Contents 1. The structure equations 1 1.1. CayleyDickson - PDF document

Contents 1. The structure equations 1 1.1. Cayley–Dickson construction 1 1.2. Spin(7) and the octonions 1 1.3. The standard basis 2 1.4. Lie algebra of Spin(7) 3 1.5. Maurer–Cartan form 4 1.6. The first structure equations 5 1.7. The group G 2 5 2. The six-sphere 6 2.1. The G 2 -action 6 2.2. The almost complex structure 7 1. The structure equations 1.1. Cayley–Dickson construction. We assume familiarity with quaternions H . Octonions O are pairs of quaternions equipped with the multiplication ( a, b ) · ( c, d ) := ( ac − ¯ (1) db, da + b ¯ c ) . We have a unit 1 = (1 , 0), but the product is not associative. If we write ε = (0 , 1) ∈ O then every octonion can be written x = ( a, b ) = a + bε with a, b ∈ H . There is also an involution a + bε := ¯ a − bε and it satisfies xy = ¯ y ¯ x . The standard metric on R 8 can then be expressed as � x, y � = 1 2( x ¯ y + y ¯ x ) and is compatible with the product in these sense that (2) � xy, zy � = � x, z �� y, y � . This has many useful consequences, but we shall only need (3) � u, v � = 0 ⇒ ( xu ) v = − ( x ¯ v )¯ u, u ( vx ) = − ¯ v (¯ ux ) (4) � u, v � = 0 ⇒ u ¯ v = − v ¯ u ( xw ) w = xw 2 (5) 1 The real part of an octonion is Re x = 2 ( x + ¯ x ). The kernel of Re are the imaginary octonions Im O . This space is 7-dimensional and we identify S 6 = { x ∈ Im O | � x, x � = 1 } as the imaginary units (slightly wrong, but standard terminology). For the standard basis 1 = e 0 , e 1 , . . . , e 7 of O = R 8 the multiplication (1) can be memorized using the following sundial (courtesy of Jost Eschenburg): 1.2. Spin(7) and the octonions. Just like Spin(3) can be regarded as unit quater- nions S 3 = SU (2), so can the group Spin(7) be represented using octonions. This leads to a convenient description of the Lie algebra spin (7) by matrices. Every u ∈ S 6 induces an endomorphism J u : O → O by J u ( x ) = xu . Since u 2 = − 1 these are almost complex structures on O . By (2) they are orthogonal. 1

2 e 3 e 2 e 4 e 1 e 5 e 7 e 6 The sundail shows e 1 e 2 = e 4 . The dial may be turned to get further relations, for example e 2 e 3 = e 5 . Moreover, e i e j = − e j e i when i � = j and e i e i = − 1. Using the universal property of the Clifford algebra we get a representation J : C ℓ (7) → End R ( O ) , u �→ J u . It is not faithful since the volume element ω = e 1 · · · e 7 gets mapped to J ( ω ) = J e 1 ◦ · · · ◦ J e 7 = id (verify on the basis e i using the sundial). The even part C ℓ 0 (7) ∼ = C ℓ (6) is a 64-dimensional simple algebra of matrices, and so the restriction of J is automatically an isomorphism. Recall the definition Spin(7) = { x 1 · · · x 2 n ∈ C ℓ × (7) | x i ∈ S 6 , n ∈ N } . Lemma 1.1. The restriction J : Spin(7) → Aut R ( O ) is a faithful representation, the spinor representation . In particular J : Spin(7) ∼ = � J u | u ∈ S 6 � for the gener- ated subgroup. Proof. Since Spin(7) ⊂ C ℓ 0 (7) injectivity of J is clear. To prove the second state- ment, it remains to compute the image. By definition J (Spin(7)) ⊂ � J ( S 6 ) � . Con- versely, any J u with u ∈ S 6 belongs to J (Spin(7)) because J u = J u ◦ J ω = J ( u · ω ) � 1.3. The standard basis. We view O = R ⊗ R O ⊂ C ⊗ R O embedded as the real part. Re( u ) , Im( u ) , u for u ∈ C ⊗ R O always refers to the real part, imaginary part, complex conjugation with respect to the C -factor. By complexifying the spinor representation we shall regard Spin(7) as a linear subgroup of Aut C ( C ⊗ R O ). In H ⊂ O we have elements j = e 2 , k = e 3 . We shall write e 1 = jk instead of the imaginary unit, because we need ‘ i ’ to denote the complex structure on the complexification C ⊗ R O (lying in the first factor). Recall ε = (0 , 1) = e 4 . Definition 1.2. The standard basis of C ⊗ R O is N = 1 N = 1 ¯ 2(1 − iε ) 2(1 + iε ) F 1 = j ¯ ¯ F 1 = jN N (6) F 2 = k ¯ ¯ F 2 = kN N F 3 = ( kj ) ¯ ¯ F 3 = ( kj ) N N (note that the conjugation only takes place in the C -factor) √ These vector are orthogonal to each other and all have length 1 / 2. Also J ε N = Nε = 1 2 ( ε + i ) = iN etc., and so ( C ⊗ R O ) 1 , 0 = � N, F 1 , F 2 , F 3 � , ( C ⊗ R O ) 0 , 1 = � ¯ N, ¯ F 1 , ¯ F 2 , ¯ F 3 � , where the complex structure J ε on O is understood.

3 1.4. Lie algebra of Spin(7) . Using the standard basis, Spin(7) ⊂ Aut C ( C ⊗ R O ) may be regarded as a linear subgroup of GL 8 ( C ). Our next goal is to explicitly understand which subalgebra of C 8 × 8 the Lie algebra spin (7) is. Lemma 1.3. L = { J ε ◦ J u | u ∈ Im O , � ε, u � = 0 } is a 6 -dimensional subspace of spin (7) , viewed as a subset of End C ( C ⊗ R O ) . In particular [ L, L ] ⊂ spin (7) . Proof. The exponential map of the linear group Spin(7) ⊂ Aut C ( C ⊗ R O ) is the matrix exponential. For an imaginary unit u orthogonal to ε the series expansion shows exp( tJ ε J u ) = cos( t ) id + sin( t ) J ε J u . Here we use J ε J u = − J u J ε and ( J ε J u ) 2 = − id from (3). The right hand side belongs to Spin(7) because it is J − ε J cos( t ) ε − sin( t ) u and cos( t ) ε − sin( t ) u ∈ S 6 . Since L is a vector space, we see that u may have any length. � To determine the commutator, we must understand the transformations J ε J u . The orthogonal complement of ε in Im O can be parameterized by a = ( a 1 , a 2 , a 3 ) ∈ C 3 by setting u = 2 Re( a 1 F 1 + a 2 F 2 + a 3 F 3 ) ∈ Im O . Then we can represent J ε J u in the basis ( N, F, ¯ N, ¯ F ). The result of the compu- tation is the following skew Hermitian 8 × 8 matrix  − ia t  0 0 0 0 0 ia [ − i ¯ a ]   (7) J ε J u =  a t  0 i ¯ 0 0   − i ¯ a [ ia ] 0 0  a 3 − a 2  0  for a ∈ C 3 (note [ a ] t = − [ a ]). In par- − a 3 a 1 using the notation [ a ] = 0  a 2 − a 1 0 ticular, the Lie subgroup belonging to L consists completely of complex antilinear transformations of ( O , J ε ). � � �� κ � 0 � Lemma 1.4. [ L, L ] = � κ ∈ su (4) . � 0 κ ¯ Proof. Let u = 2 Re( aF ) and v = 2 Re( bF ), using vector notation. Then letting � − i Im � a, b � ( b × a ) t � κ a,b = ba ∗ − ab ∗ + i Im � a, b � F 3 a × ¯ ¯ b we have � κ a,b � 0 [ J ε J u , J ε J v ] = 2 . 0 κ a,b ¯ From this we see ‘ ⊂ ’ in the statement of Lemma 1.4. For the converse, we compute κ a,b for ( a, b ) = ( e 1 , e 2 ) , ( ie 1 , ie 2 ) , ( e 1 , e 3 ) , ( ie 1 , ie 3 ) , ( e 2 , e 3 ) , ( ie 2 , ie 3 )       0 0 0 1 0 0 − 1 0 0 1 0 0 0 0 ∓ 1 0 0 0 0 ∓ 1 − 1 0 0 0        ,  ,       0 ± 1 0 0 1 0 0 0 0 0 0 ∓ 1     − 1 0 0 0 0 ± 1 0 0 0 0 ± 1 0

4 and for ( a, b ) = ( ie 1 , e 1 ) , ( ie 2 , e 2 ) , ( ie 3 , e 3 )       − i 0 0 0 − i 0 0 0 − i 0 0 0 0 − i 0 0 0 i 0 0 0 i 0 0        ,  ,       0 0 i 0 0 0 − i 0 0 0 i 0     0 0 0 i 0 0 0 i 0 0 0 − i and finally for ( a, b ) = ( ie 1 , e 2 ) , ( e 1 , ie 2 ) , ( ie 1 , e 3 ) , ( e 1 , ie 3 ) , ( ie 2 , e 3 ) , ( e 2 , ie 3 )       0 0 0 i 0 0 − i 0 0 i 0 0 0 0 ± i 0 0 0 0 ± i i 0 0 0        ,  ,  ,       0 ± i 0 0 − i 0 0 0 0 0 0 ± i    i 0 0 0 0 ± i 0 0 0 0 ± i 0 These 15 matrices are linearly independent over the reals. Since dim R su (4) = 15 we must have equality in the statement of the lemma. � Because dim R spin (7) = dim R so (7) = 21 we now see: Proposition 1.5. We have L ∩ [ L, L ] = { 0 } , dim R L = 6 , dim R [ L, L ] = 15 . Con- sequently, spin (7) = L ⊕ [ L, L ] is the Lie subalgebra of matrices  − a t  ic − b ∗ 0 b D a [¯ a ]   (8)   − b t 0 − a ∗ − ic   ¯ ¯ a ¯ [ a ] b D where a, b ∈ C 3 , c ∈ R , and D ∈ C 3 × 3 satisfy D + D ∗ = 0 , tr D + ic = 0 . 1.5. Maurer–Cartan form. Definition 1.6. Let G be a Lie group. Then Maurer–Cartan form is the g -valued 1-form φ ( X ∈ T g G ) := ( ℓ g − 1 ) ∗ X. Thus it is the unique left-invariant g -valued 1-form with φ e = id g . If we introduce a basis of g , then the corresponding components of φ g determine a basis of T ∗ g G . This gives a global frame of T ∗ G . Definition 1.7. Let α ∈ Ω p ( M ; K m × k ) and β ∈ Ω q ( M ; K k × n ) be matrix-valued forms. Their wedge product is � ( α ∧ β )( X 1 , . . . , X p + q ) = sgn( σ ) α ( X σ (1) , . . . , X σ ( p ) ) · β ( X σ ( p +1) , . . . , X σ ( p + q ) ) σ ∈ Sh p,q using the matrix multiplication ‘ · ’. Ordinary matrices (in particular the unit ma- trix) are embedded as constant matrix-valued functions on M . This wedge product is associative and unital, but not graded commutative. We have ( α ∧ β ) t = ( − 1) pq β t ∧ α t . (9) When G ⊂ GL n ( C ) is a linear subgroup, we may write φ = g − 1 · dg as a matrix- matrix multiplication, where dg : T g G ⊂ C n × n denotes the inclusion of tangent spaces. We always have the Maurer–Cartan equation g − 1 · dg · g − 1 � dφ = d ( g − 1 dg ) = − � (10) dg = − φ ∧ φ.