Sums of squares, the octonions, and (7, 3, 1)

Transcription

1 Sums of squares, the octonions, and (7, 3, 1) Ezra Virginia Tech MD/DC/VA Spring Section Meeting Stevenson University April 14, 2012

2 What To Expect Sums of squares Normed algebras (7, 3, 1) 1-, 2- and 4-square identities and their algebras 8 squares The octonions The connection with (7, 3, 1)

3 A question about sums of squares Sums of squares For which n can the product of two sums of n squares always be written as a sum of n squares? Answer (A. Hurwitz, 1898): For n = 1, 2, 4 and 8 and for no other positive integers. Theorem: Each sums-of-squares identity is associated with a normed algebra over the real numbers.

4 Normed algebras A real algebra is a vector space over R that has a vector multiplication that distributes over vector addition. A normed algebra is a real algebra A equipped with a mapping N : A R such that N(uv) = N(u)N(v) for all u, v A. The real numbers R and the complex numbers C both have such norms. We ll meet the other two shortly.

5 The (7, 3, 1) block design The (7, 3, 1) block design is: B B B B B B B a set V of 7 items, and a collection of 7 subsets of V called blocks, such that each block contains three items, each item is in three blocks, and each pair of items is in exactly one block together.

6 The incidence matrix for (7, 3, 1) The incidence matrix for (7, 3, 1) is the 7 7 (0, 1) matrix M with M ij = 1 if and only if B i contains the jth object: B B B B B B B Remember this, because it s important.

7 One and two squares One square The identity: a 2 b 2 = (ab) 2 The algebra: R, the real numbers The norm: N(r) = r 2. Two squares The identity: (a 2 + b 2 )(c 2 + d 2 ) = (ac bd) 2 + (ad + bc) 2, due to Diophantus (3rd century) and Brahmagupta (7th century) The algebra: C = {a + bi a, b R, i 2 = 1}, the complex numbers The norm: N(a + bi) = a 2 + b 2 The origins of C: solution of the cubic, differential equations, the geometric complex plane

8 Four squares The identity, due to Euler (1748): (a a a a 2 4)(b b b b 2 4) = = (a 1 b 1 a 2 b 2 a 3 b 3 a 4 b 4 ) 2 + (a 1 b 2 + a 2 b 1 + a 3 b 4 a 4 b 3 ) 2 +(a 1 b 3 a 2 b 4 + a 3 b 1 + a 4 b 2 ) 2 + (a 1 b 4 + a 2 b 3 a 3 b 2 + a 4 b 1 ) 2 The algebra: the quaternions H = {a 1 + a 2 i + a 3 j + a 4 k a 1, a 2, a 3, a 4 R, i 2 = j 2 = k 2 = ijk = 1} The norm: N(a 1 + a 2 i + a 3 j + a 4 k) = a a2 2 + a2 3 + a2 4 The origins: W. R. Hamilton, who failed in an attempt to define a multiplication on R 3, and then succeeded in defining a noncommutative multiplication on R 4.

9 Eight squares (a1 2 + a2 2 + a2 3 + a2 4 + a2 5 + a2 6 + a2 7 + a2 8 ) (b1 2 + b2 2 + b2 3 + b2 4 + b2 5 + b2 6 + b2 7 + b2 8 ) = (a 1 b 1 a 2 b 2 a 3 b 3 a 4 b 4 a 5 b 5 a 6 b 6 a 7 b 7 a 8 b 8 ) 2 +(a 1 b 2 + a 2 b 1 + a 3 b 4 a 4 b 3 + a 5 b 6 a 6 b 5 a 7 b 8 + a 8 b 7 ) 2 +(a 1 b 3 a 2 b 4 + a 3 b 1 + a 4 b 2 + a 5 b 7 + a 6 b 8 a 7 b 5 a 8 b 6 ) 2 +(a 1 b 4 + a 2 b 3 a 3 b 2 + a 4 b 1 + a 5 b 8 a 6 b 7 + a 7 b 6 a 8 b 5 ) 2 +(a 1 b 5 a 2 b 6 a 3 b 7 a 4 b 8 + a 5 b 1 + a 6 b 2 + a 7 b 3 + a 8 b 4 ) 2 +(a 1 b 6 + a 2 b 5 a 3 b 8 + a 4 b 7 a 5 b 2 + a 6 b 1 a 7 b 4 + a 8 b 3 ) 2 +(a 1 b 7 + a 2 b 8 + a 3 b 5 a 4 b 6 a 5 b 3 + a 6 b 4 + a 7 b 1 a 8 b 2 ) 2 +(a 1 b 8 a 2 b 7 a 3 b 6 + a 4 b 5 a 5 b 4 a 6 b 3 + a 7 b 2 + a 8 b 1 ) 2 Due to F. Degen (1818), J. T. Graves (1843) and A. Cayley (1845)

10 Eight squares, continued The algebra: the octonions O, where O = {a 0 + a 1 e a 7 e 7 a 0,..., a 7 R, e 2 t = 1}; the e t are called the octonion units The norm: N(a 0 + a 1 e a 7 e 7 ) = a a2 1 + a2 2 + a2 3 + a2 4 + a2 5 + a2 6 + a2 7 The origins: J. T. Graves, who received Hamilton s letter announcing the quaternions, and then went him one better by defining a noncommutative and nonassociative multiplication on R 8. The multiplication is given by the following table.

11 Multiplying the octonion units 1 e 1 e 2 e 3 e 4 e 5 e 6 e e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 1 e 1 1 e 4 e 7 e 2 e 6 e 5 e 3 e 2 e 2 e 4 1 e 5 e 1 e 3 e 7 e 6 e 3 e 3 e 7 e 5 1 e 6 e 2 e 4 e 1 e 4 e 4 e 2 e 1 e 6 1 e 7 e 3 e 5 e 5 e 5 e 6 e 3 e 2 e 7 1 e 1 e 4 e 6 e 6 e 5 e 7 e 4 e 3 e 1 1 e 2 e 7 e 7 e 3 e 6 e 1 e 5 e 4 e 2 1 This looks very strange... Multiplication table for the octonion units.

12 The table as a matrix of signs of the e i... but looks are deceptive. H =

13 The matrix, altered slightly... Remove the top row and left column, replace each 1 by a zero, and move the bottom row to the top. Here s the result:

14 ... becomes the incidence matrix for (7, 3, 1) B B B B B B B I told you it was important.

15 Octonion multiplication, explained The following are the seven blocks of the (7, 3, 1) design, with the given internal orderings: (1, 2, 4), (2, 3, 5), (3, 4, 6), (4, 5, 7), (5, 6, 1), (6, 7, 2), and (7, 1, 3). For distinct i, j {1, 2, 3, 4, 5, 6, 7}, define e i e j = e k = e j e i, where (i, j, k) is the unique block containing i and j, in the given internal ordering. What is e 4 e 6? The relevant block is (3, 4, 6), so e 4 e 6 = e 3. What is e 5 e 1? The relevant block is (5, 6, 1), so e 5 e 1 = e 1 e 5 = e 6. That s why the octonion units is a name for (7, 3, 1). But wait there s another reason.

16 O is another name for (7, 3, 1) Seven quaternion subalgebras of O O contains seven complex subalgebras C n = R e n and seven quaternion subalgebras H n = R e t, e u, e v, where {t, u, v} is a block in (7, 3, 1) Each H n contains three of the C k. Each C k is contained in three of the H n. Each pair {C k, C m } is contained in a unique H n together. The (7, 3, 1) design sits inside O The above design of subalgebras of O has the same incidence matrix as (7, 3, 1). The two designs are the same!

17 THANK YOU!