* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download PDF
Birkhoff's representation theorem wikipedia , lookup
Structure (mathematical logic) wikipedia , lookup
Polynomial ring wikipedia , lookup
Cartesian tensor wikipedia , lookup
Bra–ket notation wikipedia , lookup
Homomorphism wikipedia , lookup
Group (mathematics) wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Modular representation theory wikipedia , lookup
Boolean algebras canonically defined wikipedia , lookup
Heyting algebra wikipedia , lookup
Homological algebra wikipedia , lookup
Laws of Form wikipedia , lookup
Universal enveloping algebra wikipedia , lookup
Complexification (Lie group) wikipedia , lookup
Basis (linear algebra) wikipedia , lookup
Linear algebra wikipedia , lookup
Geometric algebra wikipedia , lookup
Exterior algebra wikipedia , lookup
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 created: h2013-03-21i by: hCWooi version: h37185i Privacy setting: h1i hDefinitioni h17A75i h17D05i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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: 2 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. 3