U.U.D.M. Project Report 2009:9
Examensarbete i matematik, 30 hp
Handledare och examinator: Ernst Dieterich Juni 2009
Department of Mathematics Uppsala University
Four-dimensional absolute valued algebras
Love Forsberg
Abstract
An absolute valued algebra is a real algebra, endowed with a multiplicative norm. A classical result of Albert (1947) states that every finite dimensional absolute valued algebra has dimension 1, 2, 4, or 8. The absolute valued algebras of dimension 1 or 2 are well understood, and those of dimension 4 have been exhaustively described by Ram´ırez (1999), and by Calder´on and Mart´ın (2005).
We present a categorical interpretation of the latter results by prov- ing that the category of all 4-dimensional absolute valued algebras decomposes into four full subcategories, each of which is equivalent to the category of the SO(3)-action on SO(3) × SO(3) by simultaneous conjugation. A classification of all four-dimensional absolute valued algebras is easily derived from that insight.
Acknowledgment
I want to thank my advisor Ernst Dieterich for offering advice every time I needed it, and many more.
Also I’d like to thank my friends at Uppsala University for being there, especially Valentina and Johan, for help with proof reading.
Contents
1 Introduction 1
2 Generalities 2
3 Decomposition of A2,A4 and A8 3
4 The main categories 6
5 The main functors 7
5.1 Fi :E S3×S3 → Wi . . . 7 5.2 G :E S3×S3 →SO(3) SO(3) × SO(3) . . . 8
6 Useful lemmata 8
7 Morphisms between Wi(a, b) and Wi(c, d) 9
8 Properties of the functors 12
8.1 Fi . . . 12 8.2 G . . . 12
9 Classifying A4 12
10 Automorphism groups of four dimensional absolute-valued
algebras 13
11 A4 as a two step reduction 14
1 Introduction
The study of division algebras has come a long way since Albert proved that all finite dimensional absolute valued algebras are isomorphic to isotopes of
R,C,HorO[1]. So what is a division algebra? To know this we need to know what an algebra is. There are several definitions commonly used, but we chose a fairly general definition. An algebra A = (A, ∗) over a field k is a vector space over k equipped with a k−bilinear product ∗ : A × A → A. The algebra is said to have the same dimension as the underlying vector space.
In every algebra the maps x 7→ a ∗ x and x 7→ x ∗ a, denoted La and Ra respectively, are linear maps. A division algebra is a nonzero algebra A in which La and Ra are invertible maps, for all nonzero a ∈ A.
So far, all ground fields have been possible, but we now restrict ourselves toR. Algebras overR are commonly called real algebras. A famous result of Hopf [17], Kervaire [19], Bott and Millnor [3] states that every real division algebra has dimension 1, 2, 4 or 8. There is, up to isomorphism, only one real division algebra of dimension 1, namely R itself. Dieterich and others have found cross-sections for the isoclasses of the two-dimensional real division al- gebras [14][4][16][18][8], but in dimensions 4 and 8 the problem of classifying the real division algebras is so far only partially understood. Certain
subcategories of real division algebras with extra structure are classified, such as the flexible real division algebras, completed by Darp¨o [9][10], four- dimensional quadratic real division algebras, completed by Dieterich [11][12][13], and absolute valued algebras with a one-sided unity or a central idempotent, completed by Cuenca, Darp¨o and Dieterich [7]. We will consider real division algebras with the extra structure of being absolute valued algebras.
An absolute valued algebra is a nonzero real algebra (A, ∗) equipped with a norm || · || such that ||x ∗ y|| = ||x|| · ||y|| for all x, y ∈ A. It turns out that all finite dimensional absolute valued algebras are division algebras, but that not all division algebras are absolute valued algebras.
Ram´ırez has presented a rather big set of 4-dimensional absolute valued algebras exhausting all isoclasses, together with a workable neccesary and sufficient criterion for when two algebras in her set are isomorphic [2]. She did not, however, remove superfluous algebras in order to create a cross- section. Calder´on and Mart´ın worked with her results and came up with a list almost free from redundancies, however with more complicated criteria for isomorphism in the redundant cases [5]. Both lists are parametrized by subsets of S3×S3, which itself has the structure of a topological group, while no structure is found in the lists.
On the last day before print the author of this thesis learned of one more, considerably earlier, supposed classification (complete or almost complete)
by Stampfli-Rollier [20]. However, due to shortage of time and lack of understanding of german of behalf of the author of this thesis, it’s up to the reader to verify how complete the classification is.
What I want to achieve is a structural understanding that gives some insight into the classification of four dimensional absolute valued algebras.
One way to work with structure is by using category theory. We create the categories Dn and An of the n−dimensional division algebras and absolute valued algebras respectively. The morphisms in these categories are under- stood to be the algebra isomorphisms.
2 Generalities
For all real vector spaces V we may consider the general linear group GL(V ) of the bijective linear operators, and the orthogonal group O(V ) of orthogonal operators. The latter can be partitioned into O+(V ), the subgroup formed by orthogonal linear operators with determinant 1, and O−(V ), the subset formed by orthogonal linear operators with determinant −1. The group O+(V ) is also commonly known as the special orthogonal group SO(V ). We also use SO(n), the group of n × n orthogonal matrices.
In every real algebra A with unity e, one can define imA, the imaginary elements, as imA := {a ∈ A \Re|a∈Re} ∪ {0}. In general imA is a cone, but not a subspace. However, due to Frobenius, if A is quadratic, imA is a subspace that complements Re[15].
For every algebra (A, ∗) and every pair (σ, τ ) ∈ GL(A)2there is an isotope of A with same space as A and multiplication (x, y) 7→ σ(x) ∗ τ (y). The isotope is denoted Aσ,τ.
It is obvious that division algebras (over any field) are free from zero divisors. However, for finite dimensional division algebras the converse is also true. Let (A, ∗) be a finite dimensional nonzero algebra without zero divisors, and let a, b ∈ A \ {0}. Assume La = Lb, that is ax = bx, for all x ∈ A. Because ∗ is bilinear it is equivalent with (a − b)x = 0 for all x ∈ A.
But A has no zerodivisors and is nonzero this implies a − b = 0. Thus La is injective, but since A is finite dimensional, injective linear operators are bijective. A similar argument holds for Ra.
Using this simple criterion, we can verify that isotopes of a finite
dimensional division algebra are again division algebras. Let (A, ∗) be a division algebra, (Aσ,τ, ×) an isotope of A, and x, y ∈ A. Then x × y = 0 iff σ(x) ∗ τ (y) = 0. A is a division algebra, so σ(x) ∗ τ (y) = 0 implies σ(x) = 0 or τ (y) = 0. Since σ, τ are bijective linear operators σ(x) = 0 is equivalent to x = 0, and τ (y) = 0 is equivalent to y = 0.
2
It should be noted that Albert used a more general notion of isotopy, but the same isotopes are gained, up to isomorphism.
Isotopes Aσ,τ with σ, τ ∈ O(A) are, in this thesis, called orthogonal iso- topes. Since orthogonal maps preserve norms, we can easily verify that orthogonal isotopes of absolute valued algebras are absolute valued. Let (Aσ,τ, ×) be an orthogonal isotope of an absolute valued algebra (A, ∗)and let x, y ∈ A. Then ||x × y|| = ||σ(x) ∗ τ (y)|| = ||σ(x)|| · ||τ (y)|| = ||x|| · ||y||
A handy way to work with multiplication in H is by using that imH is equipped with crossproducts ×. Let α1 + u, β1 + v ∈ R⊕ imH. Their product can be written as (α1 + u)(β1 + v) = (αβ− < u, v >)1 + αv + βu + u × v.
Let x = γ1 + w, where < ·, · > is understood as the euclidean scalar product.
Then x¯x = (γ2− < w, −w >)1 + γ(−w) + γw + w × (−w) = (γ2+ < w, w >
)1 = ||x||21. Therefore x−1 = ||x||x¯2 for all nonzero x in H. This gives the nice form x−1 = ¯x for all x ∈ S3. Because H is associative and absolute valued, 1 ∈ S3 and all x ∈ S3 has inverses in S3, S3 forms a group (with respect to the inherited multiplication).
There will be quite a few situations where conjugation in a group is in- volved. Therefore, we introduce the notation κx : H → H, y 7→ xyx−1, for conjugating y ∈ H with x ∈ G, where G ⊂ H forms a group by the inherited multiplication inherited from an associative multiplicative structure in H.
3 Decomposition of A
2, A
4and A
8Lemma 3.1. Let A ∈ {R,C,H,O}, and a ∈ A \ {0}. Then det(Ra2) > 0 and det(La2) > 0.
Proof. Ra2(x) = x(aa) = (xa)a = R2a(x), ∀x ∈ A, where the middle equality comes from A being alternative. Thus Ra2 = R2a. But since A is a division algebra, right multiplication with any nonzero element is invertible and thus det(Ra) 6= 0. This in turn implies det(Ra)2 > 0, since R is the ground field. But det(Ra2) = det(R2a) = det(Ra)2. A similar argument holds for left multiplication.
Let A ∈ {C,H,O}, and n = dim A. For all (i, j) ∈ {−, +}2 we define the full subcategories
Anij := {B ∈An|B ˜→Aσ,τ for some σ ∈ Oi(A), τ ∈ Oj(A)}
of An.
Theorem 3.2. For all n ∈ {2, 4, 8}, the equality `
(i,j)∈{−,+}2Anij = An of categories holds. Also, the categories An−+ and An+− are isomorphic.
Proof. Albert showed that every finite dimensional absolute valued algebra is isomorphic to an isotope of R,C,H or O[1]. Thus the union of the object classes of the Anij is the object class of An for all n ∈ {2, 4, 8}.
To show that the object classes of the Anij are pairwise disjoint, we assume that Anij ∩ Ankl 6= ∅. Let D be an algebra in the intersection.
Then, by definition, there are algebras Aa,b ∈ Anij and Ac,d ∈ Ankl such that Aa,b←D ˜˜ →Ac,d. Let ∗ and × denote the algebra multiplications in Aa,b and Ac,drespectively, let ϕ be an isomorphism ϕ : Aa,b→A˜ c,d, and let · denote the multiplication in A.
ϕ(x ∗ y) = ϕ(x) × ϕ(y), ∀x, y ∈ A ⇐⇒
ϕ(a(x) · (b(y))) = c(ϕ(x)) · d(ϕ(y)), ∀x, y ∈ A ⇐⇒
a(x) · b(y) = ϕ−1(c(ϕ(x)) · d(ϕ(y))), ∀x, y ∈ A
We specify x = ϕ−1(c−1(1)) and y = ϕ−1(d−1(1)) respectively, where 1 is the unity in A. To simplify notation, we set c1 := a(ϕ−1(c−1(1))) and c2 := b(ϕ−1(d−1(1))). We get
c1· b(y) = ϕ−1dϕy, ∀y ∈ A, and a(x) · c2 = ϕ−1cϕx, ∀x ∈ A, respectively.
We may rephrase these identities as
Lc1b(y) = ϕ−1dϕy, ∀y ∈ A, and Rc2a(x) = ϕ−1cϕx, ∀x ∈ A, respectively.
Since the equations hold at every point in A, the operators are equal. By taking the determinants of the linear operators, and using the
multiplicativity of determinants we get det(Lc1) det(b) = det(ϕ−1) det(d) det(ϕ) and det(Rc2) det(a) = det(ϕ−1) det(c) det(ϕ), or equivalently det(Lc1) det(b) = det(d) and det(Rc2) det(a) = det(c).
Because imA is orthogonal toR1, and A has dimension at least 2, we may write x ∈ A as x = ||x||(cos(α)1 + sin(α)u) for some purely imaginary u of unit length, and some α ∈ [0, 2π]. Thus
x2 = ||x||2(cos2(α)1 + cos(α) sin(α)u + sin(α) cos(α)u − sin2(α)1) =
||x||2(cos2(α)1 − sin2(α)1 + 2 sin(α) cos(α)u) = ||x||2(cos(2α)1 + sin(2α)u).
Thus p||x||(cosα21 + sinα2u) is a square root of x as above, and all x ∈ A are squares. In particular, c1 and c2 are squares, so we may use the previous
4
lemma to obtain sign(det(a)) = sign(det(c)) and sign(det(b)) = sign(det(d)).
Thus (i, j) = (k, l).
Because An is a groupoid and the Anij are closed under isomorphisms, there are no morphisms between algebras in Anij and algebras in Ankl if (i, j) 6= (k, l). Thus we have finished the proof of the first part.
For the second claim we need the notion of an opposite algebra. Let (B, ∗) be an algebra over any field. Its opposite algebra is the algebra with the same space as B but with multiplication (x, y) 7→ x∗opy := y ∗x. As soon as domains and codomains contain corresponding algebras and morphisms, we may form an op functor that takes algebras to their opposites and algebra morphisms to themselves, viewed as linear maps: Assume ϕ is an algebra morphism between (B, ∗) and (C, ×) Then ϕ(x ∗op y) = ϕ(y ∗ x) = ϕ(y) × ϕ(x) = ϕ(x) ×opϕ(y).
To prove the last claim we need to show that if B ∈Anij then Bop ∈Anji. By definition, if B ∈Anij there are σ ∈ Oi(A), τ ∈ Oj(A) such that B ˜→Aσ,τ. Using the arguments above we get that Bop→(A˜ σ,τ)op. We take a look at the multiplication in (Aσ,τ)op: (x, y) 7→ σ(x) ·op τ (y) = τ (y) · σ(x). This is the same as the multiplication in (Aop)τ,σ. Thus we only need to show that A ˜→Aop.
Let α1 + u, β1 + v ∈ R1 ⊕ imA be general elements in A. Let < ·, · >
denote the standard scalar product in Rn and let × denote the zeromap if A = C, or the cross product in imA otherwise. Then (α1 + u)(β1 + v) = (αβ− < u, v >)1 + αv + βu + u × v. Since < ·, · > is bilinear and symmetric, and × is bilinear and antisymmetric we see that the conjugation in A is an isomorphism between A and Aop:
(α1 + u)(β1 + v) = (αβ− < u, v >)1 + αv + βu + u × v = (αβ− < u, v >)1 − αv − βu − u × v =
(βα− < −v, −u >)1 + β(−u) + α(−v) + (−v) × (−u) = (β1 − v)(α1 − u) = (β1 + v)(α1 + u)
Thus the second claim is proven.
We say that algebras in the same subcategoryAnij have equal sign type.
Notice that this construction has potential for generalization. By
replacing Oi(A) and Oj(A) with GLi(A) and GLj(A) and taking n−dimensional real division algebras instead of absolute valued algebras, we get a statement about the class of n−dimensional real division algebras that are isotopic to A, closed under isomorphism. The proof is completely analogous to the one above. Also, since every real division algebra is isotopic to some real division
algebra with unity, we see that we can do the construction whenever we have a n−dimensional unital real division algebra A such that det(Ra), det(La) > 0 for all a ∈ A. (We need to modify the proof for the second part, but assum- ing an isomorphism between (Aa,b)op and Ac,d we may use the same strategy, evaluating at inverse images of the unity, as in the first claim.)
4 The main categories
To deal withAnwe need a few more categories. Ram´ırez’ results contain some structure on the classifying set on four dimensional absolute valued algebras, that we want to take with us to categories. We therefore use a construction that takes a group action (G, M ) and creates the categoryGM whose objects are the elements of M and whose morphisms are triples (g, x, y) : x → y such that gx = y.
We define a couple of categories heavily inspired by Ram´ırez and then work out some relations between them and certain other categories. Since we are mainly interested in the objects, the morphisms in these categories will be taken as algebra isomorphisms.
H Certain orthogonal isotopes H(a, b) ofH, with multiplication (x, y) 7→
axyb for some (a, b) ∈S3×S3.
H∗ Certain orthogonal isotopes H∗(a, b) ofH, with multiplication (x, y) 7→
axb¯y for some (a, b) ∈S3×S3.
∗H Certain orthogonal isotopes∗H(a, b) ofH, with multiplication (x, y) 7→
¯
xayb for some (a, b) ∈S3×S3. H*
Certain orthogonal isotopes H*
(a, b) ofH, with multiplication (x, y) 7→
a¯x¯yb for some (a, b) ∈ S3×S3.
From a theorem by Cayley [15, page 215] it easy to se that all algebras in any one of the above categories have the same sign type, and that the categories have different sign type. Ram´ırez proves that any four-dimensional algebra is isomorphic to an algebra contained in one of the categories [2]. For enumerability, we will call the categories W1through W4 from top to bottom.
Also, we will denote an object in Wi by Wi(a, b). It should be clear (by sign type arguments) that i 6= j implies that Wi(a, b) can not be isomorphic to Wj(c, d).
Ram´ırez mentiones, without proof, a neccesary and sufficient criterion for the existance of isomorphisms between Wi(a, b) and Wi(c, d) and we will
6
therefore study the set S3×S3 as a category. We define the group
E := C2 × C2 × (S3/ < −1 >) and the group action of E on S3 ×S3 as (ε, δ, [p])(a, b) = (εpa¯p, δpb¯p). We need to check if it is welldefined, but [p] = [p0] ⇒ p = ±p0 and px¯p = (−p)x(−p) solves the potential problem. It satisfies the axioms of a group action:
(1, 1, [1])(a, b) = (1 ∗ 1 ∗ 1 ∗ a ∗ ¯1, 1 ∗ 1 ∗ 1 ∗ b ∗ ¯1) = (a, b), ∀(a, b) ∈S3 ×S3
(ε, δ, [p])((ε0, δ0, [q])(a, b)) = (ε, δ, [p])(ε0qa¯q, δ0qb¯q) = (εpε0qa¯q ¯p, δpδ0qb¯q ¯p) = (εε0pqapq, δδ0pqbpq) =
(εε0, δδ0, [pq])(a, b), ∀(ε, δ, [p]), (ε0, δ0, [p0]) ∈ H, (a, b) ∈S3×S3
Now we define the category HS3×S3 as the one induced by the group action (E,S3×S3) using the construction described earlier.
For several reasons, among them a geometric understanding, we will relate
S3×S3 to SO(3) × SO(3). We will use a group action to define this category aswell. First we consider the action of SO(3)×SO(3) on itself by conjugation.
Then we identify the diagonal subgroup {(x, x)|x ∈ SO(3)} with SO(3) for notational reasons and consider the category induced by the action of SO(3) on SO(3) × SO(3).
5 The main functors
5.1 F
i:
E S3×
S3→ W
iFrom now on, it is understood that i ∈ {1, 2, 3, 4} inFi, to simplify notation.
We define the functor Fi :E S3 ×S3 → Wi by it’s action on objects and morphisms:
Fi(a, b) = Wi(a, b),Fi(ε, δ, [p]) = εδκp.
Proof that it is welldefined is needed for the action on morphisms as it is not clear that εδκp is uniquely defined from (ε, δ, [p]) and that it is a morphism between Wi(a, b) and Wi(εpa¯p, δpb¯p). We then need to check the axioms of functors. The uniqueness follows from the facts that [p] = [p0] ⇒ p = ±p0 and px¯p = (−p)x(−p). The other part to check, in order forFi to be welldefined, is straightforward verification:
H: εδκp(x ◦ y) = εδκp(axyb) = εδpaxyb¯p = εδpa¯pxy = εδpa¯ppx¯ppy ¯ppb¯p = εδεcpx¯ppy ¯pδd = c(εδpx¯p)(εδpy ¯p)d = εδκp(x) ∗ εδκp(y)
∗H: εδκp(x ◦ y) = εδκp(¯xayb) = εδp¯xayb¯p = εδp¯x¯ppa¯ppy ¯ppb¯p = εδp¯x¯pεcpy ¯pδd = εδp¯x¯pcεδpy ¯pd = εδκp(x) ∗ εδκp(y)
H∗: εδκp(x ◦ y) = εδκp(axb¯y) = εδpaxb¯y ¯p = εδpa¯ppx¯ppb¯pp¯y ¯p = εδεcpx¯pδdp¯y ¯p = εδcpx¯pdεδp¯y ¯p = εδκp(x) ∗ εδκp(y)
H*
: εδκp(x ◦ y) = εδκp(a¯x ¯yb) = εδpa¯x ¯yb¯p = εδpa¯pp¯x¯pp¯y ¯ppb¯p = εδεcp¯x¯pp¯y ¯pδd = cεδp¯x¯pεδp¯y ¯pd = εδκp(x) ∗ εδκp(y)
A straightforward check that Fi satisfies the axioms for a functor:
Fi(1, 1, [1]) = 1 ∗ 1 ∗ κ1 = identity Fi((ε, δ, [p])(ε0, δ0, [p0])) =Fi(εε0, δδ0, [pp0]) = εε0δδ0κpp0 = εδκp◦ ε0δ0κp0 =Fi(ε, δ, [p]) ◦Fi(ε0, δ0, [p0]) where ◦ reads as composition of maps.
5.2 G :
E S3×
S3→
SO(3)SO(3) × SO(3)
Definition of G by action on objects and morphisms:
G (a, b) = (κa, κb),G (ε, δ, [p]) = κ0p,
where κais understood as a function on the three-dimensional space imHfor a ∈H.
Because of the multiple uses of κxas conjugation with x we need to clarify what κ0p does:
κ0p(ϕ, ψ) = (κp◦ ϕ ◦ κp¯, κp ◦ ψ ◦ κp¯)
Straight from the definition, there seems to be a problem with codomains in thatG ((ε, δ, [p])(a, b)) = G (εpa¯p, δpb¯p) = (κεpa ¯p, κδpb ¯p) andG (ε, δ, [p])(G (a, b)) = κ0p(κa, κb) = (κpa ¯p, κpb ¯p). However κx = κ−x, so there is no problem.
To proveG is a functor we need to prove some properties. It takes identi- ties to identities: G (1, 1, [1]) = κ01 = identity. Since G acts forgetfully about ε, δ we will simplify notation by ommitting them. G ([p][q]) = G ([pq]) = κ0pq = κ0p ◦ κ0q=G ([p]) ◦ G ([q]).
6 Useful lemmata
Lemma 6.1. Let G be a group, and Z(G) it’s center. Then axb = cxd, ∀x ∈ G ⇒ ∃z ∈ Z(G) : a = zc ∧ b = z−1d.
8
Proof. We reformulate the identity:
axb = cxd, ∀x ∈ G ⇐⇒
c−1ax = xdb−1, ∀x ∈ G
In particular, this should hold for x = e, which gives c−1a = db−1. Let z := c−1a. Then zx = xz, for all x ∈ G and thus z ∈ Z(G). Moreover, z = c−1a ⇐⇒ a = zc and z = db−1 ⇐⇒ b = z−1d.
Lemma 6.2. Let G be a group. If there exists a, b, c, d, f and g ∈ G such that axbyc = dyf xg, for all x, y ∈ G then G is abelian.
Proof. Assume there exists a, b, c, d, f and g ∈ G such that axbyc = dyf xg, for all x, y ∈ G. We rewrite the identity as d−1axby = yf xgc−1. By evalu- ating in y = e we get d−1axb = f xgc−1, for all x ∈ G. We use the previuos lemma to obtain d−1a = zf and b = z−1gc−1 for some z ∈ Z(G). Thus zf xby = yf xzb, for all x, y ∈ G, or, since z commutes with everything, f xby = yf xb, for all x, y ∈ G. Now we evaluate in x = f−1 to obtain f f−1by = yf f−1b, for all y ∈ G, which gives by = yb, for all y ∈ G. Thus b ∈ Z(G) and we may shorten our identity to f xy = yf x, for all x, y ∈ G.
Evaluating at x = e gives f y = yf, for all y ∈ G. We may once again shorten our identity and obtain xy = yx, for all x, y ∈ G.
7 Morphisms between W
i(a, b) and W
i(c, d)
The purpose of this section is to prove results that were published without proof by Ram´ırez[2] in order to use them later, when we prove certain nice properties of the functors.
Theorem 7.1. Wi(a, b) and Wi(c, d) are isomorphic precicely if there exists ε, δ ∈ {±1} and p ∈ S3 such that (c, d) = (εpa¯p, δpb¯p). In that case, all isomorphisms are on the form εδκp.
Proof. We begin by assuming ϕ is an isomorphism from Wi(a, b) to Wi(c, d).
It is on the form x 7→ pxq or x 7→ p¯xq for some p, q ∈S3[2]. We prove that the latter is out of question. Essentially, we will study the formal requirements for isomorphism, and then use lemma 6.2, the fact thatS3 is not abelian and that xy = ¯y ¯x to obtain a contradiction. Let ∗, × denote the multiplications of Wi(a, b) and Wi(c, d) respectively.
H: ϕ(x ∗ y) = ϕ(axyb) = p¯b¯y ¯x¯aq should equal ϕ(x) × ϕ(y) = cp¯xqp¯yqd for all x, y ∈H. We may switch ¯x for x and ¯y for y to obtain the following
equation: (p¯b)yx(¯aq) = (cp)x(qp)y(qd), ∀x, y ∈ H. ButS3 is a subset of
H that forms a group, so lemma 6.2 is applicable. It states that S3 is an abelian group, which is not true. Contradiction.
H∗: ϕ(x ∗ y) = ϕ(axb¯y) = py¯b¯x¯aq = (p)y(¯b)¯x(¯aq) but ϕ(x) × ϕ(y) = (p¯xq) ∗ (p¯yq) = cp¯xqd¯qy ¯p = (cp)¯x(qd¯q)y(¯p).
∗H: ϕ(x ∗ y) = ϕ(¯xayb) = p¯b¯y¯axq = (p¯b)¯y(¯a)x(q) but ϕ(x) × ϕ(y) = (p¯xq) ∗ (p¯yq) = ¯qx¯pcp¯yqd = (¯q)x(¯pcp)¯y(qd).
H*
: ϕ(x ∗ y) = ϕ(a¯x¯yb) = p¯byx¯aq = (p¯b)yx(¯aq) but ϕ(x) × ϕ(y) = (p¯xq) ∗ (p¯yq) = c¯qx¯p¯qy ¯pd = (c¯q)x(¯p¯q)y(¯pd).
Thus ϕ(x) = pxq for some p, q ∈ S3. Now we study the equations that need to hold for ϕ to be an isomorphism:
1. H(a, b) ˜→H(c, d)
paxybq = ϕ(axyb) = ϕ(x∗y) = ϕ(x)×ϕ(y) = (pxq)×(pyx) = cpxqpyqd, ∀x, y ∈H
Evaluating at y = 1 and x = 1 respectively gives (a) (pa)x(bq) = (cp)x(qpqd), ∀x ⇒Lemma (6.1)
∃ε ∈ {±1} : pa = εcp ∧ bq = εqpqd (b) (pa)y(bq) = (cpqp)y(qd), ∀y ⇒Lemma (6.1)
∃δ ∈ {±1} : pa = δcpqp ∧ bq = δqd
By studying εcp = pa = δcpqp we find q = εδ ¯p, and thus (c, d) = (εpa¯p, δpb¯p), and ϕ(x) = εδpx¯p.
2. ∗H(a, b) ˜→∗H(c, d)
p¯xaybq = . . . = ¯q ¯x¯pcpyqd, ∀x, y ∈H Evaluating at y = 1 and x = 1 respectively gives
(a) (p)¯x(aybq) = (¯q)¯x(¯pcpqd), ∀x ⇒Lemma (6.1)
∃ε ∈ {±1} : p = ε¯q ∧ aybq = ε¯pcpqd (b) (pa)y(bq) = (¯q ¯pcp)y(qd), ∀y ⇒Lemma (6.1)
∃δ ∈ {±1} : pa = δ ¯q ¯pcp ∧ bq = δqd
Substituting q = ε¯p in (b) gives pa = δεp¯pcp = εδcp and
bε¯p = δε¯pd ⇐⇒ pb = δdp and thus (c, d) = (εpa¯p, δpb¯p), and ϕ(x) = εδpx¯p.
10
3. H∗(a, b) ˜→H∗(c, d)
paxb¯yq = . . . = cpxqd¯q ¯y ¯p, ∀x, y ∈H
Evaluating at y = 1 and x = 1 respactively gives (a) (pa)x(bq) = (cp)x(qd¯q ¯p), ∀x ⇒Lemma (6.1)
∃ε ∈ {±1} : pa = εcp ∧ bq = εqd¯q ¯p (b) (pab)¯y(q) = (cpqd¯q)¯y(¯p), ∀y ⇒Lemma (6.1)
∃δ ∈ {±1} : pab = δcpqd¯q ∧ q = δ ¯p Substituting q = δ ¯p in (a) gives pa = εcp and
bδ ¯p = εδ ¯pdδ ⇐⇒ pb = εδdp, and thus (c, d) = (εpa¯p, δpb¯p), and ϕ(x) = εδpx¯p.
4. H*
(a, b) ˜→H* (c, d)
pa¯x¯ydq = . . . = c¯q ¯x¯p¯q ¯y ¯pd, ∀x, y ∈H Evaluating at y = 1 and x = 1 respectively gives
(a) (pa)¯x(bq) = (c¯q)¯x(¯p¯q ¯pd), ∀x ⇒Lemma (6.1)
∃ε ∈ {±1} : pa = εc¯q ∧ bq = ε¯p¯q ¯pd (b) (pa)¯y(bq) = (c¯q ¯p¯q)¯y(¯pd)∀y ⇒Lemma (6.1)
∃δ ∈ {±1} : pa = δc¯q ¯p¯q ∧ bq = δ ¯pd
Comparing pa we get εc¯q ∧ bq = δc¯q ¯p¯q ⇐⇒ q = εδ ¯p.
Thus pa = εcεδp = δcp and bεδ ¯p = εεδ ¯pd ⇐⇒ pb = εdp, giving (c, d) = (εpa¯p, δpb¯p), and ϕ(x) = εδpx¯p.
Conversly we assume (c, d) = (εpa¯p, δpb¯p) and prove that εδκp is an isomor- phism between Wi(a, b) and Wi(c, d):
H: ϕ(x∗y) = ϕ(axyb) = εδpaxby ¯p = εpa¯pεδpb¯pεδpy ¯pδpb¯p = cεδpx¯pεδpy ¯pd = ϕ(x) × ϕ(y)
H∗: ϕ(x∗y) = ϕ(axb¯y) = εδpaxb¯y ¯p = εpa¯pεδpx¯pδpb¯pεδp¯y ¯p = cεδpx¯pdεδp¯y ¯p = ϕ(x) × ϕ(y)
∗H: ϕ(x∗y) = ϕ(¯xayb) = εδp¯xayb¯p = εδp¯x¯pεpa¯pεδpy ¯pδpb¯p = εδp¯x¯pcεδpy ¯pd = ϕ(x) × ϕ(y)
H*
: ϕ(x∗y) = ϕ(a¯x¯yb) = εδpa¯x¯yb = εpa¯pεδp¯x¯pεδp¯y ¯pδpb¯p = cεδp¯x¯pεδp¯y ¯pd = ϕ(x) × ϕ(y)
8 Properties of the functors
LetF : C → D be a functor. It is called essentially surjective if every object y ∈D is isomorphic to an object y0 =F (x) for some object x ∈ C .
It is called full if, for every morphism ϕ : F (x) → F (y) is an image of some ψ : x → y.
It is called faithful if, for all morphisms ϕ, ψ : x → y, ϕ 6= ψ implies F (ϕ) 6= F (ψ).
A functor is an eqivalence of categories if it is essentially surjective, full and faithful. They are useful since equivalent categories have corresponding crossections of isoclasses.
8.1 F
iThe functorsFiare surjective. They are full because of the previous theorem.
Let (ε, δ, [p]), (ε0, δ0, [p0]) be two morphisms (a, b) → (c, d). Assume κp = κp0. Then pa¯p = p0ap0 and pb¯p = p0bp0, but (εpa¯p, δpb¯p) = (c, d) = (ε0p0ap0, δ0p0bp0) which leads to ε = ε0 and δ = δ0 and thus (ε, δ, [p]) = ε0, δ0, [p0]). ThusFi are faithfull, and Fi are equivalences of categories.
8.2 G
There is a 2-1 homomorphism between S3 and SO(3), sending x to κx[6].
Because of it, G is surjective as well as full. For faithfullness all we need to remember is the trick above that recovers ε and δ from [p] in a morphism between (fixed) (a, b) and (c, d). Thus G is an equivalence of categories.
9 Classifying A
4Since Fi and G are equivalences of categories, classifying A4 is equivalent to classifying SO(3)SO(3) × SO(3).
By a proper rotation we mean one which is not the identity. Since all proper rotations of R3 have an axis and an angle of rotation we will denote them as pairs (u, α) where u ∈ S2 is a vector in the axis of rotation and α is the angle of rotation, counted clockwise around u. For completeness we include improper rotations with the understanding (u, 0) = (v, 0) for all u, ∈S3.
It is easy to verify that (u, α)(v, β)(u, α)−1 = (u, α)(v, β)(u, −α) = ((u, α)v, β).
This means that by conjugation, the axis of rotation can change, but that the angle of rotation remains the same. Thus, if the axis is fixed as a line under
12
some conjugation, the angle is either untouched or opposite (corresponding to the fact (u, α) = (−u, −α)). Let a = (u, α), b = (v, β) ∈ SO(3), and let a be proper. If a = bab−1, then v = ±u. If a = −bab−1, then v is orthogonal to u, and β = π2.
Thus, if the axes of two rotations are fixed under some simultaneous conjugation, the axis of the conjugating rotation is either parallel to or or- thogonal to the axes respectively.
We now begin constructing a crossection in different cases. Let a = (u, α), b = (v, β) be two rotations in R3.
If at least one of a, b is improper, we may choose any axis of rotation for the last one. Also, we may take the (possibly) proper rotation into its inverse by conjugation. Thus a crossection is given by {((ε, α), (ε, 0))|α ∈ [0, π]} ∪ {((ε, 0), (ε, β))|β ∈ [0, π]}.
If both a and b are proper, we need to distinguish if u and v are parallel, orthogonal, or neither. Let us first take the case where they are parallel.
The pair (a, b) can be conjugated to any axis. Once that is done, however, the only conjugations that fix the axis are either parallel to or orthogonal to the common axis of a and b. A crossection is given by {((ε, α), (ε, β))|α ∈ ]0, π], β ∈]0, 2π[}.
If a and b are proper, and u and v are orthogonal, the pair (a, b) can be conjugated to have any pair of orthogonal axes. Once that is done, both angles can be conjugated independently into their inverses (by a rotation by π2 around the axis of the other rotation). Thus a crossection is given by {((ε, α), (ε0, β))|α, β ∈]0, π]}
If a and b are proper, but u and v are neither parallel, nor ortogonal, there is an angle γ between the axes of rotation. The only way to conjugate the axes into themselves are by identity, and conjugation by a rotation with axis orthogonal to both u and v. Thus a crossection is given by {((ε, α), (cos γε + sin γε0, β))|α ∈]0, π], β ∈]0, 2π[}
10 Automorphism groups of four dimensional absolute-valued algebras
The automorphism groups are easy to calculate in SO(3)SO(3) × SO(3). We need to split into cases. Let (a, b) ∈ SO(3)2. To prove the following claims, all we need to remember is that if one proper rotation a leaves another proper rotation b untouched (b = aba−1) under conjugation, then either a and b have the same axes of rotation, or they are orthogonal, and the angles of a,b are
π
2,π respectively. For readability, we will call the rotations about an axis
orthogonal to some axis l, and with angle π the flippings of l. Thus, the flippings of l are precicely the rotations that act as multiplication with −1 in l.
If both a, b are improper rotations the automorphism group is the whole SO(3).
If one of a, b is improper, and the other has an angle of rotation which is not π the automorphism group consists of all rotations sharing axis with the proper rotation.
If one of a, b is improper, and the other has angle of rotation π, then the automorphism group consists of all rotations sharing axis with the proper rotation, and the flippings of the same axis.
If both a, b are proper, share axis of rotation, and have angles which are not both π, the automorphism group consists of all rotations around the shared axis.
If both a, b are proper, share axis of rotation, and both have angles of rotation π, the automorphism group consists of the rotations around the shared axis, and the flippings of the shared axis.
If both a, b are proper, their axes are ortogonal, and have angles which are not π, the automorphism group is trivial.
If both a, b are proper, their axes are othogonal, one has angle which is not π and the other has angle π, the automorphism group consists of the identity and the rotation by π2 around the axis of the general rotation.
If both a, b are proper, their axes are neither parallel nor orthogonal, and they have angles which are not both π, the automorphism group is trivial.
If both a, b are proper, their axes are neither parallel nor orthogonal, and they both have angles π, the automorphism group consists of the identity and a rotation around an axis orthogonal to both the axes of a and b, by an angle π2.
11 A
4as a two step reduction
There is a common pattern to a lot of the so far classified subcategories of real division algebras. Let D be the category of real division algebras, and C a full subcategory. The pattern is to
1. establish an equivalence of categories from some category B, of repre- sentations of a certain restricted quiver, to C ,
2. establish an equivalence of categories from some categoryA , of certain geometrical objects to B, and
14
3. create a classification in A .
This pattern has been noted by Dieterich, who has not yet published anything on the subject (of this particular pattern).
The classification of A4 as above qualifies into this pattern. Here, the role of B is played by four copies of the representations of the quiver
R3
SO(3) 66 ii SO(3)
with the restriciton that an equivalence between representations comes from SO(3).
The role of A is also played by four copies of SO(3) × SO(3), this time viewed as geometric objects rather than linear maps.
References
[1] A. A. Albert, Absolute valued real algebras, Ann. of Math. (2) 48, (1947) 495-501.
[2] M. Ram´ırez, On four dimensional absolute-valued algebras, Proceedings of the International Conferance on Jordan Structures (Universidad de M´alaga, M´alaga), 1999, 169-173.
[3] Bott, R. and J. Milnor, On the parallelizability of the spheres. Bull. Amer.
Math. Soc. 64 (1958), 87-89.
[4] I. Burdujan, Types of nonisomorphic two-dimensional real division alge- bras, Proceedings of the National Conferance on Algebra (Romanian), An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat. 31 (1985), suppl. 102-105.
[5] A. Calder´on and C. Mart´ın. Two-graded absolute valued algebras, J. of Alg. 292 (2005) 492-515.
[6] J. H.Conway, D. A.Smith, On quaternions and Octonions, A K Peters, Wellesley (Massachusetts), 1970.
[7] J.A. Cuenca Mira, E. Darp¨o, E. Dieterich, Classification of the Finite Dimensional Absolute Valued Algebras having a Non-Zero Central Idem- potent or a One-Sided Unity, U.U.D.M. Report 2009:6 (2009).
[8] D. ˇZ. Dokovi´c and K. Zhao, Non-associative Real Division Algebras with Large Automorphism Group University of Waterloo, Canada (2003).
J.Algebra 282, no 2, (2004), 758-796.
[9] E. Darp¨o, On the Classification of real flexible division algebras, Collo- quium mathematicum 105(1) (2006) 1-17.
[10] E. Darp¨o, Normal forms for theG2- action on the real symmetric 7 × 7- matrices by conjugation. U.U.D.M. Report 2005:28 (2005).
[11] E. Dieterich, Zur Klassifikation vierdimensionaler reeller Divisionsalge- bren. Math. Nachr. 194 (1998), 13-22.
[12] E. Dieterich, Quadratic division algebras revisited (remarks on an arictle by J.M. Osborn). Proc. Amer. Math. Soc. 128 (2000), 3159-3166.
[13] E. Dieterich and J. ¨Ohman, On the classification of 4-dimensional quadratic division algebras over square-ordered fields. J. London Math.
Soc (2) 65 (2002), 285-302.
[14] E. Dieterich, Classification, Automorphism Groups and Categorical Structure of the Two-Dimensional Real Division Algebras, Journal of Algebra and its Applications (october 2005) 517-538.
[15] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert, Numbers, Springer Verlag, New York, 1991.
[16] E. Gottschling, Die zweidimensionalen reellen Divisionsalgebren, Semi- narber. Fachb. Math. FernUniversit¨at-GHS in Hagen 63 (1998) 228-261.
[17] Hopf, H. Ein topologisher Beitrag zur reellen Algebra. Comment. Math.
Helv. 13 (1940), 219-239.
[18] M. H¨ubner and H.P.Petersson, Two-dimensinal real division algebras revisited, Beitr¨age zur Algebra und Geometrie 45 (2004) 29-36.
[19] M. Kervaire, Non-parallelizability of the n−sphere for n > 7, Proc. Natl.
Acad. Sci. USA 44 (1958), 280-283.
[20] C. Stampfli-Rollier, 4-dimensionale Quasikompositionsalgebren, Arch.
Math. (Basel) 40 (6) (1983) 516-525.
16
