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octonion∗
CWoo†
2013-03-21 19:22:46
Let H be the quaternions over the reals R. Apply the Cayley-Dickson construction to H once, and we obtain an algebra, variously called Cayley algebra,
the octonion algebra, or simply the octonions, over R. Specifically the construction is carried out as follows:
1. Form the vector space O = H ⊕ Hk; any element of O can be written as
a + bk, where a, b ∈ H;
2. Define a binary operation on O called the multiplication in O by
(a + bk)(c + dk) := (ac − db) + (da + bc)k,
where a, b, c, d ∈ H, and c is the quaternionic conjugation of c ∈ H. When
b = d = 0, the multiplication is reduced the multiplication in H. In
addition, the multiplication rule above imply the following:
a(dk) = (da)k
(1)
(bk)c = (bc)k
(2)
(bk)(dk) = −db.
(3)
In particular, in the last equation, if b = d = 1, k2 = −1.
3. Define a unary operation on O called the octonionic conjugation in O by
a + bk := a − bk,
where a, b ∈ H. Clearly, the octonionic conjugation is an involution (x =
x).
4. Finally, define a unary operation N on O called the norm in O by N (x) :=
xx, where x ∈ O. Write x = a + bk, then
N (x) = (a + bk)(a − bk) = (aa + bb) + (−ba + ba)k = aa + bb ≥ 0.
It is not hard to see that N (x) = 0 iff x = 0.
∗ hOctonioni
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The above four (actually, only the first two suffice) steps makes O into an 8dimensional algebra over R such that H is embedded as a subalgebra.
With the last two steps, one can define the inverse of a non-zero element x ∈ O
by
x
x−1 :=
N (x)
so that xx−1 = x−1 x = 1. Since x is arbitrary, O has no zero divisors. Upon
checking that x−1 (xy) = y = (yx)x−1 , the non-associative algebra O is turned
into a division algebra.
Since N (x) ≥ 0 for any
p x ∈ O, we can define a non-negative real-valued function
k·k on O by kxk = N (x). This is clearly well-defined and kxk = 0 iff x = 0.
In addition, it is not hard to see that, for any r ∈ R and x ∈ O, krxk = |r|kxk,
and that k·k satisfies the triangular inequality. This makes O into a normed
division algebra.
Since the multiplication in H is noncommutative, O is noncommutative. In
fact, if we write H = C ⊕ Cj, where C are the complex numbers and j2 = −1,
then B = {1, i, j, ij} is a basis for the vector space H over R. With the introduction of k ∈ O, we quickly check that k anti-commute with the non-real basis
elements in B:
ik = −ki,
jk = −kj,
(ij)k = −k(ij).
Furthermore, one checks that i(jk) = (ji)k = −(ij)k, so that O is not associative.
Since O = H ⊕ Hk, the set {1, i, j, ij, k, ik, jk, (ij)k}(= B ∪ Bk) is a basis for O
over R. A less messy way to represent these basis elements is done the following
assignment:
basis element
1
i
j
ij
k
ik
jk
(ij)k
basis element rewritten
i0
i1
i2
i3
i4
i5
i6
i7
P7
Any element x of O can thus be expressed uniquely as n=0 rn in , where rn ∈ R.
Using Equations (1)-(3) above, one can form a multiplication table for these
basis elements in ’s:
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row×column
i1
i2
i3
i4
i5
i6
i7
i1
-1
i3
-i2
i5
-i4
-i7
i6
i2
-i3
-1
i1
i6
i7
-i4
-i5
i3
i2
-i1
-1
i7
-i6
i5
-i4
i4
-i5
-i6
-i7
-1
i1
i2
i3
i5
i4
-i7
i6
-i1
-1
-i3
i2
i6
i7
i4
-i5
-i2
i3
-1
-i1
i7
-i6
i5
i4
-i3
-i2
i1
-1
Other well known properties of the octonions are
1. xy = y x for any x, y ∈ O.
2. N (xy) = N (x)N (y) so that O is a composition algebra. It also proves
that the product of sums of eight squares is a sum of eight squares.
3. O is an alternative algebra. As a result, any two elements of O generate
an associative algebra. If fact, the algebra is isomorphic of one of R, C,
and H. This is the consequence of Artin’s Theorem.
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