k G Isomorphy classes of -involutions of 2 3 1 John Hutchens 0 [email protected] 2 n Department of Mathematics a J North Carolina State University 3 Raleigh, NC 27695 1 January 15, 2013 ] R G . Abstract h t Isomorphy classes of k-involutions have been studied for their corre- a m spondence with symmetric k-varieties, also called generalized symmetric spaces. Asymmetrick-varietyof ak-group GisdefinedasGk/Hk where [ θ : G → G is an automorphism of order 2 that is defined over k and θ 4 Gk and Hk are the k-rational points of G and H = G , the fixed point v group ofθ,respectively. Thisisacontinuation ofpaperswrittenbyA.G. 4 Helminckandcollaborators[4],[3],[2],[1]expandingonhiscombinatorial 7 classification overcertain fields. Resultshavebeenachievedforgroupsof 8 type A, B and D. Here we begin a series of papers doing the same for 1 algebraic groups of exceptional type. . 1 1 1 Introduction 2 1 : The problem of identifying all isomorphy classes of symmetric k-varieties is de- v i scribedby Helminck in[5]. There he notesthatisomorphyclassesofsymmetric X k-varieties of algebraic groups and isomorphy classes of their k-involutions are r in bijection. In the following we provide a classification of isomorphy classes of a k-involutions for the split type of G over certain fields. 2 The main result of this paper is an explicit classification of k-involutions of the split formofG where k =R,C,Q,Q , andF , where q >2. We do this by 2 p q finding explicit elements of Aut(G ), where G = Aut(C) and C is always the 2 2 split octonion algebra over a given field of characteristic not 2. TheresultsfromthispaperrelymostheavilyontheworksofJacobson[6]on composition algebras,Lam’s presentation of quadratic forms [7], and Helminck et. al. on symmetric spaces and k-involutions of algebraic groups. A k-involution is an automorphism of order exactly 2, that is defined over the field k. The isomorphy classes of these k-involutions are in bijection with the quotient spaces G /H , where G and H are the k-rational points of the k k k k 1 groups G and H =Gθ ={g ∈G | θ(g)=g} respectively, these quotient spaces will be called symmetric k-varieties or generalized symmetric spaces. ThegroupofcharactersandrootspaceassociatedwithatorusT aredenoted ∗ by X (T) and Φ(T) respectively. We will also denote by A− ={a∈A | θ(a)=a−1}◦, θ and by − − I (A )={a∈A | θ◦Inn(a) is a k-involution}. k θ θ In the following we introduce a characterization of k-involutions of an alge- braic group of a specific type given by Helminck. The full classification can be completed with the classification of the following three types of invariants, [4], ∗ ∗ (1) classificationofadmissibleinvolutionsof(X (T),X (A),Φ(T),Φ(A)),where T is a maximal torus in G, A is a maximal k-split torus contained in T (2) classificationoftheG -isomorphyclassesofk-involutionsofthek-anisotropic k kernel of G − (3) classification of the G -isomorphy classes of k-inner elements a∈I (A ). k k θ For this paper we do not consider (2) since our algebraic groups will be k-split. We mostly focus on (3) and refer to (1) when appropriate, though Helminck has provided us with a full classification of these [4]. The main result is an explicit description of the k-inner elements up to isomorphy,whichcompletestheclassificationofk-involutionsforthesplitgroup ∗ ∗ of type G . Each admissible involution of (X (T),X (A),Φ(T),Φ(A)) can be 2 lifted to a k-involutionθ of the algebraic group. This lifting is not unique. The involutions that induce the same involution as θ when restricted to ∗ ∗ − (X (T),X (A),Φ(T),Φ(A)) are of the form θ◦Inn(a) where a∈A . This set θ ofelementssuchthatθ◦Inn(a)isak-involutionformthesetofk-innerelements − associated with the involution θ and are denoted I (A ). k θ Yokota wrote about k-involutions, θ, and fixed point groups, Gθ, for alge- braic groups of type G for k = R,C. In fact the elements of G = Aut(C) 2 2 we call I correspondto the γ maps in [13], which are a conjugation with t(±1,±1) respectto complexificationat different levels within the octonion algebrataken over R or C [13] . We found these using different methods than in [13], and show the correspondence. In section3.1we find k-involutionsthat come fromconjugationby elements inamaximalk-splittorusandusingaresultofJacobsonshowthey areisomor- phic in Proposition 2.2.8. This will take care of all cases where Aut(C) = G 2 is taken over a field whose structure permits a split octonion algebra and only split quaternion algebras, namely k =C and F when p>2. p Overotherfields thereis the possibility ofdivisionquaternionalgebras,and thisfactusingProposition2.2.3givesusanotherisomorphyclassofk-involutions when we take k = R,Q and Q . In section 3.2 we find the k-involution θ for p which our maximal k-split torus is a maximal (θ,k)-split torus and in Lemma 2 2.4.1 we find a representative for the only other possible isomorphy class of k-involutions over these fields. In section 3.3 we summarize the main results of the paper in Theorem2.4.3 and give the full classification of k-involutions when k =C,R,Q and F when p q p ≥ 2 and q > 2. We finish up by giving descriptions of the fixed point groups of isomorphy classes of k-involutions. 1.1 Preliminaries and recollections Most of our notation is borrowed from [11] for algebraic groups, [4] for k- involutions and generalized symmetric spaces, [7] for quadratic forms and [12] for octonion and quaternion algebras. The letter G is reserved for an arbitrary reductive algebraic group unless it is G , which is specifically the automorphism group of an octonion algebra. 2 When we refer to a maximal torus we use T and any subtorus is denoted by anothercapitalletter,usuallyA. LowercaseGreeklettersarefieldelementsand other lowercaseletters denote vectors. We use Z(G) to denote the center of G, Z (T)to denotethe centralizerofT inG,andN (T)todenote the normalizer G G of T in G. ByAut(G)wemeantheautomorphismgroupofG,andbyAut(C)wemean the linear automorphisms of the composition algebra C. The group of inner automorphisms are denoted Inn(G) and the elements of Inn(G) are denoted by I where g ∈G and I (x)=gxg−1. g g We define a θ-split torus, A, of an involution, θ, as a torus A⊂G such that θ(a) = a−1 for all a ∈ A. We call a torus (θ,k)-split if it is both θ-split and k-split. − Let A be a θ-stable maximal k-split torus such that A is a maximal (θ,k)- θ − split torus. By [1] there exists a maximal k-torus T ⊃ A such that T ⊃ θ − A is a maximal θ-split torus. The involution θ induces an involution θ ∈ θ ∗ ∗ Aut(X (T),X (A),Φ(T),Φ(A)). It was shown by Helminck [4] that such an involution is unique up to isomorphy. For T ⊃A a maximal k-torus, an ∗ ∗ θ ∈Aut(X (T),X (A),Φ(T),Φ(A)) is admissible if there exists an involution θ˜∈ Aut(G,T,A) such that θ˜| = θ, T A− is a maximal (θ,k)-split torus, and T− is a maximal θ˜-split torus of G. θ θ This will give us the set of k-involutions on G that extend from involutions ∗ on the group of characters, X (T). As for the k-inner elements they are defined as follows; if θ is a k-involution − and A is a maximal θ-split torus then the elements of the set, θ I (A−)= a∈A− (θ◦I )2 =id, (θ◦I )(G )=G , k θ θ a a k k n (cid:12) o (cid:12) arecalledk-innerelements ofθ. Somecompositionsθ◦I willnotbeisomorphic a − in the group Aut(G) for different a ∈ I (A ), though they will project down k θ to the same involution of the group of characters of a maximal torus fixing the − characters associated with a maximal k-split subtorus for all a∈I (A ). k θ 3 1.2 Split octonion algebra ThroughoutthispaperweuseN tobeaquadraticformofacompositionalgebra and h , i to be the bilinear form associated with N. The capital letters C and D denote composition algebras and composition subalgebras respectively. The compositionalgebraswewillrefertoalwayshaveanidentity,e. Thereisananti- automorphism on a composition algebra that resembles complex conjugation denoted by,¯, which will have specific but analogous definitions depending on the dimension of the algebra. If {e,a,b,ab} is a basis for D, a quaternion algebra such that a2 = α and b2 =β,thenwedenoteitsquadraticfromby, α,β ∼=h1,−α,−β,αβi. Wewill k (cid:16) (cid:17) often refer to a quaternion algebra over a field k by the 2-Pfister form notation of its quadratic form. Since a composition algebra is completely determined by its quadratic form and its center, ke, there is no risk of ambiguity. By k we are referring to an arbitrary field and by using the blackboardbold F we refer to a specific field. We consider only fields that do not have characteristic 2. We always consider the octonion algebra as a doubling of the split quater- nionsthoughtofasM (k),the2×2matricesoverourgivenfield. Theoctonion 2 algebrasweconsiderareanorderedpairofthesewithanextendedmultiplication that will be described in the next section. Forthefollowingresultswereferyouto[12]fortheproofs. Intheseupcoming resultsV isavectorspaceoverafield,k,char(k)6=2,equippedwithaquadratic form N :V →k and the associated bilinear form h , i:V ×V →k, such that 1. N(αv)=α2N(v), 2. hv,wi=N(v+w)−N(v)−N(w), where, v,w ∈V and α∈k. A composition algebra, C, willbe avectorspacewith identity, e, andN and h , i as above such that N(xy) = N(x)N(y). Note that ke is a composition algebra in a trivial way. Proposition, [12] 1.2.1. When C is a composition algebra with D ⊂ C a finite dimensional subalgebra of C with C 6= D, then we can choose a ∈ D⊥ with N(a) 6= 0, then D⊕Da is a composition algebra. The product, quadratic form, and complex conjugation are given by 1. (x+ya)(u+va)=(xu+αv¯y)+(vx+yu¯)a, for x,y,u,v∈D, α∈k∗ 2. N(x+ya)=N(x)−αN(y) 3. x+ya=x¯−ya. The dimension of D⊕Da is twice the dimension of D and α=−N(a). We often use this decomposition above in the results that follow, and a theorem of Adolf Hurwitz gives us the possible dimensions of such algebras. 4 Theorem, (A. Hurwitz), [12] 1.2.2. Every composition algebra can be ob- tained from iterations of the doubling process starting from ke. The possible dimensions of a composition algebra are 1,2,4, or 8. A composition algebra of dimension 1 or 2 is commutative and associative, a composition algebra of dimension 4 is associative and not commutative, a composition algebra of di- mension 8 is neither commutative nor associative. Corollary, [12] 1.2.3. Any doubling of a split composition algebra is again a split composition algebra. There are 2 general types of composition algebras. If there are no zero divisors we call the composition algebra a division algebra, and otherwise we call it a split algebra. It follows from the definition that a composition algebra is determined completely by its norm, and we have the following theorem. Theorem, [12] 1.2.4. In dimensions 2,4, and 8 there is exactly one split com- position algebra, over a given field k, up to isomorphism. We cantakeasasplitquaternionalgebra,D,overafieldk tobe M (k), the 2 2×2matricesoverk. MultiplicationinD willbe typicalmatrixmultiplication, our quadratic form will be given by x x x x N 11 12 =det 11 12 =x x −x x , (cid:18)(cid:20)x21 x22(cid:21)(cid:19) (cid:18)(cid:20)x21 x22(cid:21)(cid:19) 11 22 12 21 and bar involution will be given by x x x −x 11 12 = 22 12 . (cid:20)x21 x22(cid:21) (cid:20)−x21 x11 (cid:21) Elements of our split octonion algebra will have the form x x y y (x,y)= 11 12 , 11 12 . (cid:18)(cid:20)x21 x22(cid:21) (cid:20)y21 y22(cid:21)(cid:19) Since all split octonions over a given field are isomorphic, we can take α=1 in our composition algebra doubling process. The multiplication, quadratic form, and octonion conjugation are given by the following; (x,y)(u,v)=(xu+v¯y,vx+yu¯), (1.1) N (x,y) =det(x)−det(y), (1.2) (cid:0) (cid:1) (x,y)=(x¯,−y), (1.3) with x,y,u,v ∈ M (k) = D. The basis of the underlying vector space is taken 2 to be {(E ,0),(0,E )} , where the E are the standard basis elements ij ij i,j=1,2 ij for 2×2 matrices, and so our identity element in C is e=(E +E ,0). 11 22 5 2 Automorphisms of G 2 2.1 Some results on G 2 It is well known that the automorphism group of a k-split octonion algebra, C, over a field, k, is a k-split linear algebraic group of type G over k. We 2 can compute a split maximal torus for Aut(C), where C = M (k),M (k) as 2 2 above. Here we collect some some known results, and again(cid:0)we refer to [12](cid:1). Theorem 2.1.1. The following statements concerning G=Aut(C) are true; 1. There is a maximal k-split torus T ⊂G of the form T = diag(1,βγ,β−1γ−1,1,γ−1,β,β−1,γ) β,γ ∈k∗ . (cid:8) (cid:12) (cid:9) (cid:12) 2. The center of Aut(C) contains only the identity. 3. For any composition algebra, C, over k. The only nontrivial subspaces of C left invariant by Aut(C) are ke and e⊥. 4. All automorphisms of Aut(C) are inner automorphisms. 2.2 k-involutions of G Remark 2.2.1. If θ ∈Inn(G) and θ =I is a k-involution then t2 ∈Z(G). t Since groups of type G have a trivial center, the problem of classifying 2 k-involutions for Aut(C), where C is a split octonion algebra, is the same as classifying the conjugacy classes of elements of order 2 in Aut(C) that preserve the k-structure of Aut(C). Remark 2.2.2. The involutions that are of the form I where t t ∈T = diag(1,βγ,β−1γ−1,1,γ−1,β,β−1,γ) β,γ ∈k∗ (β,γ) (cid:8) (cid:12) (cid:9) have (β,γ)=(1,−1),(−1,1) or (−1,−1). (cid:12) Proof. We set diag(1,βγ,β−1γ−1,1,γ−1,β,β−1,γ)2 = id and exclude (β,γ) = (1,1) which corresponds to the identity map. UsingtheabovestatementandthefollowingresultofJacobsonwecanshow thatallk-involutionsgivenbyconjugationbyelementscomingfromthemaximal k-split torus T are isomorphic. Proposition,[6]2.2.3. LetC beanoctonionalgebraoverk,thentheconjugacy class of quadratic elements, t ∈ G = Aut(C) such that t2 = id are in bijection with the isomorphism classes of quaternion subalgebras of C. In particular if t ∈ Aut(C) has order 2, then it leaves some quaternion subalgebra D elementwise fixed giving us the eigenspace corresponding to the ⊥ eigenvalue1. ThenD isthe eigenspacecorrespondingtothe eigenvalue−1. If gtg−1 =s for some g ∈G, then s has order 2 and g(D)=D′, D′ a quaternion subalgebra elementwise fixed by s, and D ∼=D′. 6 Corollary2.2.4. LetC beanoctonionalgebraoverk andD andD′ quaternion subalgebras of C. If s,t∈G=Aut(C) are elements of order 2 and s,t fix D,D′ elementwise respectively, then s∼=t if and only if D ∼=D′ over k. Corollary 2.2.5. For s,t ∈ Aut(C), It ∼= Is if and only if s and t leave isomorphic quaternion subalgebras invariant. Wecanbeginlookingforelementsoforder2inthek-splitmaximaltoruswe have computed, which in our case will be t ∈ T, t2 = id. Solving the following equation diag(1,βγ,β−1γ−1,1,γ−1,β,β−1,γ)2 =id, we obtain the elements t where (β,γ)=(±1,±1), and (β,γ)6=(1,1). (β,γ) Lemma 2.2.6. IgIεIg−1 =Igǫg−1 Proof. We can apply the left hand side to an element y ∈G, I I I−1(y)=g ε(g−1yg)ε−1 g−1 g ε g =(g(cid:0)εg−1)y(gεg−1(cid:1))−1 =I(gεg−1)(y). Proposition 2.2.7. If ε2 =ε2 =id and ε ,ε ∈T a maximal torus of G when 1 2 1 2 Z(G)={id}, then Iε1 ∼=Iε2 if and only if ε1 =nε2n−1 for some n∈NG(T). Proof. If nε2n−1 = ε1 for n ∈ NG(T) then Iε1 ∼= Iε2 via the isomorphism Inn(I ). n I INowI−w1e, tlehtenIεw1e∼=havIeε2b,yaLnedmmsoat2h.2er.6e,exists a g ∈ G such that Iε1 = g ε2 g (cid:0) (cid:1) (gε g−1)−1ε y =y(gε g−1)−1ε , 2 1 2 1 for all y ∈ G and so (gε g−1)−1ε ∈ Z(G) = {id}. Thus we have that ε−1 = 2 1 1 (gε g−1)−1, so ε = gε g−1. So now notice S = gTg−1 is a maximal torus 2 1 2 containingε . ThegroupZ (ε )containsSandT,sothereexistsanx∈Z (ε ) 1 G 1 G 1 such that xSx−1 =T. We know that S =gTg−1 so xgTg−1x−1 =xgT(xg)−1 =T, which has xg ∈N (T). We notice that G IxgIε2Ix−g1 =Ixgε2(xg)−1 =Ixgε2g−1x−1 =Ixε1x−1 =I , ε1 which from the previous argument we have (xg)ε (xg)−1 =ε . 2 1 7 Using the previous proposition it is possible to find elements n,m∈N (T) G such that t(−1,−1) =n t(−1,1) n−1 =m t(1,−1) m−1. (cid:0) (cid:1) (cid:0) (cid:1) Itisalsopossibletoshow,andperhapsmoreillustrative,thattheyleaveiso- morphic quaternion subalgebras invariant, and thus by Corollary 2.2.5 provide us with isomorphic k-involutions. Proposition 2.2.8. t(−1,−1) ∼=t(−1,1) ∼=t(1,−1). Proof. LetG=Aut(C)⊃T = diag(1,βγ,β−1γ−1,1,γ−1,β,β−1,γ) β,γ ∈k∗ , then G is an algebraic group ov(cid:8)er the field k, char(k) 6= 2. The autom(cid:12) orphism (cid:9) (cid:12) t(−1,−1) leaves the split quaternion subalgebra (M2(k),0) elementwise fixed. The element of order 2, t(1,−1) =diag(1,−1,−1,1,−1,1,1,−1), leaves the quaternion subalgebra, 1 1 1 1 k ,0 k 0, k 0, k ,0 , (cid:18)(cid:20) 1(cid:21) (cid:19) (cid:18) (cid:20)−1 (cid:21)(cid:19) (cid:18) (cid:20)1 (cid:21)(cid:19) (cid:18)(cid:20) −1(cid:21) (cid:19) M M M e a b ab | {z } | {z } | {z } | {z } elementwise fixed. Notice that (b−a)(e+ab) = (0,0), and so the quaternion subalgebra is split. And t(−1,1) =diag(1,−1,−1,1,1,−1,−1,1), leaves the quaternion algebra 1 1 1 1 k ,0 k ,0 k 0, k 0, , (cid:18)(cid:20) 1(cid:21) (cid:19) (cid:18)(cid:20) −1(cid:21) (cid:19) (cid:18) (cid:20) 1(cid:21)(cid:19) (cid:18) (cid:20) −1(cid:21)(cid:19) M M M e a b ab | {z } | {z } | {z } | {z } elementwise fixed. Notice that (ab+b)(e+a) = (0,0), and so the quaternion subalgebra is split. Since over a given field k every split quaternion subalgebra is isomorphic, we have that t(−1,−1) ∼=t(−1,1) ∼=t(1,−1). Corollary 2.2.9. It(−1,−1) ∼=It(−1,1) ∼=It(1,−1). Sofromnowonwerefertoarepresentativeofthecongruenceclasscontaining I , I , and I as I , when there is no ambiguity. t(−1,−1) t(−1,1) t(1,−1) t Lemma 2.2.10. There is only one isomorphy class of k-involutions when k is a finite field of order greater than 2, complex numbers, p-adic fields when p>2, or when k is a complete, totally imaginary algebraic number field. Proof. In these cases only split quaternion algebras exist, [8], [12]. 8 In [13] Yokota talks about the maps γ,γ , and γ and shows are they iso- C H morphic,andthattheyarealsoisomorphictoanycompositionofmapsbetween them. In his paper he defines a conjugation coming from complexification. In particular we can look at γ , which is the complexification conjugation on the H quaternion level of an octonion algebra over R. If we take u+vc ∈ H ⊕Hc ⊥ where u,v ∈H and c∈H his map is γ (u+vc)=u−vc, H which in our presentation of the octonion algebra would look like, u u v v u −u v −v γ 11 12 , 11 12 = 11 12 , 11 12 , H(cid:18)(cid:20)u21 u22(cid:21) (cid:20)v21 v22(cid:21)(cid:19) (cid:18)(cid:20)−u21 u22 (cid:21) (cid:20)−v21 v22 (cid:21)(cid:19) and corresponds to our map I . t(−1,1) 2.3 Maximal θ-split torus Rather than trying to find a maximal θ-split torus, where θ ∼= It, and then computing its maximal k-splitsubtorus,we find ak-involutionθ thatsplits our already maximal k-split torus of the form T = diag(1,βγ,β−1γ−1,1,γ−1,β,β−1,γ) β,γ ∈k∗ . (cid:8) (cid:12) (cid:9) It is straight forwardto check that (cid:12) 1 1  1   1  s= , 1 1  M  1  1      is an element of Aut(C), where C is the split octonion algebra described above over a field k, char(k)6=2. It is immediate that T is a I -split torus. s Proposition 2.3.1. T is a maximal (I ,k)-split torus. s Proof. Notice firstthat ift∈T that I (t)=t−1, andnextthat T is k-splitand s is a maximal torus. Proposition 2.3.2. Is ∼=It Proof. The element s is an automorphism of order 2 of C, our split octonion algebra described above, that leaves the following quaternion algebra fixed ele- mentwise, 1 1 1 1 k ,0 k ,0 k 0, k 0, . (cid:18)(cid:20) 1(cid:21) (cid:19) (cid:18)(cid:20)1 (cid:21) (cid:19) (cid:18) (cid:20) 1(cid:21)(cid:19) (cid:18) (cid:20)1 (cid:21)(cid:19) M M M e a b ab | {z } | {z } | {z } | {z } Notice that (b+ab)(e+a+b+ab) = 0, and so the quaternion subalgebra is split. 9 2.4 Another isomorphy class of k-involutions over certain fields − We have seen that our maximal torus T =T , and so we can look at elements I s ofT fork-innerelementsofI thatwillgiveusnewconjugacyclassesoverfields s for whichquaterniondivisionalgebrascanexist. The fields we are interestedin include the real numbers, 2-adics, and rationals. Lemma 2.4.1. For C a split octonion algebra over a field k =R,Q ,Q, 2 s·t(1,−1) ∈Aut(C), leaves a quaternion division subalgebra elementwise fixed. Proof. The element s·t(1,−1) ∈ Aut(C) leaves the following quaternion subal- gebra elementwise fixed, 1 1 1 1 k ,0 k 0, k 0, k ,0 . (cid:18)(cid:20) 1(cid:21) (cid:19) (cid:18) (cid:20) −1(cid:21)(cid:19) (cid:18) (cid:20)1 (cid:21)(cid:19) (cid:18)(cid:20)−1 (cid:21) (cid:19) M M M e a b ab | {z } | {z } | {z } | {z } All basis elements are such that xx¯=1, and so have a norm isomorphic to the 2-Pfister form −1,−1 , where k =R,Q ,Q, which corresponds to a quaternion k 2 division algebra(cid:0) over e(cid:1)ach respective field. Moreover,over k =R or Q2 there is only one quaternion division algebra up to isomorphism. Lemma 2.4.2. For C a split octonion algebra over a field k = Q and p > 2 p s·t(−Np,−pNp−1) leaves a divison quaternion algebra elementwise fixed. Proof. The element s·t(−Np,−pNp−1) leaves the following quaternion subalgebra elementwise fixed, 1 −N p N k ,0 k 0, p k ,0 k 0, p , (cid:18)(cid:20) 1(cid:21) (cid:19) (cid:18) (cid:20)1 (cid:21)(cid:19) (cid:18)(cid:20)1 (cid:21) (cid:19) (cid:18) (cid:20) −p(cid:21)(cid:19) M M M e a b ab | {z } | {z } | {z } | {z } with Q∗/(Q∗)2 = {1,p,N ,pN }. This algebra is ismorphic to p,Np , which p p p p (cid:16) Qp (cid:17) is a representative of the unique ismorphy class of quaternion division algebras for a given p. Theorem 2.4.3. Let θ = I and G = Aut(C) where C is a split octonion s algebra over a field k, then 1. when k = R,Q,Q ; θ and θ◦I are representatives of 2 isomorphy 2 t(1,−1) classes of k-involutions of G. In the cases k =R or Q these are the only 2 cases, but this is not true for k =Q. 2. when k =Q and p>2, we have two isomorphy classes of k-involutions. p 10