# Triality > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Triality > Markdown URL: https://mediated.wiki/source/Triality.md > Source: https://en.wikipedia.org/wiki/Triality > Source revision: 1354551902 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Relationship between certain vector spaces}} {{for|the concept of triality in linguistics|Grammatical number#Trial}} {{no footnotes|date=July 2017}} 133px|right|thumb|The automorphisms of the Dynkin diagram D4 give rise to triality in Spin(8). In [mathematics](/source/mathematics), '''triality''' is a relationship among three [vector space](/source/vector_space)s, analogous to the [duality](/source/duality_(mathematics)) relation between [dual vector space](/source/dual_vector_space)s. Most commonly, it describes those special features of the [Dynkin diagram](/source/Dynkin_diagram) D4 and the associated [Lie group](/source/Lie_group) [Spin(8)](/source/Spin(8)), the [double cover](/source/Double_covering_group) of 8-dimensional rotation group [SO(8)](/source/SO(8)), arising because the group has an [outer automorphism](/source/outer_automorphism) of order three. There is a geometrical version of triality, analogous to [duality in projective geometry](/source/Duality_(projective_geometry)). Of all [simple Lie group](/source/simple_Lie_group)s, Spin(8) has the most symmetrical Dynkin diagram, D4. The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The [symmetry group](/source/symmetry_group) of the diagram is the [symmetric group](/source/symmetric_group) ''S''3 which acts by permuting the three legs. This gives rise to an ''S''3 group of outer automorphisms of Spin(8). This [automorphism group](/source/automorphism_group) permutes the three 8-dimensional [irreducible representation](/source/irreducible_representation)s of Spin(8); these being the ''vector'' representation and two [chiral](/source/chirality_(mathematics)) ''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''8}}—consists over the real numbers of [Euclidean 8-vectors](/source/Euclidean_space) and is generally known as the "defining module", while the chiral spin representations are also known as ["half-spin representations"](/source/spinor), and all three of these are [fundamental representation](/source/fundamental_representation)s. No other connected Dynkin diagram has an automorphism group of order greater than 2; for other D''n'' (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](/source/Tits_building) associated with the group. For [special linear group](/source/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 D4 diagram also gives rise to the [Steinberg group](/source/Steinberg_group_(Lie_theory)) [3D4](/source/3D4). ==General formulation== A duality between two vector spaces over a field {{mvar|F}} is a non-degenerate [bilinear form](/source/bilinear_form) : V_1\times V_2\to F, 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](/source/linear_functional) on the other. Similarly, a triality between three vector spaces over a field {{mvar|F}} is a non-degenerate [trilinear form](/source/multilinear_form) : V_1\times V_2\times V_3\to F, 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''''i''}} in each {{math|''V''''i''}} on which the trilinear form evaluates to 1, we find that the three vector spaces are all [isomorphic](/source/isomorphism) to each other, and to their duals. Denoting this common vector space by {{mvar|V}}, the triality may be re-expressed as a [bilinear multiplication](/source/algebra_over_a_field) : V \times V \to V where each {{math|''e''''i''}} corresponds to the identity element in {{mvar|V}}. The non-degeneracy condition now implies that {{mvar|V}} is a [composition algebra](/source/composition_algebra). It follows that {{mvar|V}} has dimension 1, 2, 4 or 8. If further {{math|1=''F'' = ['''R'''](/source/real_number)}} and the form used to identify {{mvar|V}} with its dual is [positive definite](/source/definite_quadratic_form), then {{mvar|V}} is a [Euclidean Hurwitz algebra](/source/Hurwitz's_theorem_(composition_algebras)), and is therefore isomorphic to ['''R'''](/source/real_numbers), ['''C'''](/source/complex_numbers), ['''H'''](/source/quaternions) or ['''O'''](/source/octonions). Conversely, composition algebras immediately give rise to trialities by taking each {{math|''V''''i''}} equal to the algebra, and [contracting](/source/tensor_contraction) 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](/source/Triple_product), may be related to the 4-dimensional triality (on [quaternion](/source/quaternion)s) ==References== * [John Frank Adams](/source/John_Frank_Adams) (1981), ''Spin(8), Triality, F4 and all that'', in "Superspace and supergravity", edited by Stephen Hawking and Martin Roček, Cambridge University Press, pages 435–445. * [John Frank Adams](/source/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 | author1-link=Max-Albert Knus |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](/source/Jacques_Tits) | zbl=0955.16001 | series=Colloquium Publications | publisher=[American Mathematical Society](/source/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](/source/Graduate_Texts_in_Mathematics) | first=Robert | last=Wilson | authorlink=Robert Arnott Wilson | publisher=[Springer-Verlag](/source/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 --- Adapted from the Wikipedia article [Triality](https://en.wikipedia.org/wiki/Triality) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Triality?action=history)). 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