define the ''[[associator]]'' by
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| :<math>\displaystyle{[a,b,c]=a(bc) - (ab)c.}</math>
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| It is trilinear and vanishes identically if {{mvar|A}} is associative. Since {{mvar|A}} is an alternating algebra
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| {{math|1=[''a'', ''a'', ''b''] = 0}} and {{math|1=[''b'', ''a'', ''a''] = 0}}. Polarizing it follows that the associator is antisymmetric in its three entries. Note also that {{mvar|a}}, {{mvar|b}} or {{mvar|c}} lie in {{math|'''R'''}} then the {{math|1=[''a'', ''b'', ''c''] = 0}}. This implies that {{math|''M''<sub>3</sub>(''A'')}} has certain commutation properties. In fact if {{mvar|X}} is a matrix in {{math|''M''<sub>3</sub>(''A'')}} with real entries on the diagonal then
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| :<math>\displaystyle{[X,X^2]=aI,}</math>
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| with {{mvar|a}} in {{mvar|A}}. In fact if {{math|1=''Y'' = [''X'', ''X''<sup>2</sup>]}}, then
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| :<math>\displaystyle{y_{ij}=\sum_{k,\ell} [x_{ik},x_{k\ell},x_{\ell j}].}</math>
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| Since the diagonal entries of {{mvar|X}} are real, the off diagonal entries of {{mvar|Y}} vanish. Each diagonal
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| entry of {{mvar|Y}} is a sum of two associators involving only off diagonal terms of {{mvar|X}}. Since the associators are invariant under cyclic permutations, the diagonal entries of {{mvar|Y}} are all equal.
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| Let {{math|''H''<sub>''n''</sub>(''A'')}} be the space of self-adjoint elements in {{math|''M''<sub>''n''</sub>(''A'')}} with product {{math|1=''X''∘''Y'' = {{sfrac|1|2}}(''X Y'' + ''Y X'')}} and inner product {{math|1=(''X'', ''Y'') = Tr<sub>'''R'''</sub>(''X Y'')}}.
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| '''{{smallcaps|Theorem.}}''' {{math|''H''<sub>''n''</sub>(''A'')}} is a [[Euclidean Jordan algebra]] if {{mvar|A}} is associative (the real numbers, complex numbers or quaternions) and {{math|1=''n'' ≥ 3}} or if {{mvar|A}} is nonassociative (the octonions) and {{math|1=''n'' = 3}}.
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| The [[exceptional object|exceptional]] Jordan algebra {{math|''H''<sub>3</sub>('''O''')}} is called the '''[[Albert algebra]]''' after [[Abraham Adrian Albert|A.A. Albert]].
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| To check that {{math|''H''<sub>''n''</sub>(''A'')}} satisfies the axioms for a Euclidean Jordan algebra, note that the real trace defines a symmetric bilinear form with {{math|1=(''X'', ''X'') = ∑ ‖ ''x''<sub>''ij''</sub> ‖<sup>2</sup>}}. So it is an inner product. It satisfies the associativity property {{math|1=(''Z''∘''X'', ''Y'') = (''X'', ''Z''∘''Y'')}} because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators {{math|''L''(''X'')}} defined by {{math|1=''L''(''X'')''Y'' = ''X''∘''Y''}}:
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| :<math>\displaystyle{[L(X),L(X^2)]=0.}</math>
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| This is easy to check when {{mvar|A}} is associative, since {{math|''M''<sub>''n''</sub>(''A'')}} is an associative algebra so a Jordan algebra with {{math|1=''X''∘''Y'' = {{sfrac|1|2}}(''X Y'' + ''Y X'')}}. When {{math|1=''A'' = '''O'''}} and {{math|1=''n'' = 3}} a special argument is required, one of the shortest being due to {{harvtxt|Freudenthal|1951}}.<ref>
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| See:
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| *{{harvnb|Faraut|Koranyi|1994|pp=88–91}}
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| *{{harvnb|Postnikov|1986}}</ref>
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| In fact if {{mvar|T}} is in {{math|''H''<sub>3</sub>('''O''')}} with {{math|1=Tr ''T'' = 0}}, then
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| :<math>\displaystyle{D(X) = TX -XT}</math>
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| defines a skew-adjoint derivation of {{math|''H''<sub>3</sub>('''O''')}}. Indeed
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| :<math>\displaystyle{\mathrm{Tr}(T(X(X^2)) -T(X^2(X)))=\mathrm{Tr}\,T(aI)=\mathrm{Tr}(T)a=0,}</math>
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| so that
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| :<math>\displaystyle{(D(X),X^2)=0.}</math>
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| Polarizing yields:
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| :<math>\displaystyle{(D(X),Y\circ Z)+(D(Y),Z\circ X)+ (D(Z),X\circ Y)=0.}</math>
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| Setting {{math|1=''Z'' = 1}}, shows that {{mvar|D}} is skew-adjoint. The derivation property {{math|1=''D''(''X''∘''Y'') = ''D''(''X'')∘''Y'' + ''X''∘''D''(''Y'')}} follows by this and the associativity property of the inner product in the identity above.
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| With {{mvar|A}} and {{mvar|n}} as in the statement of the theorem, let {{mvar|K}} be the group of automorphisms of {{math|1=''E'' = ''H''<sub>''n''</sub>(''A'')}} leaving invariant the inner product. It is a closed subgroup of {{math|1=[[orthogonal group|O]](''E'')}} so a compact Lie group. Its Lie algebra consists of skew-adjoint derivations. {{harvtxt|Freudenthal|1951}} showed that given {{mvar|X}} in {{mvar|E}} there is an automorphism {{mvar|k}} in {{mvar|K}} such that {{math|''k''(''X'')}} is a diagonal matrix. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies the Jordan condition, since Jordan products by real diagonal matrices commute on {{math|''M''<sub>''n''</sub>(''A'')}} for any non-associative algebra {{mvar|A}}.
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| To prove the diagonalization theorem, take {{mvar|X}} in {{mvar|E}}. By compactness {{mvar|k}} can be chosen in {{mvar|K}} minimizing the sums of the squares of the norms of the off-diagonal terms of {{math|''k''(''X'')}}. Since {{mvar|K}} preserves the sums of all the squares, this is equivalent to maximizing the sums of the squares of the norms of the diagonal terms of {{math|''k''(''X'')}}. Replacing {{mvar|X}} by {{math|''k X''}}, it can be assumed that the maximum is attained at {{mvar|X}}. Since the [[symmetric group]] {{math|''S''<sub>''n''</sub>}}, acting by permuting the coordinates, lies in {{mvar|K}}, if {{mvar|X}} is not diagonal, it can be supposed that {{math|''x''<sub>12</sub>}} and its adjoint {{math|''x''<sub>21</sub>}} are non-zero. Let {{mvar|T}} be the skew-adjoint matrix with {{math|(2, 1)}} entry {{mvar|a}}, {{math|(1, 2)}} entry {{math|−''a''*}} and 0 elsewhere and let {{mvar|D}} be the derivation ad {{mvar|T}} of {{mvar|E}}. Let {{math|1=''k''<sub>''t''</sub> = exp ''tD''}} in {{mvar|K}}. Then only the first two diagonal entries in {{math|1=''X''(''t'') = ''k''<sub>''t''</sub>''X''}} differ from those of {{mvar|X}}. The diagonal entries are real. The derivative of {{math|''x''<sub>11</sub>(''t'')}} at {{math|1=''t'' = 0}} is the {{math|(1, 1)}} coordinate of {{math|[''T'', ''X'']}}, i.e. {{math|1=''a''* ''x''<sub>21</sub> + ''x''<sub>12</sub> ''a'' = 2(''x''<sub>21</sub>, ''a'')}}. This derivative is non-zero if {{math|1=''a'' = ''x''<sub>21</sub>}}. On the other hand the group {{math|''k''<sub>''t''</sub>}} preserves the real-valued trace. Since it can only change {{math|1=''x''<sub>11</sub>}} and {{math|1=''x''<sub>22</sub>}}, it preserves their sum. However on the line {{math|1=''x'' + ''y'' = }}constant, {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} has no local maximum (only a global minimum), a contradiction. Hence {{mvar|X}} must be diagonal.
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| ==See also==
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| *[[Composition algebra]]
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| *[[Multiplicative quadratic form]]
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| *[[Radon–Hurwitz number]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| The algebraic theory of spinors and Clifford algebras|publisher=Columbia University Press|year= 1954}}
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| *{{citation|last=Eckmann|first= Beno|title= Gruppentheoretischer Beweis des Satzes von Hurwitz–Radon über die Komposition quadratischer Formen|journal= Comment. Math. Helv.|volume= 15|year=1943|pages= 358–366|url=http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=GDZPPN002053705}}
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| * {{citation|last=Eckmann|first= Beno|title=Hurwitz–Radon matrices and periodicity modulo 8|
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| journal=Enseign. Math.|volume= 35|year=1989|pages=77–91|url=http://retro.seals.ch/digbib/view?rid=ensmat-001:1989:35::244&id=&id2=&id3=}}
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| * {{citation|last=Eckmann|first= Beno|title=
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| Topology, algebra, analysis—relations and missing links|journal=
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| *{{citation|last=Faraut|first= J.|last2= Koranyi|first2= A.|title= Analysis on symmetric cones|series= Oxford Mathematical Monographs|publisher= Oxford University Press|year= 1994|isbn= 0198534779}}
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| *{{citation|last=Freudenthal|first=Hans|title=Oktaven, Ausnahmegruppen und Oktavengeometrie|year=1951|
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| publisher= Mathematisch Instituut der Rijksuniversiteit te Utrecht}}
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| *{{citation|last=Freudenthal|first=Hans|title=Oktaven, Ausnahmegruppen und Oktavengeometrie|journal= Geom. Dedicata|volume=19|year=1985|
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| pages=7–63|url=http://link.springer.com/article/10.1007%2FBF00233101}} (reprint of 1951 article)
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| *{{citation|last=Hurwitz|first=A.|title=Über die Composition der quadratischen Formen von beliebig vielen Variabeln|journal= Goett. Nachr.|year= 1898|pages= 309–316|url=http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=GDZPPN002498200}}
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| *{{citation|url=http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=GDZPPN002269074|last=Hurwitz|first=A.|title=Über die Komposition der quadratischen Formen|year=1923|journal=Math. Ann.|volume=88|pages=1–25}}
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| On an algebraic generalization of the quantum mechanical formalism|journal=Ann. of Math.|volume= 35| year=1934|pages= 29–64|jstor=1968117}}
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| *{{citation|last=Lee|first= H. C.|title=Sur le théorème de Hurwitz-Radon pour la composition des formes quadratiques|journal=Comment. Math. Helv.|volume= 21|year=1948|pages= 261–269|url=http://retro.seals.ch/digbib/view?rid=comahe-002:1948:21::22}}
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| * {{citation| last=Porteous | first=I.R. | authorlink=Ian R. Porteous | title=Topological Geometry | publisher=Van Nostrand Reinhold | year=1969 | isbn=0-442-06606-6 | zbl=0186.06304 }}
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| *{{citation|last=Postnikov|first=M.|title= Lie groups and Lie algebras. Lectures in geometry. Semester V |publisher=[[Mir Publishers|Mir]]|year= 1986}}
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| *{{citation|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg|
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| year=1922|volume=1|pages= 1–14|title=Lineare scharen orthogonaler matrizen|first=J.|last= Radon|url=http://link.springer.com/article/10.1007%2FBF02940576}}
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| * {{citation | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}
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| * {{citation | first=Richard D. | last=Schafer | year=1995 | origyear=1966 | zbl=0145.25601 | title=An introduction to non-associative algebras | publisher=[[Dover Publications]] | isbn=0-486-68813-5 }}
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| * {{citation|title=Compositions of quadratic forms|volume=33|series=De Gruyter Expositions in Mathematics|first=Daniel|last= Shapiro|publisher=Walter de Gruyter|year= 2000|isbn=311012629X}}
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| ==Further reading==
| |
| *{{citation|last=Baez|first= John C.|title= The octonions|journal= Bull. Amer. Math. Soc.|volume= 39| year=2002|pages= 145–205|url=http://math.ucr.edu/home/baez/octonions/|doi=10.1090/S0273-0979-01-00934-X}}
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| * {{citation|last=Conway| first=John H.|last2= Smith|first2= Derek A.|title=
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| On quaternions and octonions: their geometry, arithmetic, and symmetry|publisher=A K Peters|year= 2003|isbn= 1568811349}}
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| *{{citation|title=Hypercomplex numbers. An elementary introduction to algebras |year=1989 |publisher=[[Springer-Verlag]] | first1=I.L. | last1=Kantor | first2=A.S. | last2=Solodovnikov | others=Trans. A. Shenitzer |page=121 |chapter=Normed algebras with an identity. Hurwitz's theorem. |edition=2nd |isbn=0-387-96980-2 | zbl=0669.17001}}
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| * [[Max Koecher]] & [[Reinhold Remmert]] (1990) "Composition Algebras. Hurwitz's Theorem — Vector-Product Algebras", chapter 10 of ''Numbers'' by [[Heinz-Dieter Ebbinghaus]] et al., Springer, ISBN 0-387-97202-1
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| *{{citation |pages=21 ''ff'' |chapter=Appendix to Chapter 1. Composition algebras | first=Daniel B. | last=Shapiro |title=Compositions of quadratic forms |isbn=3-11-012629-X |year=2000 |publisher=Walter de Gruyter | zbl=0954.11011 | series=de Gruyter Expositions in Mathematics | volume=33}}
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| *{{citation| first = T. A.| last = Springer| authorlink = T. A. Springer
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| | coauthors = F. D. Veldkamp| year = 2000| title = Octonions, Jordan Algebras and Exceptional Groups| publisher = [[Springer-Verlag]]| isbn = 3-540-66337-1}}
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| [[Category:Non-associative algebras]]
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| [[Category:Quadratic forms]]
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| [[Category:Representation theory]]
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