Triality

2025-09-17T02:59:51Z

2601:152:202:6D10:9981:FAC9:98A2:BC74: /* General formulation */

{{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 D4 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]] D4 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, D4. 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''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]] 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''8}}—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''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]] 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 D4 diagram also gives rise to the [[Steinberg group (Lie theory)|Steinberg group]] [[3D4|3D4]].

==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''''i''}} in each {{math|''V''''i''}} 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''''i''}} 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 [[real numbers|'''R''']], [[complex numbers|'''C''']], [[quaternions|'''H''']] or [[octonions|'''O''']].

Conversely, composition algebras immediately give rise to trialities by taking each {{math|''V''''i''}} 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, F4 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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Triality