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| {{for|the concept of triality in linguistics|Grammatical number#Trial}}
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| [[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).]]
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| 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|Lie]] [[Group (mathematics)|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]].
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| 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 (geometric)|vector]] representation and two [[chirality (mathematics)|chiral]] [[spin representation]]s. These automorphisms do not project to automorphisms of SO(8). The vector representation – the natural action of SO(8) (hence Spin(8)) on <math>K^8</math> – is also known as the "defining module", while the chiral spin representations are also known as "half-spin representations", and all three of these are [[fundamental representation]]s.
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| No other 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.
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| Roughly speaking, symmetries of the Dynkin diagram lead to automorphisms of the [[Bruhat-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".
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| The exceptional 3-fold symmetry of the <math>D_4</math> diagram also gives rise to the [[Steinberg group (Lie theory)|Steinberg group]] [[³D₄]].
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| ==General formulation==
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| A duality between two vector spaces over a field '''F''' is a [[nondegenerate]] [[bilinear map]]
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| :<math> V_1\times V_2\to \mathbb F,</math> | |
| i.e., for each nonzero vector ''v'' in one of the two vector spaces, the pairing with ''v'' is a nonzero [[linear functional]] on the other.
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| Similarly, a triality between three vector spaces over a field '''F''' is a nondegenerate [[multilinear map|trilinear map]]
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| :<math> V_1\times V_2\times V_3\to \mathbb F,</math>
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| i.e., each nonzero vector in one of the three vector spaces induces a duality between the other two.
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| By choosing vectors ''e''<sub>''i''</sub> in each ''V''<sub>''i''</sub> on which the trilinear map 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 ''V'', the triality may be reexpressed as a bilinear multiplication
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| :<math> V \times V \to V</math>
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| where each ''e''<sub>''i''</sub> corresponds to the identity element in ''V''. The nondegeneracy condition now implies that ''V'' is a [[division algebra]]. It follows that ''V'' has dimension 1, 2, 4 or 8. If further '''F''' = '''R''' and the identification of ''V'' with its dual is given by positive definite inner product, ''V'' is a [[normed division algebra]], and is therefore isomorphic to '''R''', '''C''', '''H''' or '''O'''.
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| Conversely, the normed division algebras immediately give rise to trialities by taking each ''V''<sub>''i''</sub> equal to the division algebra, and using the inner product on the algebra to dualize the multiplication into a trilinear form.
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| 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).
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| ==References==
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| * [[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.
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| * [[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.
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| ==External links==
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| *[http://math.ucr.edu/home/baez/octonions/node7.html Spinors and Trialities] by John Baez
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| *[http://homepages.wmich.edu/~drichter/zometriality.htm Triality with Zometool] by David Richter
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| [[Category:Lie groups]]
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| [[Category:Spinors]]
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The author is recognized by the title of Numbers Lint. Playing baseball is the hobby he will by no means stop doing. Years ago we moved to North Dakota and I adore each day living here. My working day occupation is a librarian.
Here is my web blog - meal delivery service (continue reading this..)