U.U.D.M. Project Report 2011:1 Generation of the classical groups SO(4) and SO(8) by means of unit quaternions and unit octonions Karin Nilsson Examensarbete i matematik, 15 hp Handledare och examinator: Ernst Dieterich Januari 2011 Department of Mathematics Uppsala University Generation of the classical groups SO(4) and SO(8) by means of unit quaternions and unit octonions Karin Nilsson January 10, 2011 Abstract The unit circle in the complex plane, viewed as a group, is well-known to be isomorphic to the matrix group SO(2). We explain how this isomorphism can be viewed as a special case of a more general context relating the unit sphere in a unital absolute valued algebra A to the group SO(A). In case A is the quaternion algebra, this yields an explicitly described set of matrices generating SO(4), such that every matrix in SO(4) has length at most two. In case A is the octonion algebra, it yields an explicitly described set of matrices generating SO(8), such that every matrix in SO(8) has length at most seven. Acknowledgements I would like to thank my supervisor Ernst Dieterich for introducing me to the subject as well as helping me through it, dedicating a lot more time than I could ever ask for. I have learnt a lot and I am greatful for it. I would also like to thank my closest friends and family for always motivating me. 1 Contents 1 Introduction 3 2 A connection between O(n) and the Euclidean space E 3 3 A description of the group O(E) 5 4 The algebras R; C; H and O 7 4.1 The algebra R ............................ 7 4.2 The algebra C ............................ 7 4.3 The algebra H ........................... 8 4.4 The algebra O ........................... 8 4.4.1 Multiplication laws in O .................. 9 4.5 Two important results . 11 5 The mappings L and R 12 5.1 A description of S(A)........................ 12 6 Properties of the mappings L and R 15 6.1 Case A = C ............................. 15 6.2 Case A = H ............................. 16 6.3 Case A = O ............................. 16 7 The case Im(A) 17 8 Description of O(A) 20 8.1 The case A = R ........................... 20 8.2 The case A = C ........................... 20 8.3 The case A = H ........................... 20 8.4 The case A = Im(H)........................ 21 8.5 The case A = O ........................... 21 8.5.1 Isotopies and companions . 22 8.6 The case A = Im(O)........................ 25 2 1 Introduction The goal of this thesis is to describe O(n) and SO(n) for n = 1; 2; 3; 4; 7 and 8 by using properties of the algebras R, C, H and O. The orthogonal group O(n) is the group of real n × n matrices A such that A−1 = AT . As we will see it is possible to view O(n) as the group of n×n all length-preserving linear operators on R , i.e. the group O(E) for an Euclidean vector space E with dim(E) = n. The special orthogonal group SO(n) is the subgroup of O(n) of all n × n real matrices with determinant 1. We will se that SO(n) is isomorphic to the n×n group of all length-preserving linear operators on R with determinant 1, i.e. the group SO(E) for an n-dimensional Euclidean vector space E. Soon we will notice that O(E) is generated by reflections, and as it turns out theese reflections can, for all of the n, be describe solely using leftmultipli- cation by a unit La : E ! E (x 7! ax where kak = 1), rightmultiplication by a unit Ra : E ! E (x 7! xa where kak = 1) and the konjugation κ: E ! E (x 7! x¯ = 2hx; 1i − x). On the other hand, by a result from Ernst Dieterich and Erik Darp¨oin [4], we know that La;Ra 2 SO(n) for n 2 f2; 4; 8g. We will arrive at a constructive description of O(E) and SO(E). 2 A connection between O(n) and the Eu- clidean space E Let E be a Euclidean vector space, i.e. a finite dimensional vector space endowed with a scalar product h ; i. We have the following definition. Definition 2.1. L(E) = ff : E ! E j f is linearg GL(E) = ff 2 L(E) j f is invertibleg O(E) = ff 2 L(E) j kf(x)k = kxkg SO(E) = ff 2 L(E) j kf(x)k = kxk and det(f) = 1g It can be proved that GL(E), O(E) and SO(E) are groups under compo- sition and that SO(E) < O(E) < SO(E). Let V be a finte dimensional vector space. Let f 2 L(V ) and let e = (e1; :::; en) be a basis of V. Then we define the matrix [f]e = [[f(e1)]e:::[f(en)]e]n×n as the matrix-representation for f and e, where the k:th column is the matrix of f(ek) for the basis e. I.e. if f(ek) = a1ke1 + ::: + ankek, 2 3 a1k 6 : 7 6 7 we define [f(ek)] = 6 : 7, where aij 2 R 8i; j 2 f1; :::; ng. 6 7 4 : 5 ank 3 Proposition 2.2. For V, an Euclidean space, and e = (e1; :::en) an ON-base n×n in V, the mapping ' : L(V ) ! R , '(f) = [f]e is bijective. Proof. For a choosen basis, an operator f 2 L(V ) is uniquely determined by f(e1); :::; f(en) and so [f]e as well is uniquely determined for each f 2 L(V ). Conversely to every matrix corresponds an uniquely determined linear operator, namely the one taking ek 7! f(ek). So ' is bijective. Definition 2.3 (The determinant of a linear operator). Let a = (a1; :::; an) be a basis for a finite dimensional vector space, let f 2 L(V ) and define the determinant det(f) of f as det(f) := det([f]a) The determinant is independent of basis. To see this let a; b be two differ- n×n −1 ent basis for V. Then there exists a T 2 R such that [f]a = T [f]bt and −1 −1 −1 det([f]a) = det(T [f]bT ) = det(T )det([f]b)det(T ) = det(TT )det([f]b) = det([f]b). The next proposition enables us to study O(V) and SO(V) instead of directly looking at O(n) and SO(n). But first we need a lemma. Lemma 2.4. Let f 2 L(V ), for a finite dimensional vector space V , let A = '(f) (where ' is the mapping defined in proposition 2.2) and let e = (e1; :::en) T be an ON-basis for V. Then hf(ei); f(ej)i = (A A)ij Proof. P P hf(ei); f(ej)i = h µ Aµieµ; ν Aνjeνi = 2 2 ha1ie1 + ::: + anien; a1je1 + ::: + anjeni = a1ia1je1 + ::: + anianjen = P P P µ,ν AµiAνjhei; eji = µ,ν AµiAνjδµν = µ AµiAµj = P T T µ(A )iµAµj = (A A)ij: n×n Proposition 2.5. The mapping ' : L(V ) ! R , '(f) = [f]e induces the following isomorphisms of groups: (i) GL(V )! ~ GL(n) (ii) O(V )! ~ O(n) (iii) SO(V )! ~ SO(n) Proof. We will prove that the intended mappings are well defined. Then, as ' is bijective, so will 'jGL(v) (for case (i)), 'jO(V ) (for case (ii)) and 'jSO(v) (for case (iii)) be. (i) ()) Let f 2 GL(V ). Then there exists a g 2 GL(V ) such that fg = gf = I. But [f]e[g]e = [fg]e = In = [gf]e = [g]e[f]e, i.e [f]e 2 GL(n) (() Let A 2 GL(n). Then there exists a B 2 GL(n) such that AB = BA = In. Now due to proposition 2.2 there exists unique linear operators f, g, h1 and h2 such that A = [f]e, B = [g]e, h1 = [AB]e and h2 = [BA]e. Then fg = h1 = I = h2 = gf and therefore f 2 GL(V ). 4 (ii) ()) Let f 2 O(V ) and let e = (e1; :::; en) be an ON-basis for V. Then hf(ei); f(ej)i = hei; eji = δij, where the first equality holds because f is orthogonal and the second because e is an ON-basis. Now by lemma 2.4 T T T hf(ei); f(ej)i = ([f]e [f]e)ij, so that we have δij = ([f]e [f]e)ij, i.e [f]e [f]e = In and so [f]e 2 O(n) (() Let A 2 O(n) and f 2 L(V ) such that A = '(f). Then hei; eji = δij = T (A A)ij = hf(ei); f(ej)i, where the first equality holds because e is an ON- basis, the second equality holds because A is orthogonal and the last equality comes from lemma 2.4. This means that hf(ei); f(ej)i = hei; eji, which is equivalent to kf(x)k = kxk 8x 2 V so that f 2 O(V ). (iii) Above we defined det(f) := det([f]a) for any basis a. This means that det(f) = 1 if and only if det([f]e) = 1. 3 A description of the group O(E) Let V be an Euclidean space. Lemma 3.1. Let f 2 O(V ). Then det(f) 2 f1; −1g T Proof. As f 2 O(V ) we have that [f]a[f]a = 1n for some basis a = (a1; :::an) T T 2 of V .