In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, triality is a relationship between three vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s, analogous to the duality

Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...

 relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D₄ and the associated Lie

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 Spin(8), the double cover of 8-dimensional rotation group SO(8)

SO(8)

In mathematics, SO is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28.-Spin:...

, arising because the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry

Duality (projective geometry)

A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...

.

Of all simple Lie group

Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...

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

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

 of the diagram is the symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

 S₃ which acts by permuting the three legs. This gives rise to an S₃ group of outer automorphisms of Spin(8). This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral

Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

 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 – 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

Fundamental representation

In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...

s.

No other Dynkin diagram has an automorphism group of order greater than 2; for other D_n (corresponding to other even Spin groups, Spin(2n)), 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 Bruhat-Tits building associated with the group. For special linear group

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

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 diagram also gives rise to the Steinberg group .

General formulation

A duality between two vector spaces over a field F is a nondegenerate bilinear map
i.e., for each nonzero vector v in one of the two vector spaces, the pairing with v is a nonzero linear functional

Linear functional

In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

 on the other.

Similarly, a triality between three vector spaces over a field F is a nondegenerate trilinear map
i.e., each nonzero vector in one of the three vector spaces induces a duality between the other two.

By choosing vectors e_i in each V_i on which the trilinear map evaluates to 1, we find that the three vector spaces are all isomorphic

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 to each other, and to their duals. Denoting this common vector space by V, the triality may be reexpressed as a bilinear multiplication
where each e_i corresponds to the identity element in V. The nondegeneracy condition now implies that V is a division algebra

Division algebra

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...

. 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

Normed division algebra

In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....

, and is therefore isomorphic to R, C, H or O.

Conversely, the normed division algebras immediately give rise to trialities by taking each V_i equal to the division algebra, and using the inner product on the algebra to dualize the multiplication into 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).

External links

• Spinors and Trialities by John Baez
• Triality with Zometool by David Richter