>Octonions? A non-associative geometric algebra Benjamin Prather Florida State University Department of Mathematics October 19, 2017 Scalar Product Spaces >Octonions? Prather Let K be a field with 1 6= −1 Let V be a vector space over K. Scalar Product Spaces Let h·; ·i : V × V ! K. Clifford Algebras Composistion Definition Algebras Hurwitz' Theorem V is a scalar product space if: Summary Symmetry hx; yi = hy; xi Linearity hax; yi = a hx; yi hx + y; zi = hx; zi + hy; zi for all x; y; z 2 V and a; b 2 K Scalar Product Spaces >Octonions? Prather Scalar Product Let N(x) = − hx; xi be a modulus on V . Spaces Clifford Algebras N(x + y) = − hx; xi − 2 hx; yi − hy; yi Composistion Algebras hx; yi = (N(x) + N(y) − N(x + y))=2 Hurwitz' Theorem 2 Summary N(ax) = − hax; axi = a N(x) hx; yi can be recovered from N(x). N(x) is homogeneous of degree 2. Thus hx; yi is a quadratic form. Scalar Product Spaces >Octonions? Prather Scalar Product Spaces Clifford Algebras (James Joseph) Sylvester's Rigidity Theorem: (1852) Composistion Algebras A scalar product space V over R, by an Hurwitz' Theorem appropriate change of basis, can be made Summary diagonal, with each term in {−1; 0; 1g. Further, the count of each sign is an invariant of V . Scalar Product Spaces >Octonions? Prather Scalar Product Spaces The signature of V ,(p; n; z), is the number of Clifford Algebras 1's, −1's and 0's in such a basis. Composistion Algebras Hurwitz' Theorem The proof uses a modified Gram-Shmidt process Summary to find an orthogonal basis. Any change of basis preserving orthogonality preserves the signs of the resulting basis. This basis is then scaled by 1=pjN(x)j. Scalar Product Spaces >Octonions? Prather Scalar Product This can be generalized to any field K. Spaces Let a ∼ b if a = k2b for some k 2 K. Clifford Algebras Composistion This forms an equivalence relation. Algebras Hurwitz' Theorem Summary The signature of V over K is then unique, up to an ordering of equivalence classes. A standard basis for V is an orthogonal basis ordered by signature. Scalar Product Spaces >Octonions? Prather Scalar Product Spaces Clifford Algebras Over Q the signature is a multi-set of products of finite Composistion subsets of prime numbers, plus −1. Algebras Hurwitz' Theorem Summary Any other q 2 Q can be put in reduced form then multiplied by the square of its denominator to get a number of this form. Scalar Product Spaces >Octonions? Prather Scalar Product Spaces Clifford Algebras For any finite field there are exactly 3 classes. Composistion Algebras Over F5, for example, −1 ∼ 1 so we need another Hurwitz' Theorem element for the third class. The choices are ±2. Summary Thus the signature uses {−1; 2; 0g. For quadratically closed fields, like C, the signature is entirely 1's and 0's. Scalar Product Spaces >Octonions? Prather Scalar Product Spaces A scalar product space is degenerate if Clifford Algebras its signature contains 0s. Composistion Algebras Hurwitz' Theorem x 2 V is degenerate if hx; yi = 0 for all y 2 V . Summary V is degenerate iff it contains a degenerate vector. Scaler product spaces will be assumed to be non-degenerate, unless stated otherwise. Inner Product Spaces >Octonions? Prather Scalar Product Spaces V is an inner product space if additionally: Clifford Algebras Positive definite Composistion Algebras N(x) ≥ 0 Hurwitz' Theorem N(x) = 0 iff x = 0 Summary In particular this restricts K to ordered fields. Note: If K is a subset of C symmetry is typically replaced by conjugate symmetry. This forces N(x) 2 R. Clifford Algebras >Octonions? Prather Scalar Product Spaces Let V , N(x) be a scalar product space. Clifford Algebras Let T (V ) be the tensor space over V . Composistion Algebras 2 Hurwitz' Theorem Let I be the ideal hx + N(x)i, Summary for all x 2 V . William Then the Clifford algebra over V is Kingdon Cl(V ; N) = T (V )=I Clifford Clifford Algebras >Octonions? Prather Scalar Product This shows Clifford algebras are initial, or the freest, Spaces among algebras containing V with x 2 = −N(x). Clifford Algebras Composistion Thus they satisfy a universal property. Algebras Hurwitz' Theorem Summary Further, if N(x) = 0 for all x this reduces to the exterior algebra over V . Scalar product spaces over R relate to geometry. Thus if V is over R or C these are geometric algebras. Clifford Algebras >Octonions? Prather Scalar Product Spaces Let ei be a standard basis for V . Clifford Algebras Let E be an ordered subset of basis vectors. Composistion Algebras Since I removes all squares of V , Hurwitz' Theorem Q E ei forms a basis for Cl(V ; N) Summary Let the product over the empty set be identified with 1. Thus dim(Cl(V )) = 2d , where d = dim(V ). Clifford Algebras >Octonions? Prather Scalar Product Spaces By construction: 2 Clifford Algebras ei = hei ; ei i = −N(ei ). Composistion Algebras Hurwitz' Theorem Since this basis is orthogonal: Summary hei ; ej i = 0 for i 6= j, and ei ej = −ej ei . Since Clifford algebras are associative, this uniquely defines the product. Clifford Algebras >Octonions? Prather Consider a vector space over Q with 2 2 2 Scalar Product N(ae + be + ce ) = a − 2b − 3c . Spaces 1 2 3 2 2 2 Clifford Algebras Thus e1 = −1, e2 = 2 and e3 = 3. Composistion Algebras Hurwitz' Theorem Let p = e1e2 + e2e3 and q = e2 + e1e2e3. Summary pq = e1e2e2 + e2e3e2 + e1e2e1e2e3 + e2e3e1e2e3 = 2e1 − e2e2e3 − e1e1e2e2e3 + e2e1e3e3e2 = 2e1 − 2e3 + 2e3 + 3e2e1e2 = 2e1 − 3e1e2e2 = 2e1 − 6e1 = −4e1 Clifford Algebras >Octonions? Prather Scalar Product Spaces Let Clp;n denote the geometric algebra with Clifford Algebras signature (p; n; 0). Let Cl represent Cl . Composistion n 0;n Algebras Hurwitz' Theorem Cl1 = C, Cl2 = H Summary − ∼ − ∼ Cl1;0 = C = R ⊕ R, Cl2;0 = Cl1;1 = H = M2(R). The algebras in second line, and all larger Clifford algebras have zero divisors. Clifford Analysis >Octonions? Prather Cl3;1 and Cl1;3 are used to create algebras over Scalar Product Spaces Minkowski space-time. Cl0;3 is sometimes used, Clifford Algebras with the scalar treated as a time-coordinate. Composistion Algebras Hurwitz' Theorem Physicists like to use differential operators. Summary This motivates active research in Clifford analysis. Clifford analysis gives us well behaved differential forms on the underlying space. One difficulty in Clifford analysis is the lack of the composition property. Unital Algebras >Octonions? Prather Definition Scalar Product Spaces A unital algebra A is a vector space with: Clifford Algebras bilinear product A × A ! A Composistion Algebras (x + y)z = xz + yz Hurwitz' Theorem x(y + z) = xy + xz Summary (ax)(by) = (ab)(xy) Multiplicative identity 1 2 A 1x = x1 = x For all x; y; z 2 A and a; b 2 K. In particular, associativity is not required. a 2 K is associated with a1 in A, in particular 1. Composition Algebras >Octonions? Prather Definition Scalar Product Spaces A composition algebra is a unital algebra A that is a Clifford Algebras scalar product space with: Composistion Multiplicative modulus Algebras Hurwitz' Theorem N(xy) = N(x)N(y) Summary Clifford algebras only give N(xy) ≤ CN(x)N(y). C, H, C−, and H− do have this property. Any associative algebra over a scalar product space will be a Clifford algebra. Can we get more by relaxing associativity? Hurwitz' Theorem >Octonions? Prather Hurwitz Theorem (1923) Scalar Product Spaces The only positive definite composition algebras Clifford Algebras Composistion over R are R, C, H and O. Algebras Hurwitz' Theorem Yes! We get precisely the octonions, O. Summary This can be generalized as follows: A non-degenerate composition algebra over any field (1 6= −1) must have dimension 1,2,4 or 8. − − − Adolf Over R this adds only C , H and O . Hurwitz Hurwitz' Theorem >Octonions? Prather Scalar Product We will follow the proof of Conway. Spaces Clifford Algebras Composistion The identities on the next slide Algebras follow from the definitions. Hurwitz' Theorem Summary Conway exhibits 2 line proofs of each from the prior, using non-degeneracy John H. for the last two. Conway (i.e. hx; ti = hy; ti for all t iff x = y) Hurwitz' Theorem >Octonions? Prather Scalar Product Spaces Let x = 2 hx; 1i − x. Clifford Algebras (Scaling) hxy; xzi = N(x) hy; zi and Composistion Algebras hxy; zyi = hx; zi N(y) Hurwitz' Theorem (Exchange) hxy; uzi = 2 hx; ui hy; zi − hxz; uyi Summary (Braid) hxy; zi = hy; xzi = hx; zyi (Biconjugation) x = x (Product Conjugation) xy = y x Hurwitz' Theorem >Octonions? Prather Now, if A contains a proper Scalar Product Spaces unital sub-algebra, H, we Clifford Algebras construct a Cayley-Dickson Composistion double, H + iH, within A. Algebras Hurwitz' Theorem Summary Let H be a proper unital Arthur Cayley and sub-algebra of A. Leonard Eugene Dickson Let i be a unit vector of A orthogonal to H. Let a, b, c, d and t be typical elements of H. Hurwitz' Theorem >Octonions? Prather Inner-product doubling: Scalar Product Spaces ha + ib; c + idi = ha; ci + N(i) hb; di Clifford Algebras Composistion ha; idi = ad; i = hib; ci = i; cb = 0 Algebras Hurwitz' Theorem hib; idi = N(i) hb; di Summary Conjugation doubling: (so ib = −ib = −b i = bi) a + ib = a − ib ib = 2 hib; 1i − ib = −ib Hurwitz' Theorem >Octonions? Prather Product doubling: Scalar Product Spaces (a + ib)(c + id) = (ac − N(i)db) + i(cb + ad) Clifford Algebras Composistion ha · id; ti = hid; ati = 0 − hit; adi = ht; i · adi Algebras Hurwitz' Theorem hib · c; ti = hib; tci = bi; tc = 0 − bc; ti Summary = bc · i; t = hi · cb; ti hib · id; ti = − hib; t · idi = 0 + hi · id; tbi = − hid; i · tbi = −N(i) hd; tbi = N(i) −db; t Hurwitz' Theorem >Octonions? Prather Theorem (Lemma 1) Scalar Product Spaces K = J + iJ is a composition algebra iff Clifford Algebras J is an associative composition algebra.