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Octonion
In mathematics, the octonions are a nonassociative extension of the quaternions. They form an 8-dimensional normed division algebra over the real numbers. The octonion algebra is often denoted O.
Lacking the desirable property of associativity, the octonions receive far less attention then the quaternions. Despite this, the octonions retain importance for being related to a number of exceptional structures in mathematics, among them the exceptional Lie groups.
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History
The octonions were discovered in 1843 by John T. Graves , a friend of William Hamilton, who called them octaves. They were discovered independently by Arthur Cayley, who published the first paper on them in 1845. They are sometimes referred to as Cayley numbers or the Cayley algebra.
Definition
The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions {1, i, j, k, l, li, lj, lk}. That is, every octonion x can be written in the form
- x = x_{0} + x_{1} i + x_{2} j + x_{3} k + x_{4} l + x_{5} li + x_{6} lj + x_{7} lk.
with real coefficients x_{a}.
Addition of octonions is accomplished by adding corresponding coefficients, as with the complex numbers and quaternions. By linearity, multiplication of octonions is completely determined by the multiplication table for the unit octonions given below.
1 | i | j | k | l | li | lj | lk |
i | −1 | k | −j | −li | l | −lk | lj |
j | −k | −1 | i | −lj | lk | l | −li |
k | j | −i | −1 | −lk | −lj | li | l |
l | li | lj | lk | −1 | −i | −j | −k |
li | −l | −lk | lj | i | −1 | −k | j |
lj | lk | −l | −li | j | k | −1 | −i |
lk | −lj | li | −l | k | −j | i | −1 |
(Note that the basis for the octonions given here is not nearly as universal as the standard basis for the quaternions, however, nearly all other choices differ from this one only in order and sign.)
Cayley-Dickson construction
A more systematic way of defining the octonions is via the Cayley-Dickson construction. Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions. Addition is defined pairwise. The product of two pairs of quaternions (a, b) and (c, d) is defined by
- (a, b)(c, d) = (ac − db^{*}, a^{*}d + cb)
where z^{*} denotes the conjugate of the quaternion z. This definition is equivalent to the one given above when the eight unit octonions are idenitifiied with the pairs
- (1,0), (i,0), (j,0), (k,0), (0,1), (0,i), (0,j), (0,k)
Fano plane mnemonic
A convenient mnemonic for remembering the products of unit octonions is given by the following diagram:
This diagram with seven points and seven lines (the circle through i, j, and k is considered a line) is called the Fano plane. Note that the lines are oriented in this diagram. The seven points correspond to the seven standard basis elements of Im(O). Note that each pair of distinct points lies on a unique line and each line runs through exactly three points.
Let (a, b, c) be an ordered triple of points lying on a given line with the order specified by the direction of the arrow. Then multiplication is given by
- ab = c and ba = −c
together with cyclic permutations. These rules together with
- 1 is the multiplicative identity,
- e^{2} = −1 for each point in the diagram
completely defines the algebraic structure of the octonions. Note that each of the seven lines generates a subalgebra of O isomorphic to the quaternions H.
Conjugate, norm, and inverse
The conjugate of an octonion
- x = x_{0} + x_{1} i + x_{2} j + x_{3} k + x_{4} l + x_{5} li + x_{6} lj + x_{7} lk
is given by
- x^{*} = x_{0} − x_{1} i − x_{2} j − x_{3} k − x_{4} l − x_{5} li − x_{6} lj − x_{7} lk.
Conjugation is an involution of O and satisfies (xy)^{*} = y^{*}x^{*} (note the change in order).
The real part of x is defined as ½(x + x^{*}) = x_{0} and the imaginary part as ½(x - x^{*}). The set of all purely imaginary octonions span a 7 dimension subspace of O, denoted Im(O).
The norm of the octonion x is defined as
The square root is well-defined here as x^{*}x = xx^{*} is always a nonnegative real number:
Note that this norm agrees with the standard Euclidean norm on R^{8}.
The existence of a norm on O implies the existence of inverses for every nonzero element of O. The inverse of x ≠ 0 is given by
It satisifies xx^{−1} = x^{−1}x = 1.
Properties
Octonionic multiplication is neither commutative:
- ij = − ji
nor associative:
- (ij)l = − i(jl)
They do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of O is isomorphic to R, C, or H, all of which are associative.
The octonions do retain one important property shared by R, C, and H: the norm on O satisfies
This implies that the octonions form a nonassociative normed division algebra. The higher-dimensional algebras defined by the Cayley-Dickson construction (e.g. the sedenions) all fail to satisfy this property. They all have zero divisors.
It turns out that the only normed division algebras over the reals are R, C, H, and O. These four algebras also form the only alternative, finite-dimensional division algebra over the reals (up to isomorphism).
Not being associative, the nonzero elements of O do not form a group. They do, however, form a quasigroup, indeed a Moufang loop.
Automorphisms
An automorphism, A, of the octonions is an invertible linear transformation of O which satisfies
- A(xy) = A(x)A(y).
The set of all automorphisms of O forms a group called G_{2}. The group G_{2} is a simply connected, compact, real Lie group of dimension 14. This group is the smallest of the five exceptional Lie groups.
See also: PSL(2,7) - the automorphism group of the Fano plane.
Related topics
External links and references
- The Octonions http://math.ucr.edu/home/baez/Octonions/octonions.html - an article by John C. Baez
- Octonion Fractals http://www.stud.ux.his.no/~onar/Octonion.html - fractals generated using octonion mathematics
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