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Torsors, Reductive Group Schemes and Extended Affine Lie Algebras

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Torsors, Reductive Group Schemes and Extended Affine Lie Algebras Torsors, Reductive Group Schemes and Extended Affine Lie Algebras Philippe Gille1 and Arturo Pianzola2,3 1UMR 8553 du CNRS, Ecole Normale Sup´erieure, 45 rue d’Ulm, 75005 Paris, France. 2Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada. 3Centro de Altos Estudios en Ciencia Exactas, Avenida de Mayo 866, (1084) Buenos Aires, Argentina. Abstract We give a detailed description of the torsors that correspond to multiloop al- gebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Ex- tended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that we take draws heavily for the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows us to find a bridge between multiloop algebras and the work of J. Tits on reductive groups over complete local fields. Keywords: Reductive group scheme, torsor, multiloop algebra. Extended Affine Lie Algebras. MSC 2000 17B67, 11E72, 14L30, 14E20. Contents 1 Introduction 1 2 Generalities on the algebraic fundamental group, torsors, and reductive group schemes 5 2.1 Thefundamentalgroup ............................. 5 2.2 Torsors ...................................... 8 2.3 An example: Laurent polynomials in characteristic 0 . .......... 9 2.4 Reductive group schemes: Irreducibility and isotropy . ............ 13 1 3 Loop, finite and toral torsors 14 3.1 Looptorsors.................................... 15 3.2 Loopreductivegroups ............................. 16 3.3 Looptorsorsatarationalbasepoint . ..... 17 3.4 Finitetorsors ................................... 20 3.5 Toraltorsors ................................... 21 4 Semilinear considerations 23 4.1 Semilinearmorphisms . .. .. .. .. .. .. .. .. 23 4.2 Semilinearmorphisms . .. .. .. .. .. .. .. .. 25 4.3 Caseofaffineschemes .............................. 25 4.4 Groupfunctors .................................. 27 4.5 Semilinear version of a theorem of Borel-Mostow . ........ 30 4.6 Existence of maximal tori in loop groups . ...... 33 4.7 VariationsofaresultofSansuc.. ..... 35 5 Maximal tori of group schemes over the punctured line 38 5.1 Twinbuildings .................................. 39 5.2 Proof of Theorem ?? ............................... 40 6 Internal characterization of loop torsors and applications 44 6.1 Internal characterization of loop torsors . ......... 44 6.2 Applications to (algebraic) Laurent series. .......... 49 7 Isotropy of loop torsors 51 7.1 Fixedpointstatements. 51 7.2 Caseofflagvarieties .............................. 55 7.3 Anisotropiclooptorsors . 58 8 Acyclicity 63 8.1 Theproof ..................................... 63 8.2 Application: Witt-Tits decomposition . ....... 65 8.3 Classification of semisimple k–loopadjointgroups. 66 8.4 Action of GLn(Z) ................................ 68 9 Small dimensions 79 9.1 Theonedimensionalcase . .. .. .. .. .. .. .. .. 79 9.2 Thetwodimensionalcase . 80 9.2.1 Classification of semisimple loop R2-groups............... 80 9.2.2 Applications to the classification of EALAs in nullity 2. ....... 85 9.2.3 Rigidity in nullity 2 outside type A. .................. 88 9.2.4 Tables................................... 90 2 10 The case of orthogonal groups 93 11 Groups of type G2 95 12 Case of groups of type F4, E8 and simply connected E7 in nullity 3 96 13 The case of PGLd 99 13.1 LoopAzumayaalgebras . 99 13.2 Theonedimensionalcase . 101 13.3 Thegeometriccase............................... 104 13.4 Loop algebras of inner type A .......................... 109 14 Invariants attached to EALAs and multiloop algebras 110 15 Appendix 1: Pseudo-parabolic subgroup schemes 111 15.1 The case of GLn,Z. ................................ 112 15.2Thegeneralcase ................................. 115 16 Appendix 2: Global automorphisms of G–torsors over the projective line116 1 Introduction To our good friend Benedictus Margaux Many interesting infinite dimensional Lie algebras can be thought as being “finite dimensional” when viewed, not as algebras over the given base field, but rather as algebras over their centroids. From this point of view, the algebras in question look like “twisted forms” of simpler objects with which one is familiar. The quintessential example of this type of behaviour is given by the affine Kac-Moody Lie algebras. Indeed the algebras that we are most interested in, Extended Affine Lie Algebras (or EALAs for short), can roughly be thought of as higher nullity analogues of the affine Kac-Moody Lie algebras. Once the twisted form point of view is taken the theory of reductive group schemes developed by Demazure and Grothendieck [SGA3] arises naturally. Two key concepts which are common to [GP2] and the present work are those of a twisted form of an algebra, and of a multiloop algebra. At this point we briefly recall what these objects are, not only for future reference, but also to help us redact a more comprehensive Introduction. *** 3 Unless specific mention to the contrary throughout this paper k will denote a field of characteristic 0, and k a fixed algebraic closure of k. We denote k–alg the category of associative unital commutative k–algebras, and R and object of k–alg. Let n 0 and m > 0 be integers that we assume are fixed in our discussion. Consider≥ the 1 1 1 1 ± m ± m Laurent polynomial rings R = Rn = k[t1± ,...,tn± ] and R′ = Rn,m = k[t1 ,...,tn ]. For convenience we also consider the direct limit R′ = limRn,m taken over m which ∞ in practice will allow us to “see” all the Rn,m at the same−→ time. The natural map R R′ is not only faithfully flat but also ´etale. If k is algebraically closed this → extension is Galois and plays a crucial role in the study of multiloop algebras. The explicit description of Gal(R′/R) is given below. Let A be a k–algebra. We are in general interested in understanding forms (for the fppf-topology) of the algebra A R, namely algebras over R such that ⊗k L (1.1) S A S (A R) S L ⊗R ≃ ⊗k ≃ ⊗k ⊗R for some faithfully flat and finitely presented extension S/R. The case which is of most interest to us is when S can be taken to be a Galois extension R′ of R of Laurent polynomial algebras described above.1 Given a form as above for which (1.1) holds, we say that is trivialized by S. L L The R–isomorphism classes of such algebras can be computed by means of cocycles, just as one does in Galois cohomology: (1.2) Isomorphism classes of S/R–forms of A R H1 S/R, Aut(A) . ⊗k ←→ fppf The right hand side is the part “trivialized by S” of the pointed set of non-abelian cohomology on the flat site of Spec(R) with coefficients in the sheaf of groups Aut(A). 1 In the case when S is Galois over R we can indeed identify Hfppf S/R, Aut(A) with the “usual” Galois cohomology set H1 Gal(S/R), Aut(A)(S) as in [Se]. Assume now that k is algebraically closed and fix a compatible set of primitive m– e th roots of unity ξm, namely such that ξme = ξm for all e > 0. We can then identify n n Gal(R′/R) with (Z/mZ) where for each e = (e1,...,en) Z the corresponding 1 ∈ 1 e m e m element e =(e , , e ) Gal(R′/R) acts on R′ via t = ξ i t . 1 ··· n ∈ i m i The primary example of forms of A R which are trivialized by a Galois L ⊗k extension R′/R as above are the multiloop algebras based on A. These are defined as follows. Consider an n–tuple σ =(σ1,...,σn) of commuting elements of Autk(A) m n satisfying σi = 1. For each n–tuple (i1,...,in) Z we consider the simultaneous ij ∈ eigenspace A = x A : σ (x) = ξmx for all 1 j n . Then A = A , i1...in { ∈ j ≤ ≤ } i1...in 1 P The Isotriviality Theorem of [GP1] and [GP3] shows that this assumption is superflous if Aut(A) is a an algebraic k–group whose connected component is reductive, for example if A is a finite dimensional simple Lie algebra. 4 and A = Ai1...in if we restrict the sum to those n–tuples (i1,...,in) for which 0 ij < mLj. ≤ The multiloop algebra corresponding to σ, commonly denoted by L(A,σσ), is de- fined by i i1 n m m L(A,σσ)= Ai1...in t ...tn A k R′ A k R′ n (i1,...,i⊕n) Z ⊗ ⊂ ⊗ ⊂ ⊗ ∞ ∈ Note that L(A,σσ), which does not depend on the choice of common period m, is not only a k–algebra (in general infinite dimensional), but also naturally an R–algebra. It is when L(A,σσ) is viewed as an R–algebra that Galois cohomology and the theory of torsors enter into the picture. Indeed a rather simple calculation shows that L(A,σσ) R′ A R′ (A R) R′. ⊗R ≃ ⊗k ≃ ⊗k ⊗R Thus L(A,σσ) corresponds to a torsor over Spec(R) under Aut(A). When n = 1 multiloop algebras are called simply loop algebras. To illustrate our methods, let us look at the case of (twisted) loop algebras as they appear in the theory of affine Kac-Moody Lie algebras. Here n = 1, k = C and A = g is a finite-dimensional simple Lie algebra. Any such is naturally a Lie algebra over 1 L R := C[t± ] and S g C S (g C R) S for some (unique) g, and some L ⊗R ≃ ⊗ ≃ ⊗ ⊗R finite ´etale extension S/R. In particular, is an S/R–form of the R–algebra g C R, L ⊗ with respect to the ´etale topology of Spec(R). Thus corresponds to a torsor over Spec(R) under Aut(g) whose isomorphism class is anL element of the pointed set 1 1 H ´et R, Aut(g) . We may in fact take S to be R′ = C[t± m ]. Assume that A is a finite-dimensional. The crucial point in the classification of forms of A R by cohomological methods is the exact sequence of pointed sets ⊗k ψ (1.3) H1 R, Aut0(A) H1 R, Aut(A) H1 R, Out(A) , et´ → et´ −→ et´ where Out(A) is the (finite constant) group of connected components of the algebraic k–group Aut(A).2 Grothendieck’s theory of the algebraic fundamental group allows us to identify 1 Het´ R, Out(A) with the set of conjugacy classes of n–tuples of commuting elements of the corresponding finite (abstract) group Out(A) (again under the assumption that k is algebraically closed).
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