Isomorphism classes of k -involutions of algebraic groups of type F 4
aa r X i v : . [ m a t h . G R ] J a n Isomorphism classes of k -involutions of algebraicgroups of type F [email protected] of Mathematics and Computer ScienceSouthern Arkansas UniversityMagnolia, AR 71753 October 3, 2018
Abstract
We continue the classification of isomorphism classes of k -involutionsof exceptional algebraic groups. In this paper we classify k -involutions forsplit groups of type F over certain fields, and their fixed point groups.The classification of k -involutions is equivalent to the classification of sym-metric k -varieties, [13]. This is a continuation of a classification initiated by Helminck et al. [4, 5, 11, 14,15]. In [11] Helminck introduces three invariants whose classification is equiva-lent to a full classification of k -involutions of reductive algebraic groups. Herewe look at the specific case when G is a split algebraic group of type F . Onereason to classify the isomorphism classes of k -involutions for an algebraic groupof a certain type has to do with their correspondence to symmetric k -varieties ofthese groups. A symmetric k -variety is a quotient space of the form G ( k ) /H ( k )where G is a reductive algebraic group, H = G θ is the fixed point group of θ an automorphism of order 2 and G ( k ) (respectfully H ( k )) are the k -rationalpoints of G (respectfully H ). For a torus S ⊂ G we denote by X ∗ ( S ) the groupof characters, and Φ( S ) the root space. We define the set I k ( S − θ ) in section 2,along with other definitions needed to make the following statement. Helminckshows that the classification of such spaces can be reduced to the classificationof the following invariants,(1) classification of admissible involutions of ( X ∗ ( T ) , X ∗ ( S ) , Φ( T ) , Φ( S )), where T is a maximal torus in G , S is a maximal k -split torus contained in T
12) classification of the G ( k )-isomorphism classes of k -involutions of the k -anisotropic kernel of G (3) classification of the G ( k )-isomorphism classes of k -inner elements a ∈ I k ( S − θ ),Yokota gave explicit descriptions of k -involutions and their fixed point groupsfor algebraic groups of type F for k = C , R . Over k = C and R our resultscorrespond to the γ and σ maps in [35]. Here we use different methods, and fitthe results into the theory described in [11].Aschbacher and Seitz give a full classification of k -involutions when k is ofeven characteristic in [1]. They provide isomorphism classes of k -involutions,and their centralizers, which we refer to as their fixed point groups. Over fieldsof odd characteristic these results are also known, and can be found in TheClassification of Finite Simple Groups by Gorenstein, Lyons, and Solomon [8].The study of such automorphisms and their relation to algebraic groups wasinitiated by Gantmacher in [7] in order to classify real simple Lie groups. In [2,3]Berger classified involutions of real groups and their symmetric spaces, whichwas also done by Helminck in [10].In the F case we end up with two main types of isomorphism classes ofinvariants of type (1). According to [11] the representatives of one isomorphismclass of k -involutions should send every element of a k -split maximal torus toits inverse, and the representatives of a second isomorphism class fixes a rank 3 k -split torus.We give the full classification of k -involutions when k = K, R , Q p and F q when p ≥ q >
2, where K is the algebraic closure of k . We finish by givingdescriptions of the fixed point groups of isomorphism classes of k -involutions,and discussing the interpretation of the isomorphism classes in terms of Galoiscohomology and their relation to Kac coordinates.I would like to thank E. Neher, S. Garibaldi, H. Petersson, and V. Chernousov,all of whom I met at the Field’s Institute workshop for exceptional groups andalgebras, for their advice and conversations about all things Jordan and excep-tional. I would particularly like to thank H. Petersson for the ongoing emailcorrespondence we have had since that workshop. Most of our notation is borrowed from [32] for algebraic groups, [11] for k -involutions and symmetric k -varieties, [28] for Galois cohomology, and [34] and[23] for Albert algebras and composition algebras.The letter G is reserved for an arbitrary reductive algebraic group. Whenwe refer to a maximal torus we use T and any subtorus is denoted by anothercapital letter, usually S . Lowercase Greek letters are field elements and otherlowercase letters usually denote vectors. Unless it is the letter γ , which refersto ( γ , γ , γ ) ∈ ( k ∗ ) or diag( γ , γ , γ ) ∈ GL ( k ). We use Z ( G ) to denote thecenter of G , Z G ( S ) to denote the centralizer of S in G .2y Aut( G ) we mean the automorphism group of G , and by Aut( A ) wemean the linear automorphisms of the Albert algebra, A . The group of innerautomorphisms are denoted Inn( G ) and the elements of Inn( G ) are denoted by I g where g ∈ G and I g ( x ) = gxg − .We define a θ -split torus , S , of an involution, θ , as a torus S ⊂ G such that θ ( s ) = s − for all s ∈ S . We call a torus ( θ, k ) -split if it is both θ -split and k -split. If S is a torus that is not necessarily θ -split, we denote by, S − θ = { s ∈ S | θ ( s ) = s − } , the elements of S split by θ . So we can say S is θ -split if and only if S = S − θ .Let S be a θ -stable maximal k -split torus such that S − θ is a maximal ( θ, k )-split torus. In [14] it is shown that there exists a maximal k -torus T ⊃ S suchthat T − θ ⊃ S − θ is a maximal θ -split torus. The involution θ induces an involution˜ θ ∈ Aut( X ∗ ( T ) , X ∗ ( S ) , Φ( T ) , Φ( S )). It was shown by Helminck and Wang [13]that such an involution is unique up to isomorphism. For T a maximal k -toruscontaining a subtorus S ,˜ θ ∈ Aut( X ∗ ( T ) , X ∗ ( S ) , Φ( T ) , Φ( S ))is admissible if there exists an involution θ ∈ Aut(
G, T, S ) such that θ | X ∗ ( T ) = ˜ θ , S − θ is a maximal ( θ, k )-split torus, and T − θ is a maximal θ -split torus of G . Thiswill give us the set of k -involutions on G that extend from involutions on thegroup of characters, X ∗ ( T ). If θ is a k -involution and S − θ is a maximal θ -splittorus then the elements of the set, I k ( S − θ ) = n s ∈ S − θ (cid:12)(cid:12) ( θ ◦ I s ) = id , ( θ ◦ I s ) ( G ( k )) = G ( k ) o , are called k -inner elements of θ . Some compositions θ ◦I s will not be isomorphicin the group Aut( G ) for different s ∈ I k ( S − θ ), though they will project downto the same involution of the group of characters of a maximal torus fixing thecharacters associated with a maximal k -split subtorus for all s ∈ I k ( S − θ ).We borrow notation for quadratic forms from the text by Lam, [19]. For a2-Pfister form we write (cid:16) ζ,ηk (cid:17) for the quadratic form q D ( x ) = x − ζx − ηx + ζηx , over a field k . Here we provide notation and background information from the theory of Jordanalgebras, especially that of Albert algebras, that we will use in our classification.We will always think of algebraic groups of type F as the automorphisms of anAlbert algebra. 3ur split Albert algebra will be isomorphic to 3 × k . The Albert algebra will havedimension 27 with respect to the field. An octonion algebra is split if it containzero divisiors. We will call the Albert algebra split if the octonion algebraassociated with the Albert algebra is split. We will let k be a field over characteristic not 2, for char( k ) = 2 see [1], and C a split composition algebra of dimension eight over k . For any fixed γ i ∈ k ∗ ,we will define A = H ( C ; γ , γ , γ ) = H ( C, γ ) be the set of 3 × γ -Hermitianmatrices. Each x ∈ A , where f i ∈ k and c j ∈ C , will look like x = h ( f , f , f ; c , c , c ) = f c γ − γ ¯ c γ − γ ¯ c f c c γ − γ ¯ c f , where ¯ x = q ( x, e ) e − x , in C with q ( , ) the bilinear form on C .We will define a product xy = 12 ( x · y + y · x ) = 12 (cid:0) ( x + y ) · − x · − y · (cid:1) with the dot indicating standard matrix multiplication.So A is a commutative, nonassociative k -algebra of 3 × e = h (1 , ,
1; 0 , , × Q : A → k , with an associated bilinear form h x, y i = Q ( x + y ) − Q ( x ) − Q ( y ) , and we have Q ( x ) = 12 (cid:0) f + f + f (cid:1) + γ − γ q ( c ) + γ − γ q ( c ) + γ − γ q ( c ) . Notice the bilinear form is nondegenerate. We denote by H ( C, γ ), the 3 × C , where C is an octonion algebra and γ is a diagonalmatrix with entries in k ∗ , as an Albert algebra . We will follow [34] and refer to the family of algebras of 3 × γ -Hermitianmatrices over a composition algebra as J -algebras . If w ∈ A and w = w then we call w an idempotent element. A J -algebra is said to be reduced if itcontains an idempotent other than zero or the identity. A J -algebra is proper if it is isomorphic to an algebra of the form H ( C, γ ), where C is a compositionalgebra. The idempotent elements of a J -algebra play the following role.4 emma 3.2.1. If w ∈ A is an idempotent element and w is not or e , then Q ( w ) = or , h w, e i = 2 Q ( w ) , e − u is idempotent, w ( e − w ) = 0 , h w, e − w i = 0 ,and Q ( e − w ) = − Q ( w ) . If Q ( w ) = then we call w a primitive idempotent . The following theoremcan be found in [17]. Theorem 3.2.2. If A is a proper reduced J-algebra, then the composition alge-bra C where A ∼ = H ( C, γ ) is uniquely determined up to isomorphism. If we fix a primitive idempotent w ∈ A , and let E = { a ∈ e ⊥ ⊂ A | wa = 0 } , and let E = { a ∈ e ⊥ ⊂ A | wa = 12 a } , then an Albert algebra has a decomposition A = kw ⊕ k ( e − w ) ⊕ E ⊕ E , called the Peirce decomposition . In this paper we are concerned with split Albert algebras, which can be con-structed using the following method due to Jacques Tits. We first need to definea sharped cubic form, ( N, , X ) on a module, X , over a ring of scalars withunity. First we will choose a base point 1 X ∈ X , such that N (1 X ) = 1. Nextwe define the quadratic map, x ∈ X , such that x = x − Tr( x ) x + Sr( x )1 X . We call Tr the trace form on X and Sr the quadratic trace. If the followingidentities hold for N ,Tr( x , y ) = N ( x, y ) (3.1) x = N ( x ) x (3.2)1 X x = Tr( y )1 X − y, (3.3)we call ( N, , X ) a sharped cubic form , where x y = ( x + y ) − x − y . The maps Tr( , ), Sr( , ), and N ( , ) are the linearizations of the respectiveforms. These will be defined explicitly in our case when needed. To get an ideaof how this works in more generality we refer the reader to [23].From a sharped cubic form ( N, , X ) we can construct a unital Jordanalgebra, J ( N, , X ), which has unit element 1 X , and a U operator defined by U x y = Tr( x, y ) x − x y. roposition, (McCrimmon) 3.3.1. Any sharped cubic form ( N, , X ) givesa unital Jordan algebra J ( N, , X ) with unit X and product xy = 12 ( x y + Tr( x ) y + Tr( y ) x − Sr( x, y )1 X ) . Finally, we relate the sharp product and sharp map to the above forms inthe following way, x y = { x, y } − Tr( x ) y − Tr( y ) x + Sr( x, y )1 X (3.4)0 = x − Tr( x ) x + Sr( x ) x − N ( x )1 X . (3.5)From here we define the first Tits construction for Jordan algebras of degree3 from associative algebras of degree 3, a special case of which will provide uswith another form of split Albert algebras isomorphic to H ( C, id), where C isa split octonion algebra.Let M be an associative algebra of degree 3 over a unital commutative ring R with a cubic norm form ( n, , M ) satisfying the following m − tr( m ) m + sr( m ) m − n ( m )1 M = 0 , (3.6)with tr( m ) = n (1 M , m ) , sr( m ) = n ( m, M ) , n (1 M ) = 1. Notice that also if m = m − tr( m ) m + sr( m )1 M we can write m · m = m · m = n ( m )1 M , (3.7) n ( m, m ′ ) = tr( m , m ′ ) , (3.8)tr( m, m ′ ) = tr( mm ′ ) . (3.9) Proposition 3.3.2.
Let M be a unital associative algebra with a base point suchthat n (1 M ) = 1 , m = m − tr( m ) m + sr( m )1 M , and m m ′ = ( m + m ′ ) − m − m ′ . Then any cubic form on M satisfies tr(1 M ) = sr(1 M ) = 3 (3.10)1 M = 1 M (3.11)sr( m, M ) = 2 tr( m ) (3.12)1 M m = tr( m )1 M − m (3.13)sr( m ) = tr( m ) (3.14)2 sr( m ) = tr( m ) − tr( m ) . (3.15)A proof of this can be found in [23]. For the remainder of the constructionwe follow Petersson’s notes from the Fields Institute workshop on exceptionalalgebras and groups, but this can also be found in [23].6 irst Tits Construction 3.3.3. Let n be the cubic norm form of a degree associative algebra M over R , and let ν ∈ R ∗ . We define a module J ( M, ν ) = M ⊕ M ⊕ M , to be the direct sum of three copies of M , and define M , N , Tr , and by M = id ⊕ ⊕ N ( m ) = n ( m ) + νn ( m ) + ν n ( m ) − tr( m m m ) (3.17)Tr( m ) = tr( m ) (3.18)Tr( m, m ′ ) = tr( m , m ′ ) + tr( m , m ′ ) + tr( m , m ′ ) (3.19) m = ( m − νm m ) ⊕ ( νm − m m ) ⊕ ( m − m m ) (3.20)( mm ′ ) = m ′ m , (3.21) where n and tr are the norm and trace on M for m = ( m , m , m ) and m ′ =( m ′ , m ′ , m ′ ) . Then ( N, , M ) is a sharped cubic form and J ( N, , M ) is aJordan algebra. We want to construct a split Albert algebra over a field k . Let us denote byMat ( k ) the 3 × k . To perform our construction we pickour associative algebra Mat ( k ) with n the determinant, tr is the typical traceof a matrix, 1 M = id, and ν = 1. For the remainder of the paper we drop the .
20) for the formula.
Proposition 3.3.4.
The algebra J (Mat ( k ) ,
1) = A ⊕ A ⊕ A is a split Albertalgebra.Proof. The algebra J (Mat ( k ) ,
1) is reduced, since it contains the primitiveidempotent, · ·· · ·· · · , , . So J (Mat ( k ) , ∼ = H ( C, γ ), for some γ . Also, J (Mat ( k ) ,
1) contains a copyof A ⊕ { } ⊕ { } ∼ = Mat ( k ) + ∼ = H ( k ⊕ k ), which contains a copy of k ⊕ k by [34]. The composition algebra k ⊕ k is a split, which implies the octonionalgebra C ⊃ k ⊕ k , and so H ( C, γ ) is split.Where Mat ( k ) + is the Jordan algebra consisting of the vector space Mat ( k )with the typical Jordan product. For our two presentations of the split Albert algebra over a given field, wewill consider two different ways to assure linear maps are automorphisms. For H ( C, γ ) we can just check the map in question is a bijection and respects theAlbert algebra multiplication. When we consider the Albert algebra in the first7its construction form, we can check that the map is a bijection and respectsthe norm and adjoint maps, and leaves the base point fixed.It turns out that, for the most part the automorphisms we want to considerhave order 2, and so we will review some results of Jacobson concerning Aut( A )found in [17]. Theorem, (Jacobson) 3.4.1.
Let A be a finite dimensional exceptional cen-tral simple Jordan algebra. Then A is reduced if and only if Aut( A ) containselements of order two. If the condition holds then any s ∈ Aut( A ) having ordertwo is either; ( I ) a reflection in a sixteen dimensional central simple subalgebra of degreethree, ( II ) the center element r w = 1 in a subgroup Aut( A ) w , w is a primitiveidempotent. Corollary 3.4.2.
An involution of type (I) leaves a subalgebra B K ⊂ A K ∼ = H ( C, γ ) where K is algebraically closed, and B K ∼ = H ( D, γ ) where D ⊂ C isa quaternion subalgebra over K . If A is split then any two automorphisms of A of order two and type (II) are conjugate in Aut( A ) . When k is either finiteor algebraically closed, then any two automorphisms of A of order two and type(I) are conjugate in Aut( A ) . F × SL gives us a k -split maximal torusin Aut( A ), a construction suggested by H. Petersson following [33]. Using theclassification from [11] we use this k -split maximal torus to find representativesof the isomorphism classes of k -involutions of Aut( A ). SL ( k ) × SL ( k ) on A ∼ = J (Mat ( k ) , Proposition 4.1.1.
The map f uv : A → A, where ( a , a , a ) (cid:0) ua u − , ua v − , va u − (cid:1) , is an automorphism of A if and only if ( u, v ) ∈ SL ( k ) × SL ( k ) .Proof. In order for f uv to be an automorphism of A ∼ = J (Mat ( k ) ,
1) we need f uv to preserve the base point, the sharp map and the norm. A map of the form f uv will clearly fix the basepoint 1 A = (id , ,
0) for any u ∈ GL ( k ). Next wecheck that f uv preserves the norm. So we look at N ( f uv ( a )) = N (cid:0) ( ua u − , ua v − , va u − ) (cid:1) = n (cid:0) ua u − (cid:1) + n (cid:0) ua v − (cid:1) + n (cid:0) va u − (cid:1) − tr (cid:0) ua u − ua v − va u − (cid:1) = n ( u ) n ( a ) n ( u − ) + n ( u ) n ( a ) n ( v − ) + n ( v ) n ( a ) n ( u − ) − tr( ua a a u − )= n ( a ) + n ( a ) + n ( a ) − tr( a a a )= N ( a ) . a − i = a i n ( a i ) − and look at the sharp map a = (cid:16) a − a a , a − a a , a − a a (cid:17) . then f uv (cid:0) a (cid:1) = f uv (cid:16) a − a a , a − a a , a − a a (cid:17) = (cid:16) u ( a − a a ) u − , u ( a − a a ) v − , v ( a − a a ) u − (cid:17) = (cid:16) ua u − − ua a u − , ua v − − ua a v − , va u − − va a u − (cid:17) = (cid:16) ua u − − ua v − va u − , ua v − − ua u − ua v − , va u − − va u − ua u − (cid:17) . Since a i = a i − tr( a i ) a i + sr( a i ) c and x − = x if and only if x ∈ SL ( k ), x (cid:16) a i (cid:17) y − = (cid:0) x − (cid:1) (cid:16) a i (cid:17) ( y ) = (cid:0) x − (cid:1) ( ya i ) = (cid:0) ya i x − (cid:1) . For a let x = y = u , a let x = u and y = v , a let x = v and y = u . k -split maximal torus Now if we consider a k -split maximal torus in SL ( k ), T ∗ = u u − u u − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u , u ∈ k ⊂ SL ( k ) , we see that it is of rank 2 and so a k -split maximal torus T ∗ × T ∗ ⊂ SL ( k ) × SL ( k ) ⊂ Aut(
A, k ) must have rank 4 and we have that T = T ∗ × T ∗ ⊂ Aut(
A, k )is a k -split maximal torus. In accordance with the action of f uv on A , an elementof our k -split maximal torus can be computed directly. k -involutions From the combinatorial classification of invariants of type (1) given in [11] weknow we should have two involutions corresponding to the following diagrams. ❡ ❡ ❡ ❡ α α α α > ✉ ✉ ✉ ❡ β > A be defined over k the first diagram corresponds to a k -involution θ : Aut( A ) → Aut( A ) that splits an entire k -split maximal torus, and thesecond diagram corresponds to a k -involution σ : Aut( A ) → Aut( A ) that splitsa rank 1 k -split torus and fixes a k -split torus of rank 3. k -inner elements We have two isomorphism classes of invariants of type (1) as set out by [11].Once we describe the G ( k )-isomorphism classes of k -inner elements for our repre-sentatives of admissible k -involutions of type (1), our task will be complete. Wewill follow [17] and call automorphisms in the equivalence class of k -involutionsof the form θ ◦I t , t ∈ I k ( T − θ ) of type (I), and those of the form σ ◦I s , s ∈ I k ( T − σ )of type (II).By 3.4.2 all elements of order 2 in Aut( A ) of type (II) are conjugate for any k where char( k ) = 2. So we have the following theorem. Theorem 4.4.1.
All k -involutions of the form σ ◦ I s , s ∈ I k ( T − σ ) are isomor-phic.Proof. These automorphisms are inner automorphisms of the form σ ◦ I s ( x ) = gxg − with g ∈ Aut( A ) being of order 2 and leaving an 11 dimensional sub-algebra fixed. So g is an automorphism of A of type (II), and so the result isimmediate from 3.4.2.The isomorphism classes of k -involutions of the form θ ◦ I t , t ∈ I k ( T − θ )are said to be of type (I) and leave a 15 dimensional subalgebra fixed that isisomorphic to H ( D, γ ) where D ⊂ C is a quaternion subalgebra of a splitoctonion algebra. In order to classify k -involutions of this type we will need thefollowing Lemma. Lemma 4.4.2.
Let A be a k -algebra and D and D ′ subalgebras of A . If t, t ′ ∈ Aut( A ) are elements of order and t, t ′ fix D , D ′ elementwise respectively, then t ∼ = t ′ if and only if D ∼ = D ′ over k .Proof. Let D , D ′ ⊂ A such that t ( a ) = a and t ′ ( a ′ ) = a ′ for all a ∈ D and a ′ ∈ D ′ , and let D and D ′ be the largest such subalgebras with respect to t and t ′ . First we will show sufficiency. Let D ∼ = D ′ , and let g ∈ Aut( A ) be such that g ( D ′ ) = D . For c ∈ A c = a + b where a ∈ D and b ∈ A − D . Since t, t ′ areof order 2 we have D (resp. D ) is the 1-space and A − D (resp.
A − D ′ ) is the( − t (resp. t ′ ). Then we have, gt ′ g − ( a + b ) = gt ′ ( a ′ + b ′ )= g ( a ′ − b ′ )= a − b = t ( a + b ) , g ( A − D ′ ) = A − D . To show necessity we start by assuming there existsa g ∈ Aut( A ) such that gt ′ g − = t , which implies that t ′ g − = g − t , and fromthis we see that t ′ g − ( a + b ) = g − t ( a + b ) t ′ ( g − ( a ) + g − ( b )) = g − ( a − b ) t ′ ( g − ( a ) + g − ( b )) = g − ( a ) − g − ( b ) . This shows us that g − ( a ) ∈ D ′ for all a ∈ D , and so D ∼ = D ′ .We say that two diagonal matrices, γ, γ ′ ∈ ( k ∗ ) , are equivalent and write γ ∼ C γ ′ if H ( C, γ ) ∼ = H ( C, γ ′ ), where C is a composition algebra. Proposition 4.4.3.
Let D ⊂ C be a quaternion subalgebra of the octonion al-gebra C , over k . If H ( D, δ ) , H ( D ′ , δ ′ ) ⊂ H ( C, γ ) then H ( D, δ ) ∼ = H ( D ′ , δ ′ ) if and only if D and D ′ are split, or D and D ′ are division algebras and δ ∼ C δ ′ ∈ k ∗ /q D ( C ) ∗ .Proof. From [34] every automorphism of a subalgebra of H ( C, γ ) extends to anautomorphism of H ( C, γ ). This fact along with 3.2.2 gives us the result.If we consider θ ◦ I t of type (I), notice that θ = I g where, g = p · ·· · p · p · , where I p : Mat ( k ) → Mat ( k ), and p ( x ) = x T . So the k -inner elements areof the form t ( u , u , v , v ) = t ∈ T − θ = { t ∈ T | θ ( t ) = t − } and θ ◦ I t is a k -involution. In this case T − θ is a maximal torus, and all elements t ∈ T aresuch that θ ◦ I t is a k -involution for all k . To check that this is true we cansimply notice that θ ◦ I t = I g ◦ I t = I gt , and ( gt ) = id for all t ∈ T = T − θ .If we let t = t ( u , u , v , v ) then we can compute A θ , the subalgebra of A fixed by the element of Aut( A ) that induces the k -involution, θ ◦ I t .We assume that our field is not of characteristic 2 and we use the quadraticform Tr( x ) from [30], to make an identification between H ( D, γ ) ⊂ H ( C, id),and A θ ⊂ J (Mat ( k ) , Theorem, (Jacobson) 4.4.4.
Let H ⊂ J and H ′ ⊂ J be reduced simplesubalgebras of degree of a reduced simple exceptional Jordan algebra J , if thereis an isomorphism H ∼ = H ′ then it can be extended to an automorphism in J . Corollary 4.4.5.
Let C be an octonion algebra and D ⊂ C a quaternion sub-algebra, then H ( D, δ ) is a subalgebra of H ( C, γ ) if and only if H ( C, γ ) ∼ = H ( C, δ ) . Corollary 4.4.6.
Let C be a split octonion algebra with quaternion subalgebra D ⊂ C then H ( D, δ ) is a subalgebra for all δ ∈ ( k ∗ ) . roof. This follows from the fact that there is only one isomorphism class ofalgebras of the form H ( C, γ ) when C is split.With this in mind we look at isomorphism classes of algebras of the form H ( D, γ ) ⊂ H ( C, id) where C is a split octonion algebra. Lemma 4.4.7.
Let C be a split octonion algebra with quadratic form q and D a quaternion subalgebra with quadratic form q D . Then (1) if k = K is algebraically closed there is one isomorphism class of the form H ( D, γ ) , (2) if k = F p where char( p ) > there is one isomorphism class of algebras ofthe form H ( D, γ ) , (3) if k = R there are isomorphism classes of algebras of the form H ( D, γ ) corresponding to D being split, and D being a division algebra with γ = id or γ = ( − , , , (4) if k = Q p there are isomorphism classes of algebras of the form H ( D, γ ) corresponding to D being split or D being a division algebra. (5) if k = Q there are an infinite number of isomorphism classes.Proof. For (1) and (2) there are only split quaternion algebras, and thereforeonly split algebras of the form H ( D, γ ). For (3) when k = R it is well known, seefor example [34] chapter 1, there are 2 isomorphism classes of quaternion alge-bras. If D is split there is one isomorphism class of algebras of the form H ( D, γ ).If D is a division algebra the isomorphism classes are determined by normclasses k ∗ /q D ( D ) ∗ , 5.8.1 [34]. If k = R then k ∗ /q D ( D ) ∗ = {± } , which give ustwo equivalency classes of diagonal matrices of the form diag( γ , γ , γ ), where γ i ∈ R ∗ . One can be represented by diag(1 , ,
1) and the other by diag( − , , q D ( D ) represents all values in Q p , see [28, 34], so k ∗ /q D ( D ) ∗ = { } and all ma-trices of the form diag( γ , γ , γ ) ∈ ( k ∗ ) are equivalent. To see (5) notice thatthere are an infinite number of isomorphism classes of quaternion division alge-bras over Q and in order for two J -algebras to be isomorphic their compositionalgebras must be isomorphic. Theorem 4.4.8.
Let t = t ( u , u , v , v ) and let gt ∈ Aut( A ) be an involutionof type (I), where A is a split Albert algebra. Then for k -involutions of the form θ ◦ I t = I gt , (1) if k = K is algebraically closed there is one isomorphism class, (2) if k = F p where char( p ) > there is one isomorphism class, (3) if k = R there are isomorphism classes, (4) if k = Q p there are isomorphism classes, if k = Q there are an infinite number of isomorphism classes.Proof. This follows from 4.4.2 and 4.4.7.In fact we can identify a representative of isomorphism classes of k -involutionsof the type θ ◦ I t for each field. For k algebraically closed or a finite field wecan take u = u = v = v = 1. To find representatives for the isomorphismclasses of k -involutions when k is R , Q p , or Q it helps to consider the quadraticform from [34]. If we consider an Albert algebra A over C , a split compositionalgebra, then A ∼ = H ( C, id). Then an element of H ( C, id) is of the form, x = f x ¯ x ¯ x f x x ¯ x f , where f i ∈ k and x l ∈ C , and ¯ is the algebra involution in C . The quadraticform described in [34] is of the form Q ( x ) = 12 ( f + f + f ) + q ( x ) + q ( x ) + q ( x ) , with q the quadratic form on C . We consider the same quadratic form on thesubalgebra H ( D, γ ) ⊂ H ( C, id). An element of H ( D, γ ) has the form y = f y γ − γ ¯ y γ − γ ¯ y f y y γ − γ ¯ y f , where f i ∈ k , y l ∈ D a quaternion subalgebra of C , and ( γ , γ , γ ) ∈ ( k ∗ ) .The quadratic form restricted to this subalgebra looks like, Q ( y ) = 12 ( f + f + f ) + γ − γ q D ( y ) + γ − γ q D ( y ) + γ − γ q D ( y ) , with q D the quadratic form on C restricted to a quaternion subalgebra D ⊂ C .From here we need two facts about equivalent quadratic forms. We know that A θ ∼ = H ( D, γ ) for some D and γ . So, if we compute Q ( a ), for a ∈ A θ , Q ( a ) == 12 ( a + a + a )+ u − u a + u v a + u v − v a + u v − a + u − u − a + u u − v a + u u − v − v a + u u − v − a + u u − a + u − v a + u − v − v a + u − v − a Proposition, [16] 4.4.9.
A quaternion algebra is completely determined by itsquadratic form.
13e can use this along with the fact that the quadratic form of a quaternionalgebra is completely determined by a 2-Pfister form, (cid:18) ζ, ηk (cid:19) , where ζ and η are the negative squares of two basis vectors in e ⊥ ⊂ D , where e is the identity element in D . It is known that (cid:18) ζ, ηk (cid:19) ∼ = (cid:18) m ζ, n ηk (cid:19) , (4.1)see [9] 1.1.2, where m, n ∈ k ∗ . It is also helpful to note here that( δγ , δγ , δγ ) ∼ ( γ , γ γ ) ∼ ( δ γ , δ γ , δ γ ) , (4.2)where δ, δ i ∈ k ∗ , [17]. Using 4.1 and 4.2 we can rewrite, Q ( a ) = 12 ( a + a + a )+ u (cid:0) a + v a + v − v a + v a (cid:1) + u − u − (cid:0) a + v a + v − v a + v a (cid:1) + u (cid:0) a + v a + v − v a + v a (cid:1) , making the identifications u γ − γ , u − u − γ − γ , u γ − γ , ζ v ,and v − v η we have equivalent quadratic forms. Proposition 4.4.10.
For the following fields, k , we can take as representativesof isomorphism classes of k -involutions of Aut( A ) to be of the form I gt where g is defined above, and t = t ( u , u , v , v )(1) k = K or k = F p where p > , t = t (1 , , , is a representative of the onlyisomorphism class, (2) k = R for D split we can choose t (1 , , − , , for the positive definite casewe can choose t (1 , , , , and for the indeterminate quadratic form we canchoose t ( − , , , , (3) k = Q we can choose t (1 , , − , for the split case and t (1 , , , for D a division algebra, (4) k = Q p with p > we can choose t (1 , , − , for D split, and t (1 , , − p, − Z p ) for D a division algebra.Proof. (1) is straight forward. (2) is well known, but this can be seen throughstraight forward computations and the fact that (1 , , ( − , , k = Q is determined by (cid:16) − , − Q (cid:17) a divisionalgebra, and a quaternion algebra determined by (cid:16) , − Q (cid:17) for the split case. (4)can be seen if we let Q ∗ p / ( Q ∗ p ) = { , p, Z p , pZ p } , where Z p is the smallest non-square in F p . There are two isomorphism classes of quaternion algebras over Q p when p >
2; one when the quaternion algebra is determined by (cid:16) , − Q p (cid:17) forthe split case, and the other is determined by (cid:16) p,Z p Q p (cid:17) for the division algebracase.For k = Q we have seen that there are an infinite number of isomorphismclasses of k -involutions of Aut( C ) where C is an octonion algebra, [15]. Thisis due to the fact that there are an infinite number of isomorphism classes ofquaternion division algebras of Q . We recall that (cid:18) − , p Q (cid:19) = (cid:18) − , q Q (cid:19) , when p and q are distinct primes both equivalent to 3 mod 4. This alone isenough to give us subalgebras of H ( C, id) of the form H ( D i , id) where D i isthe quaternion algebra with the quadratic form (cid:16) − ,p i Q (cid:17) where p i are all distinctprimes equivalent to 3 mod 4. The k -involutions of the form θ ◦ I t = I s as defined above, correspond to anelement of order 2 in Aut( A ) fixing a subalgebra of the form H ( D, γ ), where D is a quaternion subalgebra defined over k , and γ is a diagonal matrix withentries in k ∗ .This induces a decomposition of the Albert algebra A into a Jordan algebraover the quaternion algebra fixed by an element of order 2 in Aut( C ) and aquaternion multiple of the skew symmetric matrices taken over a quaternionalgebra of the same type.Let t ∈ Aut( A ) be of order 2 such that t | C = ˆ t ∈ Aut( C ) is of order 2, then s is of the form, t ( f , f f , c , c , c ) = ( f , f , f , ˆ t ( c ) , ˆ t ( c ) , ˆ t ( c )) , and fixes a quaternion subalgebra D ⊂ C that is either split or a division algebra.Then the map s fixes a subalgebra of the form H ( D, γ ) ⊂ H ( C, id) ∼ = A . Sothe algebra H ( C, id) decomposes as follows H ( D, γ ) ⊕ Skew ( D, γ ) · j, where we think of j = diag( j, j, j ) with j ∈ D ⊥ , so that C = D ⊕ Dj . Anelement of H ( C, id) can be written in the form X + Y · j, X ∈ H ( D, γ ) and Y ∈ Skew ( D, γ ). Then we can look at the elements t ∈ Aut( A ) that fix a subalgebra isomorphic to H ( D, γ ) t ( X + Y · j ) = t ( X ) + t ( Y · j ) , since t leaves H ( D, γ ) invariant and is an automorphism, it must leave Skew ( D, γ ) = H ( D, γ ) ⊥ invariant as well. This means we can think of t ∈ Aut( A ) in terms ofits action on the subalgebra and its perpendicular complement. We will renamethe map t | H ( D,γ ) := r, and the map t ( Y · j ) = s ( Y ) · j , which allows us to write, t ( X + Y · j ) = r ( X ) + s ( Y ) · j. Propsition 4.5.1.
For X ∈ H ( D, γ ) and Y, V ∈ Skew ( D, γ ) the followingare true, . X ( Y · j ) ∈ Skew ( D, γ ) , . ( Y · j )( V · j ) ∈ H ( D, γ ) .Proof. This can be shown easily through straight forward computation.We denote the products defined above by( X • Y ) · j := X ( Y · j ) , and Y ∗ V := ( Y · j )( V · j ) . Using the above notation we can then say that t (( X + Y · j )( U + V · j )) = t ( XU + Y ∗ V + ( X • V + U • Y ) · j )= r ( XU + Y ∗ V ) + s ( X • V + U • Y ) · j = r ( XU ) + r ( Y ∗ V ) + ( s ( X • V ) + s ( U • Y )) · j If we now use the fact that t ∈ Aut( A ), we can say t ( X + Y · j ) s ( U + V · j ) = ( r ( X ) + s ( Y ) · j ) ( r ( U ) + s ( V ) · j )= r ( X ) r ( U ) + s ( Y ) ∗ s ( V ) + ( r ( X ) • s ( V ) + r ( U ) ∗ s ( Y )) · j. Setting t (( X + Y · j )( U + V · j )) = t ( X + Y · j ) t ( U + V · j ), we see the following r ( XU ) = r ( X ) r ( U ) (4.3) r ( Y ∗ V ) = s ( Y ) ∗ s ( V ) (4.4) s ( X • V ) = r ( X ) • t ( V ) . (4.5)Define an algebra involution to be a map on a k -algebra A such that1. ι ( x + y ) = ι ( x ) + ι ( y ),2. ι ( xy ) = ι ( y ) ι ( x ), 16. ι ( x ) = x ,for all x, y ∈ A . We will be concerned with the case where the k -algebraMat ( D ), and ι = ι γ : Mat ( D ) → Mat ( D ) is induced by γ = diag( γ , γ , γ )with γ i ∈ k ∗ , by ι ( x ) = γ − ¯ x T γ .If we look at a typical element of the space Skew ( D, γ ) ∗ Skew ( D, γ ) ⊂ H ( D, γ ) we need to consider the product ( Y · j )( V · j ) = Y ∗ V , where Y, V ∈ Skew ( D, γ ), j ∈ D ⊥ . We have an element of the following form Y ∗ V = q ( j )2 (cid:16) γ − ( ι ( V ) · Y + ι ( Y ) · V ) T γ (cid:17) . (4.6)Now we consider the space H ( D, γ )) • Skew ( D, γ ) ⊂ Skew ( D, γ ), and lookat elements of the form X • V where X ∈ H ( D, γ ) and V ∈ Skew ( D, γ ). Forthis we will consider elements X ∈ H ( D, γ ), i.e. X = f x γ − γ ¯ x γ − γ ¯ x f x x γ − γ ¯ x f . We can look at the product X • V , where X ( V · j ) = ( X • V ) · j , and we arriveat X • V = (cid:0) V · X + ( V T · X T ) T (cid:1) . Over any field there is only one isomorphism class of k -involutions, which has asa representative conjugation by an element in Aut( A ) fixing an 11 dimensionalsubalgebra, so there is only one isomorphism class of their fixed point groupsby [11].We will take σ as defined above as our representative of this class of k -involutions. We will call the 11 dimensional subalgebra B ⊂ A , and so thesubgroup of G = Aut( A ) that leaves B invariant is G σ . Proposition 4.6.1.
The fixed point group of σ is isomorphic to Spin(
Q, E ) .Proof. The subgroup Aut( A ) w ⊂ Aut( A ) that leaves a primitive idempotent w invariant is isomorphic to Spin( Q, E ) where Q is the quadratic trace form on A and E is the 0-space of multiplication by w . The algebra E is isomorphicto the 11 dimensional algebra fixed by σ . Proposition 4.6.2.
The fixed point group of θ ◦ I t is isomorphic to Aut(Mat ( D ) , ι ) × Sp(1) where ι is an algebra involution on Mat ( D ) .Proof. Let r ∈ Aut( H ( D, γ )) for D ⊂ C a quaternion subalgebra of C and γ ∈ ( k ∗ ) . By [21], r extends uniquely to an element ˜ r ∈ Aut(Mat ( D ) , ι ), where ι = ι γ is the algebra involution in A induced by γ . If t ∈ Aut( H ( C, id)) suchthat t = id and t leaves elementwise fixed a subalgebra of the form H ( D, γ ).If X ∈ H ( D, γ ) ⊂ Mat ( D ) and Y ∈ Skew ( D, γ ) ⊂ Mat ( D ), then t ( X + Y · j ) = r ( X ) + s ( Y ) · j where j ∈ D ⊥ with q ( j ) = 0, r ∈ Aut( H ( D, γ ))17nd s ∈ L (Skew ( D, γ )). Let x, y ∈ D then t | C ( x + yj ) = r | D ( x ) + s | D ( y ) j such that r | D ∈ Aut( D ) and s | D = p ( r | D ), where p ∈ Sp(1), [15]. So s | D ∈ Aut( D ) × Sp(1), and we have s = ˜ pr ∈ Aut(Mat ( D ) , ι ) × H and det(˜ p ) = 1.But we have Aut(Mat ( D ) , ι ) × H ⊂ Aut( H ( C, id)), which has rank 4. Thesubgroup Aut( H ( D, γ )) ∼ = Aut(Mat ( D ) , ι ) is of type C , H ⊃ Sp(1) has rank1, and so H ∼ = Sp(1).These groups correspond to a description of coordinates given by Kac in [18],and what Serre calls Kac coordinates in [29]. We now provide a summary ofthe idea of Kac coordinates from [29] starting with the theory for k havingcharacteristic zero. These are also mentioned in [20]. We fix a maximal torus T , and a set of roots Φ( T ) with base ( α i ) i ∈ I , with α i ∈ X ∗ ( T ) ⊗ Z Q . We denoteby ˜ α = X i ∈ I λ i α i , the longest root. The coefficients λ i ∈ Z and λ i ≥
1. If we then take I = I ∪{ } ,and set α = − ˜ α we have X i ∈ I λ i α i = 0 . We can associate the set I to the set of vertices of the Dynkin diagram ofΦ( T ), and I to the set of vertices of the extended Dynkin diagram. Whenchar( k ) = 0 we choose a parametrization of the roots of unity. In general wewant a homomorphism ǫ : Q → K ∗ , where ker( ǫ ) = Z . For example the naturalchoice when K = C is the map ǫ ( χ ) = e πiχ .From this we can associate an element t χ ∈ T ( k ) in the following way ω ( t χ ) = ǫ X i ∈ I ξ i ( ω ) χ i ! , where ω ∈ X ∗ ( T ), and ξ i ( ω ) are the coordinates of ω with respect to ( α i ).When Z ( G ) contains only the identity this is enough to characterize t χ , andthis is the case for Aut( A ). We define the set P in the following way, P = n χ = ( χ i ) | χ i ≥ , X λ i χ i = 1 , i ∈ I o . Theorem, Kac 4.6.3.
Any element of finite order of G ( k ) is conjugate toexactly one t χ , with χ ∈ P . This is proven in [18, 20]. We can simplify the situation when looking forconjugacy classes of element of a fixed order κ . Let χ i = ρ i κ , and the equationdefining P becomes X i ∈ I λ i ρ i = κ. (4.7)In our case Aut( A ) if of type F , and choosing a popular set of roots andbase we have that ˜ α = 2 α + 3 α + 4 α + 2 α .
18o the elements of order 2 correspond to the solution ( ρ i ) i ∈ I with ρ i ∈ N suchthat ρ + 2 ρ + 3 ρ + 4 ρ + 2 ρ = 2 , and we have the solutions (0 , , , ,
0) and (0 , , , , t χ have Dynkindiagrams contained within the affine Dynkin diagram for G whose vertices cor-respond to the ρ i = 0. In our case we will delete the α vertex for one conjugacyclass, and the α vertex for another conjugacy class. ❡ ❡ ❡ ❡ ❡ α α α α α > This leaves us with the following two Dynkin diagrams. ❡ ❡ ❡ ❡ α α α α > ❡ ❡ ❡ ❡ α α α α > The first is the Dynkin diagram of a group of type A × C corresponding tothe fixed point group of type (I), Sp(1) × Aut(
M, ι ), and the second to a groupof type B corresponding to the fixed point group of type (II), Spin( Q, E ). The interpretation of conjugacy classes of k -involutions in terms of Galois coho-mology follows much the same way as [15]. If we first consider H ( Gal k , Aut(
A, K ))to be the cohomology group of Aut( A ) defined over K with coefficients in Gal k the absolute Galois group of k . In this case H ( Gal k , Aut(
A, K )) ∼ = Aut( A, k )the automorphism group of A defined over k . The group H ( Gal k , Aut(
A, K ))is the group of
K/k -forms of A . So when we consider the k -involutions Aut( A )we are looking at the subgroup of automorphisms that fix a certain subalgebraof A . These subalgebras come in two types. If t ∈ Aut( A ) is an element of order2, then t induces a k -involution I t , and t fixes a subalgebra of the form H ( D, γ )or E , the zero space of multiplication by some idempotent element in A . Ineither case, if we let B ⊂ A , the group H ( Gal k , Aut(
A, B, K )) corresponds tothe
K/k -forms of B . 19 roposition 4.7.1. Let A be a k -algebra, and B a subalgebra of A fixed byan element of order in Aut( A ) , then there is a bijection between C k , the iso-morphism classes of involutions of Aut( A ) , and H ( Gal k , Aut(
A, B, K )) the K/k -forms of B .Proof. This follows from 4.4.2.When B ∼ = H ( D, γ ) the cohomology group H ( Gal k , Aut(
A, B, K )) corre-sponds to the
K/k -forms, which is in bijection with isomorphism classes of k -involutions of Aut( A ) by 4.7.1. This also corresponds to the K/k -forms of fixedpoint groups, which we can think of as the isomorphism classes of the centralizerof the k -involution in Aut( A ). In other words the groups H ( Gal k , Aut(
A, B, K )and H ( Gal k , Z G ( ϕ )) are in bijection when ϕ is an element of order 2 in Aut( G ),where G ∼ = Aut( A ). References [1] M. Aschbacher, G.M. Seitz. Involutions in Chevalley groups over fields ofeven order.
Nagoya Math J. , volume 63: 1–91, 1976.[2] M. Berger. Sur les groupes d’holonomie homog`ene des vari´et´s `a connexionaffine et des vari´et´es riemanniennes.
Bull. Soc. Math. France
83, 279–330,1955.[3] M. Berger. Les espaces sym´etriques noncompacts.
Annales scientifiques del’ ´E.N.S. , series 3, 74(2), 85–177, 1957.[4] C.E. Dometrius A.G. Helminck, L. Wu. Involutions of SL( n, k ), ( n >
App. Appl. Math. , 90(1-2):91–119, 2006.[5] C.E. Dometrius A.G. Helminck, L. Wu. Classification of involutions ofSO( n ; k ; b ). To Appear.[6] H. Freudenthal. Beziehungen der e und e zur oktavenebene. viii. Nederl.Akad. Wetensch Prod. , 62(21):447–465, 1959.[7] F. Gantmacher. On the classification of real simple Lie groups.
Mat. Sb.
5, 217–249, 1939.[8] D. Gorenstein R. Lyons, R. Solomon.
The Classification of the Finite Sim-ple Groups, Number
3, volume 40(3) of
Mathematical Surveys and Mono-graphs
Amer. Math. Soc., Providence, 1994.[9] P. Gille T. Szamuely.
Central simple algebras and Galois cohomology , Cam-bridge University Press, New York, 2006.[10] A.G. Helminck Algebraic groups with a commuting pair of involutions andsemisimple symmetric spaces.
Adv. in Math. , 71(1), 21–91, 1988.2011] A.G. Helminck. On the classification of k -involutions. Adv. in Math. ,153(1):1–117, 2000.[12] A.G. Helminck. Symmetric k -varieties. In Algebraic Groups and theirGeneralizations: Classical Methods , volume 56, pages 233–279, Providence,1994. Amer. Math. Soc.[13] A.G. Helminck, S.P. Wang. On rationality properties of involutions ofreductive groups.
Adv. in Math. , k ). Comm. inAlg. , 30(1):193–203, 2002.[15] J.D. Hutchens isomorphism classes of k -involutions of G . submitted ,http://arxiv.org/abs/1211.1874.[16] N. Jacobson. Compositoin algebras and their automorphisms. Rend. delCir. Mat. di Palermo , 7(1):55–80, 1958.[17] N. Jacobson.
Structure and Representations of Jordan Algebras , volume 39.AMS Colloquium Publications, Providence, 1968.[18] V.G. Kac. Simple Lie groups and the Legendre symbol.
Algebra Carbondale1980 , Springer, 110–123, 1981.[19] T.Y. Lam.
Introduction to Quadratic Forms over Fields , volume 67 of
AMSGraduate Studes in Math.
Amer. Math. Soc., Providence, 2005.[20] S.P. Lehalleur. Subgroups of maximal rank of reductive groups. Soci´et´eMath´ematique de France, 2012.[21] W.S. Martindale III. Jordan homomorphisms of the symmetric elementsof a ring with involution.
J. of Algebra , volume 5, pages 232-249, 1967.[22] K. McCrimmon The Freudenthal-Springer-Tits constructions of excep-tional Jordan algebras.
Trans. Amer. Math. Soc.
A Taste of Jordan Algebras . Springer-Universitext, NewYork, 2005.[24] O.T. O’Meara.
Introduction to Quadratic Forms . Springer-Verlag, NewYork, 1963. Published in USA by Academic Press.[25] H.P. Petersson. Composition algebras over a field with a discrete valuation.
J. of Algebra , 29:414–426, 1974.[26] R.S. Pierce.
Associative Algebras . Springer-Verlag, New York, 1982.[27] J-P Serre.
A Course in Arithmetic . Springer-Verlag, New York, 1973.[28] J-P Serre.
Galois Cohomology . Springer-Verlag, Berlin, 1997.2129] J-P Serre. Coordonn´ees de Kac. Oberwolfach reports, 3, 2006.[30] T.A. Springer. The classification of reduced exceptional simple Jordanalgebras.
Nederl. Akad. Wetensch. Proc. Ser.A , 63 = Indag. Math. 22(1960), 414422.[31] T.A. Springer. The classification of involutions of simple algebraic groups.
J. Fac. Sci. Univ. Tokyo Sect. IA Math.
Linear Algebraic Groups . Birkhauser, Boston, 2nd edition,1998.[33] T.A. Springer
Jordan algebras and algebraic groups . Springer, Berline,1998.[34] T.A. Springer F.D. Veldkamp .