Composition , , and the G2

Paul Tokorcheck Department of University of California, Santa Cruz

06 April 2012

Abstract

In this talk, we will give an introduction to the group G2, by explicitly describing certain elements and its generators. We will first discuss the theory of general composi- tion algebras, Hurwitz’s Theorem, and then trim our focus to the split and look at its automorphisms. For the purposes of this talk, the field over which this construction will take place can remain general.

1 Preliminaries

In our discussion, k will be any field of not 2. It will not hurt to assume that k = R or k = C. Definition 1. A V is a unital k-algebra, which is further endowed with a N : V → k which has the following properties: (a) N is multiplicative. (b) N is nondegenerate. That is, the associated symmetric

1 B(v, w) = [N(v + w) − N(v) − N(w)] 2 is nondegenerate. In this case, the quadratic form N is called a on V.1 The nondegenerate quadratic form N places quite a bit of structure on V. In addi- tion to the bilinear form B, it leads to the following structures:

1The standard use of the term ‘norm’ here is perhaps a bit unfortunate, for N is not a norm in the analytic sense of a normed . In fact, since N maps into an arbitrary field k which will not be ordered in general, concepts such as ‘positive definite’ or ‘triangle inequality’ are unlikely to have any meaning in our context. Conversely, the standard norms placed on vector spaces are not generally quadratic.

1 Definition 2. Let V be a composition algebra with norm N. Then we have also a trace, defined on v ∈ V by; T(v) := N(v + 1) − N(v) − N(1) = 2B(v, 1) and an , defined on v ∈ V by:

v := T(v)1 − v

Lemma 1. We quickly note some facts about the identity: (a)N (1) = 1 1 1 (b)B (1, 1) = 2 [N(2 · 1) − 2N(1)] = 2 [4N(1) − 2N(1)] = 1 (c)T (1) = 2B(1, 1) = 2 (d) 1 = T(1)1 − 1 = 1 Lemma 2. We may justify the terminology used in our last definition, and verify a few other facts for all v, w ∈ V: (a) Both the trace and involution are linear. (b) The subalgebra of V fixed by the involution is equal to k1. (c)T (v)1 = v + v (d) v = v (e) vw = w · v 1 (f)B (v, w) = 2 T(vw) = B(vw, 1) (g)N (v)1 = vv (h)v 2 − T(v)v + N(v)1 = 0 Looking carefully at Lemma 2, we see that all of the other structures (N, T, and B) can be described solely in terms of the involution. However, it is still necessary for the resulting N and B to satisfy the required properties if V is to be considered a composition algebra.

2 Examples

(a) The field k itself is a composition algebra of dimension 1, with quadratic form N(v) = v2, trace given by T(v) = 2v, and trivial involution. (b) k × k. This is a two dimensional k-algebra under component-wise multiplication, with identity 1 = (1, 1). The norm form N is given by N(a, b) = ab, which is multiplicative since k is a field, making this a composition algebra. The trace is given by T(a, b) = a + b. The involution is given by (a, b) = (b, a). Note that this algebra is both associative and commutative.

(c) Mat2(k). Here the relevant quadratic form is given by the determinant, which gives rise to the usual trace and involution on Mat2(k):

 a b   d −b  = c d −c a

It is well known that the determinant is multiplicative, and the needed relations are satisfied between these structures. Note that this algebra is associative but not commutative.

2 (d) Zorn’s Oz are given as a set by:  a ~v   O := | a, d ∈ k, and ~v, w~ ∈ k3 z w~ d Addition in this set is defined in the regular way (entry-wise), and it also has a multiplication, given by:  a ~v   α ~φ   aα +~v ◦ ψ~ a~φ + δ~v − w~ × ψ~  = w~ d ψ~ δ αw~ + δψ~ +~v × ~φ dδ + w~ ◦ ~φ

 1 0  The identity is the usual 1 = , and O is also equipped with the neces- 0 1 z sary norm, which is analogous to the determinant in Mat2(k):  a ~v  N = ad −~v ◦ w~ w~ d

It follows directly from the definition that the trace T on Oz is the usual trace, defined as the sum of the diagonal entries, and the involution is

 a ~v   d −~v  = w~ d −w~ a We quickly check;

 a ~v   a ~v   a ~v   d −~v  = w~ d w~ d w~ d −w~ a  ad −~v ◦ w~ −a~v + a~v + w~ × w~  = dw~ − dw~ −~v ×~v ad −~v ◦ w~  a ~v  = N 1 w~ d At this point, it is still not clear that this quadratic norm form (determinant) is multiplicative. But given what we have just shown, for v, w ∈ Oz: N(vw)1 = (vwvw)1 = (vww · v)1 = (vN(w)v)1 = vvN(w)1 = N(v)N(w)1

The algebra Oz is therefore a composition algebra. Note that this algebra is nei- ther associative nor commutative. However, it is alternative, in the sense of the definition below. Definition 3. A k-algebra V is called alternative if, for all v, w ∈ V, we have: v(vw) = (vv)w v(ww) = (vw)w v(wv) = (vw)v Or, equivalently, V is alternative if for all v, w ∈ V, the subalgebra generated by v and w is associative. Definition 4. A composition algebra is called split if it contains zero divisors. Otherwise, it is called non-split. Since the examples (b) − (d) above contain zero divisors, they are all examples of split composition algebras. There are, however, many examples of non-split compo- sition algebras, such as the field C of complex , Hamilton’s H, or Graves’ Octonions O, all of which may be constructed using the standard Cayley- Dickson construction from R.

3 3 Morphisms and Embeddings

Definition 5. Let (V, N) and (V0, N0) be two composition algebras. We call f : (V, N) → (V0, N0) a morphism of composition algebras if it is a morphism of algebras from V to V0 which additionally preserves the norm form N. It happens that the examples of split composition algebras we have listed are each embedded in the next, as composition algebras:

k ,→ k × k ,→ Mat2(k) ,→ Oz.

For example, k is embedded in k × k via the diagonal embedding ι1 : v 7→ (v, v). The (other) diagonal embedding of k × k into Mat2(k) may be given by

 a 0  ι : (a, b) 7→ 2 0 b

The embedding Mat2(k) ,→ Oz is far from unique. We may choose any unit vector ~u ∈ k3 to yield an embedding:

 a b   a b~u  ι : 7→ ~u c d c~u d

It is easily verified that all of these maps are embeddings of algebras, and that they preserve the respective forms N. In the last map, we will often simplify matters by choosing ~u to be equal to one of the standard basis vectors~ı,~, or~k. When necessary, we ( ( )) ( ( )) ( ( )) will specify our choice of embedding by writing ι~ı Mat2 k , ι~ Mat2 k , ι~k Mat2 k , as appropriate.

4 Generating New Composition Algebras, and Classifying Them

Let (V, N) be any composition algebra, with its usual structure given by the norm form N. We form the space W := V + Vl, where at this point l is simply a formal variable. Multiplication in W is given by:

(a + bl)(c + dl) = (ac + µdb) + (da + bc)l

The involution is given by a + bl = a − bl, which has order two. Note well that there are two distinct involutions present in this expression, one on V and one on W. We also verify:

(a + bl) · (c + dl) = (a − bl)(c − dl)   = a · c + µdb + −da − bc l

= (ca + µbd) − (bc + da)l = (ca + µbd) − (bc + da)l = (ca + µbd) + (bc + da)l = (c + dl)(a + bl)

4 The involution above leads to a norm form NW (w) = ww for all w ∈ W, and therefore a trace TW and bilinear form BW constructed as we described previously. However, at this point we do not yet know that BW is non-degenerate. In fact, this will not always be the case, and thus algebras W constructed in this way may or may not be composition algebras. If we begin from k = R and choose the parameter µ to be µ = −1, then this is simply the standard Cayley-Dickson construction and we will construct C, Hamiltion’s quaternions H, and Graves’ Octonions O. If we start with an arbitrary field k (char k 6= 2) and take µ = 1, then we obtain our examples of split composition algebras (b) − (d) from before. Indeed, we see that k × k can indeed be decomposed in the proper way; given (a, b) ∈ k × k, we can write:  a + b a + b   a − b a − b  (a, b) = , + , (1, −1) 2 2 2 2 The splitting element l in this case is (1, −1). Also, we may associate k × k with its image (the diagonal matrices) in Mat2(k), and have the following decomposition:  a b   a   b    = 0 + 0 0 1 c d 0 d 0 c 1 0

 0 1  The splitting element here is l = . 1 0 Finally, we examine the algebra Oz. Here, the splitting element depends of the choice of embedding, and for the three canonical embeddings (with ~u equal to either~ı, ~, or ~k), we use (respectively) the splitting elements: !  0 ~  0 ~k  0 ~ı  J := , K := , I := ~ 0 ~k 0 ~ı 0

Explicitly, an element of Oz may be decomposed in any of the following ways:  a hv v v i   a v ~ı   v w ~ı   ~  1, 2, 3 = 1 + 2 3 0 hw1, w2, w3i d w1~ı d −v3~ı w2 ~ 0 !  a v ~   v w ~  ~k = 2 + 3 1 0 w2~ d −v1~ w3 ~k 0 ~ ! ~ !   a v3k v1 w2k 0 ~ı = ~ + ~ w3k d −v2k w1 ~ı 0 The following theorem is due to Hurwitz in the real, non-split case. The precise statement below is more general. Theorem 1. Every composition algebra is obtained by repeated doubling, starting from k1 in characteristic 6= 2. The possible dimensions of a composition algebra are 1, 2, 4, and 8. Composition algebras of dimension 1 or 2 are commutative and associative, those of dimesion 4 are associative but not commutative, and those of dimension 8 are neither commutative nor associative.

5

The Group G2 and Some Notable Subgroups

Definition 6. The group G2 is defined to be the group of an : G2 := Aut(O).

5 This definition leaves much to be desired, however, as we have not specified whether or not the octonion algebra in question is split, nor anything about the field over which it is constructed. We will continue to focus on the split Zorn’s octonions in what follows, and will drop the subscript for this section. The group SL3(k) is embedded in Aut(O) via the morphism:

θ : SL3(k) ,→ Aut(O) g 7→ θ(g) where is θ(g) is defined by:

 a ~v   a g~v  [θ(g)] = w~ d g−Tw~ d

Next, we look at a class of subgroups isomorphic to SL2(k). We begin with the par- ticular embedding ι~ı : Mat2(k) ,→ O, which corresponds to a choice of decomposition of v ∈ O as v = a + bJ. For this choice, we have a related embedding of groups

γ~ı : SL2(k) ,→ Aut(O)

g 7→ γ~ı(g), where γ~ı(g) is defined in the following way: after decomposing v ∈ O into v = a + bJ as described in the last section, identify g ∈ SL2(k) with its image under the same embedding used on a and b. In this way, the elements a, b, g all associate with each other. Finally, apply the element g:

[γ~ı(g)](v) = [γ~ı(g)](a + bJ) = (a) + (gb)J

We can verify that γ~ı(g) is actually an automorphism on O, and also that γ~ı is a morphism of groups. Likewise, these constructions can be made (and verified) the other embeddings. In this way, each embedding of Mat2(k) into O will yield a distinct copy of SL2 into Aut(O). Next, we describe another, distinct class of subgroups isomorphic to SL2(k). Similar to the last construction, for each embedding of Mat2(k) into O we can embed SL2(k) into Aut(O) by applying an element g in a different manner. Again, we describe this construction with respect to the embedding ι~ı, though there is no reason to prefer this choice.

δ~ı : SL2(k) ,→ Aut(O)

g 7→ δ~ı(g)

After decomposing an element v ∈ O into v = a + bJ as before, and identifying g ∈ SL2(k) with its image under the same embedding as a and b, we apply the element g:

−1 −1 [δ~ı(g)](a + bl) = (gag ) + (bg )l

That δ~ı(SL2(k)) ⊆ Aut(O), and that δ~ı is an injective morphism of groups, are both verifications very similar to those of the previous example. Again, all verifications hold regardless of the choice of initial decomposition.

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