Multiplication on R n

Transcription

1 Multiplication on R n 1. Division Algebra and Vector Product Basudeb Datta Basudeb Datta received his Ph D degree from the Indian Statistical Institute in Since July 1992 he has been at the Indian Institute of Science, Bangalore. His areas of interest are Topology and Geometry. In 1843 Hamilton discovered the quaternions and in the same year Graves found an algebra with 8 basis elements. So, mathematicians 'knew the division algebra structures (over JR) on lr, ]R2, ]R4 and 1R 8 in the first half of the 19th century itself. It took more than 100 years to prove that these are all the division algebras over JR. This follows from a theorem of Adams (which we will discuss in Part 2). In this part we also discuss 'vector products' on ]R3 and JR7 and Hurwitz's theorem on 'sums of squares formulae'. Division Algebra Let 1R and C denote the set of real and complex numbers respectively. Recall the multiplication rule of complex numbers: (a + bi) (c + di) = (ac - db) + (da + bc)i. (1) Identifying ]R2 with C (by (a, b) ~ a + bi) we get a m~ltiplication on]r2 namely, Gerolamo Cardano. a famous medical doctor. philosopher and mathematician. who lived in Milan, was the first person to define complex numbers in (a, b) (c, d) = (ac - db, da + be). (1') With this multiplication, ]R2 becomes an associative division algebra lover lit In this case 1 = (1,0) is the unity for the multiplication and x-i = x/llxll for all x -=F (0,0), where (Xl, X2) = (Xl, -X2) and "(Xl, X2)" = (x~ + x~)1/2. We also know (!) that ]R4 has a multiplication given by: i 2 = j2 = k 2 = -1, i j=k=-j i, j k = i = -k j, k i = j = -i k, (2),AAAAAA 8..., V V V V..., RESONANCE I January 1998

2 where 1 = (1,0,0,0), i = (0,1,0,0), j = (0,0,1,0) and k = (0,0,0,1). In this case also x-l = x/llxll, where (Xl, x2, x3, X4) = (Xl, -x2, -x3, -X4). With this multiplication and the usual vector space structure, lr4 is an associative division algebra over lr. But you have to be careful about dividing a by b on the left or on the right! This algebra is called the algebra of quaternions and is denoted by JHI. Observe that multiplicatiqn in lhi is not commutative, whereas multiplication in reals or complex numbers is commutative. So you know why it took 304 years to define quaternions. Also, a polynomial may have infinitely many zeros in lhi. Find the zeros of x in lhi (exercise). If we identify lhi with C x C via X + yi + zj + wk = (x + yi) + (z + wi)j ~ (x + yi, z + wi), then the multiplication in (2) can be obtained from the formula (x, y) (z, w) = (x z - wy, w X + y z). (2') w here here x, y, z and ware elements of C. Now, if we take x, y, z, w E lhi in (2'), then we get a multiplication on lr s Therefore, for n E {I, 2, 4} we have the following: 1 An algebra over JR. consists of a vector space V over JR., together with a binary operation of multiplication ('. ') on the set V of vectors, such that for all a in JR. and 0, f3, 'Y in V, the following conditions are satisfied: (i) (ao) f3 = a(o. (3) = 0: (af3), (ii) (0: + J3). l' = (0:. 'Y) + (f3 '1') and (iii) 0:' (f3+'y) = (0:.f3)+ (0:. 'Y)' If moreover V satisfies (iv) (0:' (3). 'Y = 0:' (f3. 'Y) for all 0:, f3, l' in V, then V is called an associative algebra over JR.. An algebra V over lr. is a division algebra over JR. if V has a unity for multiplication and each nonzero element x has a unique left inverse XL, a unique right inverse XR, with XL = xr.this is denoted by x-i. An easy way to remember multiplication in JR.8 is: epep+1ep+: = e~ = -1 for p, q E {I,..., 7} (addition in the suffix is modulo 7). Multiplication in lr 2n can be obtained from that of JR.n by the formula: (x, y). (z, w) = (x z - wy, w x + y z). There is another way to remember the multiplication in IRs by the following rule: = e3 (e4 e6) = (e4 e5) e7 = e4 (e5 e7) = (e5 e6) el = e5 (e6 el) = (e6 e7) e2 = e6 (e7 e2) = (e7 el) e3 = e7 (el e3) = e; = - -1 = -eo and (er e s ) es = es (es er) = -er In a letter to John Graves, written on October 17, 1843, WR Hamilton announced the discovery of quaternions one day after the discovery. -RE-S-O-N-A-N-C-E--I-J-a-nu-a-r-Y ~~ ,

3 John Graves found an algebra with 8 basis elements in December In 1848 he published his discovery in the Transactions of the Irish Academy, 21, p.388.0ctonions were rediscovered by Arthur Cayley in 1845 (Collected Papers I, p.127 and Xl, p ). Because of this the octonions are known as Cayley numbers! for r, s E {I, 2,. 7}, where the identification between IRs and lhi x lhi is given by eo = (1,0,.,0) 1--+ (1,0), el = (0,1,0,,0) 1--+ (i,o), e (j,0), e (0,1), e (k,o), e (0, j), e (0, -k) and e7 = (0,.,0, 1) 1--+ (0, i). Now, (el e2) e5 = e7 i= -e7 = el (e2 e5). Therefore, this multiplication in IRs is non-associative. However, in IRs also each non-zero x has unique inverse, namely, x-i = x/llxll, where (hi, h2) = (hi, -h2), i.e., (x},,xs) = (x}, -X2, -xs). IRs with this multiplication forms a non-associative division algebra, denoted by 0, called the algebra of octonions or Cayley numbers. Observation 1: Any two octonions generate an associative sub-algebra of O. If we continue further with the formula (2') to define a multiplication on IRI6 then we get (q, e2) (e5, -e7) = (0,0). Thus we get 'divisors of zero'. So we better stop here! (If x and yare two nonzero elements such that xy = 0, then they are called divisors of zero. A division algebra does not have divisors of zero (exercise).) All these four algebras lr, C, lhi and 0 are normed algebras, i.e., \Ix 'yll = IIxll'llyll, where 1111 denotes the usual Euclidean length. IR and C are commutative algebras. IR, C and lhi are associative algebras. Thus, IR n has a division algebra structure if n E {I, 2, 4, 8}. Question: Does there exist n, other than 1, 2, 4 and 8, for which IRn has a division algebra structure over IR? This question is resolved by the following theorem. Theorem A : IRn has a division algebra structure over IR if and only if n E {I, 2,4, 8}. Remark: An examination of the proof of Theorem A (which LAA~AAA 10 v V V V v RESONANCE I January 1998

4 will be given in Part 2) will show that we have actually proved the following statement: lr n has an algebra structure over lr without divisors of zero if and only if n E {I, 2, 4, 8}. Vector Product Recall the cross product (or vector product) on JR3. given by It is i x j = k = -j x i, k x i = j = -i x k, j x k = i = -k x j, ixi=jxj=kxk=o where i, j, k here are the usual basis vectors in ]R3 and bilinearity (or distributive property)is assumed. Is it in any way related to 1HI? Yes! If we take 1R 3 = Span]R( {i, j, k}) ~ lhi, then 2 x x y = imaginary part of x y. This cross product has the following properties (x x y)..l x, (x x y)..l y and 3 IIx x Yll2 = (x,x)(y,y) - (x,y)2 4for all x, y in 1R 3 (3) (4) 2 For al,..., a p in ]Rn SpanR({a1,...,ap}) denotes the subspace generated by {al,'",ap }. 3 If (z, w) = 0 then we say z is orthogonal to wand we write z 1. w. By a vector product on ]Rn we mean a continuous mapping v : ]Rn x ]Rn ---+ ]Rn with the properties (3) and (4). More precisely, we have the following definition. Vector product: A continuous map v : ]Rn x ]Rn ---+ ]Rn is called a vector product on]rn if v(x, y)..l x, li(x, y)..l y and IIv(x,y)1I 2 = (x,x)(y,y) - (x,y)2 for all x,y E ]Rn. 4 Here (,) denotes the usual inner product (also known as 'dot product') in Rn, namely, {(x},...,xn), (Yl,...,Yn») = XIYl xnyn. In,this article '.' corresponds to multiplication in the algebra sense, not the inner product. Thus, the usual 'cross product' on R3 is a vector product in this sense. The map II : R x R ---+ ]R, given by II (x, y) = 0 for all x, y E R, is clearly a vector product on R. It is also not difficult to see that this is the only vector product on nt Consider]R7 as the 'imaginary' subspace of]r8 = 0, i.e., -RE-S-O-N-A-N-C-E--I -Ja-n-u-a-rY ~~

5 R7 also has a vector product like R3 ]R7 = SpanjR({el,.,e7})' Define v(x,y) (= x x y). the imaginary part of x' y. Then v satisfies (3) and (4). So, we have a vector product on jr7 also. These vector products on ]R3 and ]R7 have the following properties: x x y = -y x x, (5) When we replace R by c we lose the ordering but we gain the Fundamental Theorem of Algebra, namely, "every nonconstant polynomial over c has a zero in c ". And hence (thanks to the commutative property), a polynomial of degree n has exactly {if you count property} n zeros. When we consider R 4 we lose commutative property but we gain something. A polynomial may have infinitely many zeros. What a gain! x x (ay+bz) = (ax) X y+(bx) x z = x x (ay)+x x (bz), (6) IIxl12 = (x, x) = -x x, (7) (x, y) = - real part of x y and (8) x y = -(x,y) + x >< y (9) for all a, b E lr and x, y, z E lr n where n = 3 or 7. Here is the vector product table on ]R7 X el e2 e3 e4 e5 e6 e7 el 0 e4 e7 -e2 e6 -e5 -e3 e2 -e4 0 e5 el -e3 e7 -e6 e3 -e7 -e5 0 e6 e2 -e4 el e4 e2 -el -e6. 0 e7 e3 -e5 e5 -e6 e3 -e2 -e7 0 el e4 e6 e5 -e7 e4 -e3 -el 0 e2 e7 e3 e6 -el e5 -e4 -e2 0 This table also gives (see (9)) the multiplication in O. Question: Does there exist integer n other than 1, 3 and 7 for which lr n has a vector product? This question is again answered by the following theorem. Theorem L: jrp has a vector product if and only if p E {I, 3, 7}. Sums of Squares Formulae We all know the sums of squares formulae (a~ + a~)(a~ + a~) = Al = alai - a2a2 and (10) ~~ R-E-S-O-N-A-N-C-E-I-J-a-nu-a-~

6 So, product of sums of squares of integers is a sum of square of two integers. (10) follows from the fact that (11) for any two complex numbers z and w. As the multiplication in lh[ is also norm preserving, if we take quaternions in place of complex numbers in (11) we get 5 = where Al = alo::l - a20::2 - a30::3 - a40::4 A2 = a10::2 + a a a40::3 (12) Similarly, from norm preserving octonian multiplication we get 5 Euler discovered formula (12) in 1748, much earlier than Hamilton's invention of quaternions, while investigating the theorem (later proved by Lagrange in 1869) that 'every positive integer is a sum of four integral squares'. Euler showed, by proving formula (12), that it is sufficient to prove the theorem for every prime. So, Euler knew quaternion multiplication! where AI,,As are given by [AI. As] = [at. as] [M] (13) eq 0'2 03 0'4 05 0'6 0'7 as [M] = os 0'3-07 0'6 0::4-0' ' as 0'7-0'4 as 0'6 0'1-0'7-0'3 0'5-0' os ' os 0' os -0'5 0'4 0'2 0'1-0'3 -as '2-0'6 0'5 0'3 0'1 Observe that, for n = 2,4 and 8, AI,..,An are linear in ai,.,an and also in 01,.,On in (10), (12) and (13) respectively. So, it is natural to ask for the values of n for which there exists an identity of the form (a~ a~)( a~ + + a~) = A~ A~, where AI,.,An are linear in -R-ES-O-N-A-N--CE--I-J-Q-nu-Q-r-Y ~~

7 al,-,, an and also in ai",, an. In 1896, A Hurwitz proved the following: Theorem S: If there exists an identity of the form (a~ +.. +a~)(a~+... +a~) = A~+... +A~, where AI,.., An are linear functions of.al,..., an and ai,.., an then n = 1,2,4 or 8. 6 A map 1 : R m x R n _ RP is called bilinear if f(alal + a2a 2, f3) = al/(al, f3)+ a2/(a2, f3) and I(a, bl /3l + b2f32) = bl/(a, /3d+b2/(a, /32) for all a, ai, a2 E Rm /3,/31,/32 E JR.n andal,a2,bl, b2 E JR.. Address for Correspondence Bosudeb Datta Department of Mathematics Indian Institute of Science Bongolore , India. dottob@moth.iisc.ernet.in In 1923, A Hurwitz proved the following stronger result (Theorem N). Any positive integer n can be expressed uniquely as: n = 2 4 0:+.8(21'+1), where 0 ~ (3 ~ 3. Let k(n) := for that n. Theorem N:If there exists a bilinear 6 map such that f : JR.m X JR.n -4 JR.n such that IIf(y,x)1I = liyll'lixli for ally E IR m and x E JR.n then m ~ k(n). Remark: In fact, Hurwitz showed (for a different proof see pages 140 and 156 in Husemoller) that for each n there is a bilinear map f : JR.m x JR.n -4 JR.n satisfying II f (y, x) II = liyll IIxll for m = k(n) but not for bigger m. Theorem N also follows from another theorem of Adams. We will discuss all the proofs in Part 2 of this article. Suggested Reading D HusemoUer. FibreBtmdla. Springer-Verlag, ~ I I' Nowadays superelastic alloys are available that behave like rubber and are able to endure huge elastic deformations - two orders of magnitude greater than ordinary metals. On the other hand, many kinds of alloys can be brought to a super-elastic state, when they flow under very low pressure like heated glues. Quantum Kaleidoscope. pp.33. Sept-Oct, ~~ ISO---NAN---Cl--I-~-n-u-a-~