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Triality
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{{Short description|Relationship between certain vector spaces}} {{for|the concept of triality in linguistics|Grammatical number#Trial}} {{no footnotes|date=July 2017}} [[Image:Dynkin diagram D4.png|133px|right|thumb|The automorphisms of the Dynkin diagram D<sub>4</sub> give rise to triality in Spin(8).]] In [[mathematics]], '''triality''' is a relationship among three [[vector space]]s, analogous to the [[duality (mathematics)|duality]] relation between [[dual vector space]]s. Most commonly, it describes those special features of the [[Dynkin diagram]] D<sub>4</sub> and the associated [[Lie group]] [[Spin(8)]], the [[Double covering group|double cover]] of 8-dimensional rotation group [[SO(8)]], arising because the group has an [[outer automorphism]] of order three. There is a geometrical version of triality, analogous to [[Duality (projective geometry)|duality in projective geometry]]. Of all [[simple Lie group]]s, Spin(8) has the most symmetrical Dynkin diagram, D<sub>4</sub>. The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The [[symmetry group]] of the diagram is the [[symmetric group]] ''S''<sub>3</sub> which acts by permuting the three legs. This gives rise to an ''S''<sub>3</sub> group of outer automorphisms of Spin(8). This [[automorphism group]] permutes the three 8-dimensional [[irreducible representation]]s of Spin(8); these being the ''vector'' representation and two [[chirality (mathematics)|chiral]] ''spin'' representations. These automorphisms do not project to automorphisms of SO(8). The vector representation—the natural action of SO(8) (hence Spin(8)) on {{math|''F''<sup>8</sup>}}—consists over the real numbers of [[Euclidean space|Euclidean 8-vectors]] and is generally known as the "defining module", while the chiral spin representations are also known as [[spinor|"half-spin representations"]], and all three of these are [[fundamental representation]]s. No other connected Dynkin diagram has an automorphism group of order greater than 2; for other D<sub>''n''</sub> (corresponding to other even Spin groups, Spin(2''n'')), there is still the automorphism corresponding to switching the two half-spin representations, but these are not isomorphic to the vector representation. Roughly speaking, symmetries of the Dynkin diagram lead to automorphisms of the [[Tits building]] associated with the group. For [[special linear group]]s, one obtains projective duality. For Spin(8), one finds a curious phenomenon involving 1-, 2-, and 4-dimensional subspaces of 8-dimensional space, historically known as "geometric triality". The exceptional 3-fold symmetry of the D<sub>4</sub> diagram also gives rise to the [[Steinberg group (Lie theory)|Steinberg group]] [[3D4|<sup>3</sup>D<sub>4</sub>]]. ==General formulation== A duality between two vector spaces over a field {{mvar|F}} is a non-degenerate [[bilinear form]] :<math> V_1\times V_2\to F,</math> i.e., for each non-zero vector {{mvar|v}} in one of the two vector spaces, the pairing with {{mvar|v}} is a non-zero [[linear functional]] on the other. Similarly, a triality between three vector spaces over a field {{mvar|F}} is a non-degenerate [[multilinear form|trilinear form]] :<math> V_1\times V_2\times V_3\to F,</math> i.e., each non-zero vector in one of the three vector spaces induces a duality between the other two. By choosing vectors {{math|''e''<sub>''i''</sub>}} in each {{math|''V''<sub>''i''</sub>}} on which the trilinear form evaluates to 1, we find that the three vector spaces are all [[isomorphism|isomorphic]] to each other, and to their duals. Denoting this common vector space by {{mvar|V}}, the triality may be re-expressed as a [[algebra over a field|bilinear multiplication]] :<math> V \times V \to V</math> where each {{math|''e''<sub>''i''</sub>}} corresponds to the identity element in {{mvar|V}}. The non-degeneracy condition now implies that {{mvar|V}} is a [[composition algebra]]. It follows that {{mvar|V}} has dimension 1, 2, 4 or 8. If further {{math|1=''F'' = [[real number|'''R''']]}} and the form used to identify {{mvar|V}} with its dual is [[definite quadratic form|positive definite]], then {{mvar|V}} is a [[Hurwitz's theorem (composition algebras)|Euclidean Hurwitz algebra]], and is therefore isomorphic to '''R''', '''C''', '''H''' or '''O'''. Conversely, composition algebras immediately give rise to trialities by taking each {{math|''V''<sub>''i''</sub>}} equal to the algebra, and [[tensor contraction|contracting]] the multiplication with the inner product on the algebra to make a trilinear form. An alternative construction of trialities uses spinors in dimensions 1, 2, 4 and 8. The eight-dimensional case corresponds to the triality property of Spin(8). == See also == * [[Triple product]], may be related to the 4-dimensional triality (on [[quaternion]]s) ==References== * [[John Frank Adams]] (1981), ''Spin(8), Triality, F<sub>4</sub> and all that'', in "Superspace and supergravity", edited by Stephen Hawking and Martin Roček, Cambridge University Press, pages 435–445. * [[John Frank Adams]] (1996), ''Lectures on Exceptional Lie Groups'' (Chicago Lectures in Mathematics), edited by Zafer Mahmud and Mamora Mimura, University of Chicago Press, {{isbn|0-226-00527-5}}. ==Further reading== * {{cite book | last1=Knus | first1=Max-Albert | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Rost | first3=Markus | author3-link=Markus Rost | last4=Tignol | first4=Jean-Pierre| author-link4=Jean-Pierre Tignol | title=The book of involutions | others=With a preface by J. Tits | zbl=0955.16001 | series=Colloquium Publications | publisher=[[American Mathematical Society]] | volume=44 | location=Providence, RI | year=1998 | isbn=0-8218-0904-0 }} * {{cite book | title=The Finite Simple Groups | volume=251 | series=[[Graduate Texts in Mathematics]] | first=Robert | last=Wilson | authorlink=Robert Arnott Wilson | publisher=[[Springer-Verlag]] | year=2009 | isbn=978-1-84800-987-5 | zbl=1203.20012 }} ==External links== *[http://math.ucr.edu/home/baez/octonions/node7.html Spinors and Trialities] by John Baez *[http://homepages.wmich.edu/~drichter/zometriality.htm Triality with Zometool] by David Richter [[Category:Lie groups]] [[Category:Spinors]]
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