The transition function of $G_2$ over $S^6$

We obtain explicit formulas for the trivialization functions of the $SU(3)$ principal bundle $G_2 \to S_6$ over two affine charts. We also calculate the explicit transition function of this fibration over the equator of the six sphere. In this way we obtain a new proof of the known fact that this fibration corresponds to a generator of $\pi_{5}(SU(3))$.


Introduction
The well-known classification of simple Lie groups shows that G 2 is the smallest among the exceptional types. Further interesting properties and applications of it are numerous. In this paper we revisit the compact real form G 2 from the viewpoint of differential geometry.
We identify G 2 with Aut O ⊂ O(7), the automorphism group of the Cayley octonions. It is a classical fact that there is a fibration p : G 2 → S 6 , which makes G 2 a locally trivial SU (3)-bundle over S 6 . It is also known that the principal SU (3)-bundles over S 6 are classified by π 5 (SU (3)) = Z.
A natural question is that to which element in π 5 (SU (3)) = Z does the fibration G 2 → S 6 correspond? In other words, what is the homotopy class of the transition function S 5 → SU (3) of the above fibration, where S 5 ⊂ S 6 is the equator of the six sphere? The answer was already used in the physics literature before the first rigorous mathematical proof appeared.
The structure of the paper is as follows. In Section 2 we give a brief introduction to the algebra of Cayley octonions and to several known facts about the group G 2 . The new results of the paper are obtained in Section 3.
Acknowledgements. The author would like to thank to Gábor Etesi and to Szilárd Szabó for several helpful comments and discussions.
2. Some known facts about G 2 2.1. Cayley octonions. To perform calculations in the group G 2 we collect some known facts about the Cayley algebra of octonions. We follow [6] and for completeness we reproduce the proofs of the required results.
Let A be an algebra over the reals. A linear mapping a →ā of A to itself is said to be a conjugation or involutory antiautomorphism ifā = a and ab =bā for any elements a, b ∈ A (the caseā = a is not excluded).
Definition 2.1 (Cayley-Dickson construction [6], [1]). Consider the vector space of the direct sum of two copies of an algebra with conjugation: A 2 = A ⊕ A. A multiplication on A 2 is defined as (a, b)(u, v) = (au −vb, bū + va).
It is easy to check, that relative to this multiplication the vector space A 2 is an algebra of dimension 2 · dim(A). This is called the doubling of the algebra A.
Remark. The correspondence a → (a, 0) is a monomorphism of A into A 2 . Therefore we will identify elements a and (a, 0) and thus assume A is a subalgebra of A 2 . If A has an identity element, then the element 1 = (1, 0) is obviously an identity element in A 2 .
A distinguished element in A 2 is e = (0, 1). It follows from the definition of multiplication that be = (0, b) and hence (a, b) = a + be for all a, b ∈ A. Thus every element of the algebra A 2 is uniquely written as a + be. Moreover, the following identities are true: (1) a(be) = (ba)e, (ae)b = (ab)e, (ae)(be) = −ba.

Lemma 2.2. If
A is a metric algebra, then A 2 is also metric.
Proof. For any a, b ∈ A (a + be)(a + be) = (a + be)(ā − be) = aā + beā − abe + bb = aā + bb, since beā = abe according to the rules of multiplication in (1). Therefore, (a+be)(a + be) ∈ R and it is obviously positive if a or b is not 0.
To iterate the Cayley-Dickson construction it is necessary to define a conjugation in A 2 . This will be done by the formula This is involutory, R-linear and is simultaneously an antiautomorphism. Using this definition, the doubling R 2 of the field R is the algebra C of complex numbers and the doubling C 2 of C is the algebra of quaternions H. In the latter case e is denoted by j and ie is denoted by k, and thus a general quaternion is of the form r = r 1 + r 2 i + r 3 j + r 4 k, where r i ∈ R, i = 1, 2, 3, 4. Due to the identities (1) ea =āe for all a ∈ A. Therefore, A 2 is not commutative if the original conjugation is not the identity mapping. In particular H is not commutative.
The doubling of the algebra of quaternions leads to an 8 dimensional algebra over the reals. By definition every octonion is of the form ξ = a + be, where a and b are quaternions. The basis of O consists of 1 and seven elements i, j, k, e, f = ie, g = je, h = ke.
The square of each of these elements is −1, and they are orthogonal to 1. It is known that the Cayley algebra is an 8 dimensional alternative division algebra [6].
The algebra O is alternative, i.e.
(ξη)η = ξ(ηη), ξ(ξη) = (ξξ)η for all ξ, η ∈ O. The associator of three elements is the trilinear map defined by The algebra is alternative precisely if for all a, b Both of these identities together imply that for an alternative algebra the associator is totally skew-symmetric (or alternative). That is, for any permutation σ. From this it follows that which means that (ab)a = a(ba).
Due to Lemma 2.2 the algebra O is metric. Next we show that it is also normed.
Lemma 2.4. The algebra O is a normed algebra with the norm generated by the metric. In particular, it is a division algebra.
Suppose v = λ + v ′ , where λ ∈ R and v ′ ∈ H ′ , and hencev ′ = −v ′ . From this it follows that because aub + būā is real and therefore commutes with v ′ .
Since O is a normed algebra with the norm induced by the metric, ab, ab = a, a b, b for all a, b ∈ O. Polarizing this first by b = x + y and then by a = u + v we obtain (9).
Proof. Due to the identities of alternativity and elasticity one has and due to the skew-symmetry of the associator this leads to
Taking into account the skew-symmetry of the associator the sums of the first two terms on each side are equal. For the same reason so are the sums of the last two terms. Therefore, Now we replace b with λb, where λ ∈ R, then we divide both sides by λ and take λ = 0. This results in which is (10) and (11) can be proved similarly.
Moreover, (12) can be proved by using again the skew-symmetry of the associator, i.e.
2.2. The subgroup SU (3). Consider the subset of the vector space O ′ consisting of elements ξ, such that |ξ| = 1. This set is a 6-dimensional sphere, which is denoted by S 6 . An automorphism Φ : O → O sends the elements i, j and e to elements ξ = Φi, η = Φj and ζ = Φe in S 6 such that η is orthogonal to ξ and ζ is orthogonal to ξ, η and ξη. The next theorem shows, that these conditions are not only necessary but also sufficient for the existence of the automorphism Φ. The statement of the following theorem is classical.
Let H be a unital (and therefore closed under conjugation) subalgebra of O other than O and let ξ be an octonion in S 6 orthogonal to H. Lemma 2.9. For any element b ∈ H the octonion bξ is orthogonal to H. In particular, bξ ⊥ 1 (since 1 ∈ H), so bξ = −bξ.
Proof. First, applying (8) to x = ξ, y = b and taking into account that ξ ⊥ b and therefore ξ ⊥b we obtain the equation which is equivalent to the first identity.
Using the identities of Lemma 2.10 we also have Proof of Theorem 2.7. It follows from the fact ξ, η ∈ S 6 , that This means that ξη ∈ O ′ , and because |ξη| = |ξ||η| = 1, one has ξη ∈ S 6 . Consequently (ξη) 2 = −1. Using the identity of alternativity ξ(ξη) = (ξξ)η = −η and (ξη)η = ξ(ηη) = −ξ. Adopting (9) to u = v = ξ, x = η and y = 1 yields to ξη, ξ = ξ, ξ η, 1 = 0, from which it follows that Similarly, it can be shown that η(ξη) = ξ and this means that multiplying any number of the elements ξ and η in any order only the elements ±1, ±ξ, ±η and ±ξη can be obtained. That is, the elements of the form constitute a 4 dimensional subalgebra H of O, which is an associative subalgebra due to Lemma 2.11. That is, the correspondences define an isomorphism of the algebra of quaternions H onto the algebra H. Because ζ is by assumption orthogonal to the elements 1, ξ, η and ξη, it is orthogonal to the entire algebra H, and therefore identities of Lemma 2.10 hold for it. From the second of these identities it follows that for the subalgebra generated by H and ζ the identity (8) is also true. Therefore, it is possible to extend linearly the isomorphism H → H to a homomorphism of a subalgebra of O onto the subalgebra generated by H and ζ by sending e to ζ.
If a nonzero homomorphism of an unital division algebra is given, then it is a monomorphism, because if ξ = 0 were mapped to 0, ξ −1 would not have a finite image. Therefore, the extended homomorphism Φ is a monomorphism of O into itself and in this way it is bijective, i.e. it is an automorphism of O. This means that we have constructed an automorphism Φ : O → O sending elements i, j, e to ξ, η, ζ (and, of course, k, f, g, h to ξη, ξζ, ηζ, (ξη)ζ respectively).
From Theorem 2.7 it follows that the group G 2 = Aut O acts transitively on S 6 , i.e. the mapping p : G 2 → S 6 defined by the formula Φ → Φi is surjective. Let us denote by K the stabilizer (isotropy) group of i. This means that Due to the standard theorem of transitive Lie group actions ( [5], Theorem 9.24) The subspace V = Span{1, i} ⊥ of the algebra O is closed under the multiplication by i and thus it can be considered as a vectorspace over the field C with basis j, e, g. The scalar product in O induces in V a Hermitian scalar product with respect to which the basis j, e, g is orthogonal. Any automorphism Φ : O → O which leaves the element i fixed, i.e. which is in the subgroup K, defines an operator V → V linear over C. This operator preserves the scalar product, and therefore it is an unitary operator. Its determinant is 1 because Aut O ⊂ SO(7) and therefore the group K is identified with some subgroup of the group SU (3). From Lemma 2.7 it also follows, that the group K coincides with the entire group SU (3). Thus it may be assumed, that Corollary 2.12. Consider the evaluation mapping p : This SU (3) action clearly carries fibers to fibers. Since SU (3) is free and transitive on itself, it behaves the same way on p −1 (i) and therefore on all of the fibers.
2.3. The subgroup of inner automorphisms. In an associative division algebra, such as the quaternions over the reals, the mapping q r : x → rxr −1 is always an automorphism for any invertible element r, which is called an inner automorphism. In a non-associative algebra it is not always true that for all x, r.
Moreover, not every invertible element generate an inner automorphism. Still, in the case of the octonions a well defined linear transformation associated to an element r can be defined because of the following lemma.
Proof. If the coordinates of r in the standard basis are (r 1 , . . . , r 8 ), then r −1 =r |r| 2 = 2r 1 −r |r| 2 . Therefore, using the identity of elasticity we have The following result classifies those elements r for which the linear map q r is an automorphism of O. For completeness, we reproduce its original proof. Proof. From (12) for a = r, b = xr −1 and c = ryr it follows that and therefore Substituting this into (14) leads us to ) · (xy · r))r = r((xy · r))r = r(xy)r 2 , for all x, y, r ∈ O.
Multiplying this with r 3 from the right we get Comparing (15) to (16) we see that in order for q r to be an automorphism r 3 must be a scalar.
3. G 2 as an SU (3)-bundle over S 6 3.1. The trivialization functions. Our aim is to determine the transition function of the fibration p : G 2 → S 6 , Φ → Φi between two charts of S 6 given by S 6 \ {S} and S 6 \ {N }. The preimage of i is the set p −1 (i) = {(i, η, ζ) : η ⊥ i, ζ ⊥ Span{i, η, iη}}. As mentioned above this is isomorphic to SU (3) and this isomorphism will be called θ 1 . From now on, elements in p −1 (i) will be considered either as orthonormal vector triples in V i = T i S 6 or as operators that leave the vector i fixed.
Proposition 3.1. The trivialization map over S 6 \ {S} is given by where ϕ(i) is the image of i under ϕ and θ ϕ(i) (ϕ) is given by (17) below.
Proof. It follows from the earlier considerations that there is a complex structure This is clearly a V i → V i mapping and J 2 Thus, there is a θ i : V i → C 3 isomorphism, that assigns to each operator Φ ∈ p −1 (i), Φ : V i → V i its matrix representation in the complex basis {j, e, g}.
As a consequence, for any ξ ∈ S 6 \ {S} and any ϕ ∈ p −1 (ξ), ϕ restricts to a mapping V i → V ξ , which is complex linear, unitary and has determinant 1. We will choose a complex orthonormal basis in V ξ and write the images of j, e and g in this basis. That is, we choose particular identifications V i ≈ C 3 , V ξ ≈ C 3 and we define θ ξ : p −1 (ξ) → SU (3) by assigning to each automorphism ϕ ∈ p −1 (ξ) the matrix of the mapping ϕ : To find a basis in V ξ we will define a translating automorphism Q ξ such that Q ξ (i) = ξ. Then, for a = Q ξ (j), b = Q ξ (e) and c = Q ξ (g) the set of vectors {a, b, c} is a complex orthonormal basis in V ξ with respect to the complex structure J ξ (v) = ξv. Particularly, Using this the trivializing map is given by Similarly, the preimage of −i is diffeomorphic to SU (3), and in this case the complex structure on V −i is given by J −i (v) = −iv. Therefore,θ −i is defined as η, j + I η, −k ζ, j + I ζ, −k ηζ, j + I ηζ, −k η, e + I η, −f ζ, e + I ζ, −f ηζ, e + I ηζ, −f η, g + I η, −h ζ, g + I ζ, −h ηζ, g + I ηζ, −h   .
As we did in the previous case, for a general pont ξ ∈ S 6 \ {N } we will choose a translating automorphismQ ξ with the property thatQ ξ (−i) = ξ and thereforẽ Q ξ (j),Q ξ (e),Q ξ (g) ∈ V ξ form a complex orthonormal bases. Then we defineθ ξ : p −1 (ξ) → SU (3) by assigning to ϕ ∈ p −1 (v) the matrix of the corresponding linear mapping from V −i onto V ξ written in the bases {j, e, g} at V −i and {ã,b,c} := {Q ξ (j),Q ξ (e),Q ξ (g)} at V ξ . Similarly as in the proof Proposition 3.1 we obtain the following morphism Finally, the analogue of Proposition 3.1 is true for this chart.
Proposition 3.4. The trivialization map over S 2 \ {N } is then given by where ϕ(i) is the image of i under ϕ andθ ϕ(i) (ϕ) is given by (18).
To summarize, if Q ξ ,Q ξ ∈ G 2 are known as functions of ξ with the property that Q ξ (i) = ξ andQ ξ (−i) = ξ, then an appropriate basis in V ξ is a = Q ξ (j), b = g ξ (e), c = Q ξ (g), which are the translations of the basis j, e, g from V i in the case of the first chart. In the case of the second chartQ ξ translates j, e, g from V −i to V ξ . Thus, we need to find elements Q ξ ∈ G 2 andQ ξ ∈ G 2 . Knowing the first one is enough, because then second is given due to the identities It will be convenient to look for Q ξ in the form of an inner automorphism generated by an element r ∈ O. The easiest is to look for a unit length octonion that induces Q ξ . For a unit length octonion r the conjugate of i with r is: , 2(r 2 r 3 + r 1 r 4 ), 2(r 2 r 4 − r 1 r 3 ), 2(r 2 r 5 + r 1 r 6 ), 2(r 2 r 6 − r 1 r 5 ), 2(r 2 r 7 − r 1 r 8 ), 2(r 1 r 7 + r 2 r 8 )).
Since r ξ ir ξ = ξ = (0, x 2 , . . . , x 8 ) is needed, the following system of equations is to be sold: From Theorem 2.14 it follows that r 1 = 1 2 is required. The general solution for an arbitrary ξ ∈ S 6 \ S of this system of equations is.

3.2.
The transition function over the equator. As in the previous sections we cover the base space S 6 with two trivializing charts given by S 6 \ {S} and S 6 \ {N }. We are interested in the transition function between the two trivializations over the equator. This is enough to reconstruct the whole fibration, since the equator is a deformation retract of the intersection of the charts. The equator S 5 can be identified with a submanifold of where the coordinate functions u, v and w are the duals of j, e and g respectively.
Proposition 3.5. The transition function between the two trivializations of the principal SU (3)-bundle G 2 → S 6 at the equator is From now on we assume that any ξ ∈ O is in the equator of S 6 , and thus x 2 = 0. In this case the solution (19) simplifies to

12ÁDÁM GYENGE
Dut to the fact that iξ = (0, 0, −x 4 , x 3 , −x 6 , x 5 , x 8 , −x 7 ) we have It is easy to check that r ξ is really a solution, because in this case due to the identity of elasticity and Lemma 2.13 we may perform the multiplication in arbitrary order: Consequently, the required automorphisms for an arbitrary ξ ∈ S 6 \ {S, N } are Once again, the trivializing maps and the transition function between the two trivializations are ). As it was discussed above, the meaning of ψ 1 is the following: (ξ, η, ζ) →   η, Q ξ j + I η, Q ξ k ζ, Q ξ j + I ζ, Q ξ k ηζ, Q ξ j + I ηζ, Q ξ k η, Q ξ e + I η, Q ξ f ζ, Q ξ e + I ζ, Q ξ f ηζ, Q ξ e + I ηζ, Q ξ f η, Q ξ g + I η, Q ξ h ζ, Q ξ g + I ζ, Q ξ h ηζ, Q ξ g + I ηζ, Q ξ h   .
The mapping Q ξ (v) = r ξ vr ξ is linear in v, because O is distributive and scalars commute with everything. Due to the construction Q ξ (x) maps the subspace V i to V ξ isomorphically.
Proof. To compute Q ξ (v) 4 groups of identities will be necessary.
(1) According to the definition of the scalar product in O and (8) (2) Similarly, Summing over the two equations this leads to (3) With essentially the same tricks one obtains (4) Once again, Putting these together, , where in the sixth equality the formulas (20), (21), (22), (23) and (24) were used, while in seventh equality the rule vξ = −ξv − 2 v, ξ was applied.
Using this result the inverse function Q −1 ξ : V ξ → V i can be calculated as well by observing that the roles of i and ξ are played by −ξ and −i respectively. Taking into account that any v ∈ V ξ is perpendicular to ξ, virtually the same calculation leads to Moreover, for an arbitrary v ∈ V i more preparation is needed.

14ÁDÁM GYENGE
(1) Applying (5) we obtain (2) By changing the order of terms in the multiplications one obtains Using (9) and the definition of multiplication it can be proved, that Therefore, and thus (3) By exchanging ξ with iξ in (27) one has (4) If a, b ∈ O ′ and a ⊥ b, then ab is orthogonal to both a and b. Thus (5) Finally, taking into account again the orthogonality assumptions and (9) As a consequence, this leads to To simplify calculation it is useful to get rid of the constant factor. According to Lemma 3.6 we have where in the fourth equality the formulas (26), (28), (29), (30) and (31) were used. To sum it up, the required transformation is given by Proof of Proposition 3.5. As mentioned earlier, the subspace V i is a complex linear space with basis j, e, g and complex structure J i : Since ξ ∈ V i , the coordinate expression of ξ in V i can be written as where u, v, w ∈ C, u i , v i , w i ∈ R for i = 1, 2. Because V i = V −i as a subspace, ξ can be expressed as a element of V −i as well. Here the basis is the same, but the complex structure is given by Therefore, the coordinate expression of the same ξ here is According to the multiplication rule of the basis vectors of O (which is represented by the Fano-plane) it is possible to compute the multiplication of ξ with the basis vectors from the left: because the resulting vector v, of which the terms are calculated here, is in V −i . Similarly, ξi = −u 1 k + u 2 j − v 1 f + v 2 e + w 1 h + w 2 g = (u 2 + u 1 I)j + (v 2 + v 1 I)e + (w 2 + w 1 I)g = (uI)j + (vI)e + (wI)g.

16ÁDÁM GYENGE
Then, Putting all together, the matrix which represents the mapping and to get matrix of the same function as a V −i → V −i mapping each complex coordinate of ξ should be conjugated: This proves the statement.
3.3. The class of G 2 . As it is know the principal SU (3)-bundles over S 6 are classified by π 5 (SU (3)). The following fact is well know, but again we included a sketch proof of it.