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On Some Algebras Related to Simple Lie Triple Systems
Journal of Algebra 219, 234᎐254 Ž1999. Article ID jabr.1999.7877, available online at http:rrwww.idealibrary.com on
On Some Algebras Related to Simple Lie Triple Systems Pilar Benito* and Cristina Draper † Departamento de Matematicas y Computacion, ´ ´ Uni¨ ersidad de La Rioja, 26004 Logrono, ˜ Spain
and Alberto Elduque ‡ Departamento de Matematicas, Uni¨ ersidad de Zaragoza, 50009 Zaragoza, Spain ´ Communicated by Georgia Benkart Received August 10, 1998
The set Hom De r T ŽT mF T, T . is determined for any simple finite dimensional Lie triple system T over a field F of characteristic zero. It turns out that it contains nontrivial elements if and only if T is related to a simple Jordan algebra. In particular this provides a new proof of the determination by Laquer of the invariant affine connections in the simply connected compact irreducible Riemannian symmetric spaces. 䊚 1999 Academic Press
1. INTRODUCTION Given a Lie group G acting smoothly and transitively on a manifold M, the isotropy subgroup H at any point p g M Žso that M , GrH ., the Lie algebra ᒄ of the group G and the Lie algebra ᒅ of H Ž ᒅ F ᒄ ., the manifold M is called a reducti¨ e homogeneous space in case ᒄsᒅ[ᒊ
Ž>.
*Supported by the Spanish DGICYT ŽPb 94-1311-C03-02. and by the Universidad de La Rioja ŽAPI-97rB05.. † Supported by a grant from the ‘‘Programa de Formacion ´ de Profesorado Universitario’’ ŽMEC. and by the Universidad de La Rioja ŽAPI-97rB05.. ‡ Supported by the Spanish DGICYT ŽPb 94-1311-C03-03. and by the Universidad de La Rioja ŽAPI-97rB05.. 234 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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for a subspace ᒊ with ŽAd h.Ž ᒊ . ; ᒊ for any h g H, which implies that w ᒅ, ᒊ x : ᒊ Žand if H is connected, these are equivalent.. In this situation Nomizu w19, Theorem 8.1x established a bijection between the set of G-invariant affine connections on M and the set of bilinear maps
␣: ᒊ=ᒊªᒊ with the property that
␣ Ž Ž Ad h . X , Ž Ad h . Y . s Ž Ad h . ␣ Ž X , Y .
᭙h g H , ᭙ X , Y g ᒊ ,
that is, with Ad H < ᒊ : AutŽ ᒊ, ␣ . Žthe automorphism group of the nonassociative algebra Ž ᒊ, ␣ ... Again the condition Ad H < ᒊ : AutŽ ᒊ, ␣ . implies the condition ad ᒅ < ᒊ : DerŽ ᒊ, ␣ ., and they are equivalent if H is connected. The torsion, curvature, geodesics, holonomy, etc. of the connection associated to Ž ᒊ, ␣ . can then be expressed and studied in terms of this nonassociative algebra. Notice too that the set of the multiplications ␣ : ᒊ = ᒊ ª ᒊ such that ad ᒅ < ᒊ : DerŽ ᒊ, ␣ . is nothing else but the vector space of the homomorphisms of ᒅ-modules from ᒊ m⺢ ᒊ into ᒊ: Hom ᒅ Ž ᒊ m⺢ ᒊ, ᒊ ., which is a purely algebraic object. Thus, the problem of determining the invariant affine connections on reductive homogeneous spaces reduces to the problem of classifying algebras with a given subgroup of its automorphism group or a given subalgebra of its Lie algebra of derivations. The same problem has arisen in studying real division algebras w1x, nonunital composition algebras w2, 20x or in the classification of some flexible Lie-admissible algebras Žsee w17x and the references therein ., a subject we will come back to later on. In some cases, the algebras that appear related to some reductive homogeneous spaces are already known algebras. This is what happens for the six- and seven-dimensional spheres, viewed as S 6 s G 2rSUŽ3. and S 7 s SpinŽ7.rG 2 , where algebras related to a so-called color algebra w3x and the simple non-Lie Malcev algebras w4x appear. In 1992, Laquer w13, 14x computed the sets Hom ᒅ Ž ᒊ m⺢ ᒊ, ᒊ . associated with the simply connected compact irreducible Riemannian symmetric spaces Žsee w7x for definitions.. For these spaces, the decomposition Ž >. is a grading of ᒄ over ⺪ 2 s ⺪r2⺪, ᒊ thus being a Žsimple. Lie triple system. Therefore the problem was to compute Hom S ŽT m⺢ T, T . for some ⺪ 2-graded simple Žthat is, without nontrivial homogeneous ideals. finite dimensional real Lie algebra L s S [ T Ž S being the even part and T the odd one.. This was done by extending scalars to the complex field and by decomposing there the tensor product T⺓ m⺓ T⺓ as a direct sum of irreducible modules. The vector space Hom S ŽT m⺢ T, T . turned out to be
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0 Žwhich amounts to saying that the only invariant affine connection is the canonical one. with a few exceptions. Our purpose in this paper is twofold: Ži. We will determine the set Hom S Ž T mF T , T . for any ⺪ 2-graded simple finite dimensional Lie algebra L s S [ T over any arbitrary field of characteristic zero, without actually decomposing T mF T Žwhen restricted to the real field, this determines the invariant affine connections on the irreducible affine symmetric spaces.. Žii. We will relate the resulting nontrivial algebras with the simple finite dimensional Jordan algebras. In a sense, we can say that the Jordan algebras are responsible for the existence of noncanonical invariant affine connections on the symmetric spaces. The close relationship of Jordan algebras with some symmetric spaces is well known Žsee, for instance, w16, Chapter VIIIx.. Our results give another aspect of this relationship. Actually, our determination of the sets Hom S ŽT mF T, T . over algebraically closed fields will be very quick, using information on the ⺪ 2-graded simple Lie algebras that is contained in the Dynkin diagrams associated with the corresponding Lie triple systems T by Faulkner w5x, with no need of a case-by-case decomposition of T mF T. Our approach is inspired by w6, Theorem 1x. This allows us to simplify some of the results in w13x and w14x. From the point of view of nonassociative algebras, a central role in what follows is played by the multiplication defined on the set J 0 of Žgeneric. trace zero elements in a Jordan algebra J, given by projecting the product in J onto J 0 Žsee Ž). in 3.2.. This procedure is not new; for example, when applied to Cayley᎐Dickson algebras it gives all the central simple non-Lie Malcev algebras; it appears too in the construction of the pseudo-octonion algebras Žsee w17x and the references therein ..
2. LIE TRIPLE SYSTEMS In this section, some facts concerning simple Lie triple systems and the Dynkin diagrams attached to them will be recalled.
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2.1 A Lie triple system is a vector space T, over a ground field F, equipped with a trilinear product w xyz x satisfying
w xxy x s 0 w xyz x q w yzx x q w zxy x s 0 the map w xy y x : T ª T , z ¬ w xyz x is a derivation of T for any x, y, z g T. If S s span²w xy y x : x, y g T :, S is a Lie subalgebra of the Lie algebra gl ŽT . Žactually of DerŽT ..; the ⺪ 2-graded Lie algebra L s S [ T Žwhere the multiplication is given by the multiplication in S Ževen part. as a subalgebra of gl ŽT ., the natural action of S on T and w t 1 , t 2 x s w t 1 t 2 y x for any t 1 , t 2 g T . is called the standard imbedding of T. Thus any Lie triple system is nothing else but the odd part of a ⺪ 2-graded Lie algebra with product w xyz x s ww x, y x, z x. The Lie triple system is simple if and only if its standard imbedding is graded simple Žfor these facts see w15x.. 2.2 Now assume that T is a simple Lie triple system over the field F and let ⌫ s ␣ g End F ŽT . : ␣ w xyz x s wŽ ␣ x . yz x s w x Ž ␣ y . z x s w xy Ž ␣ z .x ᭙ x, y, z g T 4 be its centroid. ⌫ is a field extension of F and, for any x, y g T,w xy y x g End ⌫ ŽT .; therefore the standard imbedding L is a ⌫-algebra. Moreover, the centroid ⌳ of L is a commutative ⺪ 2-graded algebra: ⌳ s ⌳ 0 [ ⌳ 1 , with ⌳ 0 s ␣ g ⌳ : ␣ S : S, ␣ T : T 4 and ⌳ 1 s ␣ g ⌳ : ␣ S : T, ␣ T : S4 , with any nonzero homogeneous element being invertible Žby the graded simplicity of L., and it easily follows that ⌳ 0 s ⌫. There are two possibilities: 2.2Ža. If there is an element 0 / u g ⌳ 1 , then ⌳ 1 s ⌫u, so that ⌳ s ⌫w u x, with 0 / u 2 s g ⌫, T s uS, and L s ⌫w u x m⌫ S. In this case S is a central simple Lie algebra over ⌫ and T s uS is the adjoint module for S; so we will refer to this case as T being of adjoint type. For any x, y, z g S, wŽ ux .Ž uy .Ž uz .x s u 2 Ž uww x, y x, z x. s Ž uww x, y x, z x.; so T is isomorphic to the Lie triple system S under the product xyz 4 s ww x, y x, z x. Notice that if u 2 g ⌫ 2 , then can be taken to be 1. 2.2Žb. ⌳ s ⌫. Then L is a simple algebra Žungraded.: otherwise for any proper ideal I of L, by graded simplicity Ž S l I . [ ŽT l I . s 0 and S Ž I . [ T Ž I . s L Žwhere S and T denote the projections on S and T respectively.; therefore S l I s T l I s 0 and S Ž I . s S, T Ž I . s T. Hence for any s g S there is a unique t s g T with s q t s g I, and also for
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any t g T there is a unique, st g S with st q t g I. Then the odd map given by s ¬ t s and t ¬ st is easily checked to be an odd element in the centroid of L, a contradiction with ⌳ s ⌫. Therefore in this case, for any extension field Kr⌫, K m⌫ L is a simple algebra Žand ⺪ 2-graded.. From now on all the algebras and systems considered will be assumed to be finite dimensional. 2.3 Lister showed in w15, Theorem 4.5x Žsee also w5, Proposition 1x. that, given any simple Lie triple system T over an algebraically closed field of characteristic zero with standard imbedding L s S [ T, either 2.3Ži. S is semisimple and T is an irreducible self-dual S-module Žthis is always the situation for the adjoint type cases. or 2.3Žii. S s Fz [ w S, S x, where z is a nonzero central element of S and w S, S x is semisimple, and T s T1 [ T2 , where T1 and T2 are irreducible dual S-modules, so that z acts as a nonzero scalar on T1 and as its negative on T2 . This is the situation that occurs after complexifying the simple Lie triple systems associated to the hermitian symmetric spaces. 2.4 Faulkner w5x associated a diagram with any such simple Lie triple system, starting with the Dynkin diagram of the semisimple Lie algebra w S, S x and adding ‘‘marked’’ nodes representing the lowest weight of the representation of w S, S x on each irreducible summand of T. These diagrams thus contain the information on the structure of S and of T as an S-module Žwhich is enough to determine L.. The resulting diagrams are exactly the affine diagrams X NŽ1. and X NŽ2. Žwhich are equipped with some numerical labels in the nodes representing the linear dependence of the roots and weights involved; see w12, Chap. 4x. with one or two marked nodes. More precisely, the diagrams that appear are exactly: 2.4ŽI. The diagrams X NŽ1. with a node labeled by 1 marked Žby the symmetry of these diagrams, in this case it does not matter which is the marked node. ᎏthese correspond to the adjoint types. 2.4ŽII. The diagrams X NŽ1. with two nodes labeled by 1 markedᎏthese correspond to case 2.3Žii. above. Here the same diagram may give different possibilities according to which two nodes are marked. The same comment applies to the next cases.
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2.4ŽIII. The diagrams X NŽ1. with a node labeled by 2 markedᎏin this case the highest weight of T as an S-module Žwhich is the negative of the lowest weight since T is self-dual. is not in the root lattice, which is equivalent to 0 not being a weight of the representation of S on T Žsee w8, Exercise 21.3x.. 2.4ŽIV. The diagrams X NŽ2. Žso XN is A N Ž N G 2., DN , or E6 . with a node labeled by 1 markedᎏin this case 0 is a weight of the representation of S on T. This implies that there are roots which are weights and, therefore, at least the short roots of every simple ideal of S are weights Žnotice that the action of S on T is faithful, so that, for any simple ideal I of S and any 0 / ¨ g T in the zero weight space, I¨ / 0.. In case 2.4ŽI. L s S [ T is isomorphic Žas ungraded algebra. to the direct sum of two copies of the simple algebra of type X N . In the remaining cases L is a simple Žungraded. Lie algebra of type X N and the semisimple Lie algebra w S, S x Žwhich is the whole S if only one marked node appears. is the semisimple Lie algebra associated with the finite Dynkin diagram obtained by removing the marked nodes. Remark. Faulkner obtained these diagrams just for the Dynkin diagram of w S, S x plus the lowest weights of the irreducible subrepresentations of w S, S x on T. V. Kac had previously arrived at the same situation in his research on automorphisms of finite order in semisimple Lie algebras over an algebraically closed field of characteristic zero Žsee w10; 11; 12, Chapter 8; 7, Chap. Xx., since a ⺪ 2-graded Lie algebra is just a Lie algebra with an automorphism of order 2 Žcharacteristic not 2.. 2.5 Among the simple Lie triple systems, the ones obtained from Jordan algebras will be of particular interest to us. Given any simple Jordan algebra J of degree n G 3 with generic trace t w9, Chap. VIx and multiplication x ( y over a field of characteristic zero, its set of trace zero elements J 0 s x g J : t Ž x . s 04 becomes a simple Lie triple system under the trilinear multiplication w xyz x s Ž y ( z .( x y y (Ž z ( x . Žthe associator of y, z, and x .. Since w xy y x s w R x , R y x, where R u¨ s u( ¨ s ¨ ( u, it follows that in this case S s w R J , R J x s Der J is the Lie algebra of derivations of the Jordan algebra J and the standard imbedding is L s Der J [ J 0 ( Der J [ R J 0 , which is the derived subalgebra of the Lie multiplication algebra of J w9, Theorem 8.3x.
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Over an algebraically closed field F of characteristic zero there are the following possibilities for the simple Jordan algebras of degree n G 3: 2.5Ži. J s Mat nŽ F .q is the algebra of n = n matrices over F under the product x ( y s 12 Ž xy q yx .. In this case J 0 s sl Ž n, F . is a simple Lie triple system of adjoint type since Der J s ad J 0 , so that the associated diagram is
2.5Žii. J is the algebra of symmetric n = n matrices over F: J s Ž H Mat nŽ F ., t . Ž t the transposition.. In this case Der J is isomorphic to the set of skew-symmetric matrices Žsee w9, Theorem 6.9x. and the Lie algebra L s Der J [ J 0 is isomorphic to sl Ž n, F .. The associated diagram is
2.5Žiii. J is the algebra of symmetric 2 n = 2 n matrices with respect to the standard symplectic involution. Again in this case Der J is isomorphic to the skew-symmetric matrices with respect to this involution w9, Theorem 6.9x, L is isomorphic to sl Ž2 l, F ., and the associated diagram is
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2.5Živ. J is the exceptional simple Jordan algebra, whose derivation algebra is the Lie algebra of type F4 w9, Sect. 9.11x. The associated diagram is then
3. ALGEBRAICALLY CLOSED FIELDS We intend to determine in this section the vector space Hom S ŽT mF T, T . for any simple Lie triple system T with standard imbedding L s S [ T, over an algebraically closed field F of characteristic zero, an assumption we will keep throughout the section. 3.1 In case T s T1 [ T2 is a sum of two irreducible S-modules Ždiagrams 2.4ŽII.., the center of S is one-dimensional: ZŽ S . s Fz, with ad z < T1 s 1, ad z < T 2 s y1. Therefore T mF T is the sum of the eigenspaces 2, 0, y2 for ad z, which forces Hom S ŽT mF T, T . s 0. 3.2 In case T is of adjoint type, then Hom S ŽT mF T, T . ( Hom S Ž S mF S, S . and there is a distinguished Žskew-symmetric. element in this latter space: the Lie multiplication in S. So at least dim Hom S ŽT mF T, T . G 1. Actually, by w6, Theorem 1 and Corollary 2x dim Hom S Ž T mF T , T . s
½
2 1
if S is of type A n Ž n G 2 . otherwise.
On the other hand, if T s J 0 is the triple system associated with a simple Jordan algebra of degree n G 3 with multiplication x ( y, then S s Der J and we obtain a nonzero Žsymmetric. element of Hom S ŽT mF T, T . by means of J 0 mF J 0 ª J 0 x m y ¬ x ⭈ y s x( y y
1 n
tŽ x( y.1
Ž ).
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Žthe projection of the element x ( y onto J 0 .. Notice that if the degree were 2, this would be zero. Besides, if J s Mat nŽ F .q, then the triple system J 0 is both of adjoint type Ž A ny1 . and of ‘‘Jordan type.’’ Therefore, for the adjoint type Ždiagrams 2.4ŽI.., Hom S ŽT mF T, T . is spanned by the ‘‘Lie bracket in S’’ for types other than A n Ž n G 2. and by the ‘‘Lie bracket in S’’ and the ‘‘Jordan product Ž).’’ for types A n Ž n G 2.. 3.3 The situation for the simple Lie triple systems with diagrams 2.4ŽIII. can be settled immediately with the following lemma Žinspired by w6, Theorem 1x.: LEMMA. Let S be a semisimple Lie algebra, H a Cartan subalgebra of S, V an irreducible self-dual module with highest weight relati¨ e to H, and ¨ and ¨ y nonzero weight ¨ ectors for and y. For any root ␣ of S relati¨ e to H let S␣ be the corresponding root space. If V0 is the weight 0-space in V, then the linear map
: Hom S Ž V mF V , V . ª ¨ g V0 : S␣ ¨ s 0 ᭙␣ H 4 ¬ Ž ¨ m ¨ y . is well defined and one-to-one. Proof. Since V is self-dual, y is the lowest weight of V and ¨ m ¨ y generates V mF V. Hence any g Hom S Ž V mF V, V . is determined by Ž ¨ m ¨ y . g V0 . But, for any root ␣ orthogonal to the weight and any x␣ g S␣ , x␣ ¨ s 0 s x␣ ¨ y , so x␣ Ž ¨ m ¨ y . s 0 and x␣ Ž ¨ m ¨ y . s 0. As a direct consequence, Hom S ŽT mF T, T . s 0 for the simple Lie triple systems with diagrams 2.4ŽIII., for which 0 is not a weight of the representation of S on T. 3.4 From the paragraphs above, only the simple Lie triple systems with diagrams 2.4ŽIV. remain to be studied. According to Lemma 3.3, we will try to bound the dimension of t g T0 : S␣ t s 0 ᭙␣ H 4 for a triple system T which is an irreducible S-module of highest weight . LEMMA. Let T be a simple Lie triple system which is an irreducible module for S s w TT᎐ x. Let H be a Cartan subalgebra of the semisimple Lie algebra S, ⌽ the associated root system, ⌬ a base of ⌽, and ⌳ the set of weights of T relati¨ e to H. For any ⭋ / ⌬⬘ : ⌬, let S⌬⬘ be the subalgebra of S generated by the root spaces S␣ , ␣ g ⌬⬘ j y ⌬⬘, let T⌬⬘ be the sum of the weight spaces T for g ⌳ l ⺪⌬⬘, and let L⌬⬘ s S⌬⬘ [ T⌬⬘.
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Then L⌬⬘ is a graded subalgebra of L s S [ T that decomposes as L⌬⬘ s Z⌬⬘ [ w L⌬⬘ , L⌬⬘ x , where Z⌬⬘ s ¨ g T0 : S␣ ¨ s 0 ᭙␣ g ⌬⬘ j y ⌬⬘4 is its center and w L⌬⬘ , L⌬⬘ x is semisimple Ž hence a direct sum of graded simple algebras.. In particular, the codimension in T0 of Z⌬⬘ equals the dimension of the zero weight space in w S⌬⬘ , T⌬⬘ x Ž the odd part of w L⌬⬘ , L⌬⬘ x.. Proof. Since S⌬⬘ s Ý ␣ g ⌬⬘ Fh␣ q Ý  g ⌽ l ⺪ ⌬⬘ S Žthe h␣ ’s as in w8, p. 37x., S⌬⬘ is a semisimple subalgebra of S with Cartan subalgebra H⌬⬘ s H l S⌬⬘ s Ý ␣ g ⌬⬘ Fh␣ . Besides, ⌬⬘ is a base of the root system of S⌬⬘: ⌽ l ⺪⌬⬘. Moreover, L⌬⬘ is clearly a graded subalgebra of L. By complete reducibility T⌬⬘ s Z⌬⬘ [ w S⌬⬘ , T⌬⬘ x, and if 0 / W is an irreducible S⌬⬘-module in w S⌬⬘ ,T⌬⬘ x, its highest weight belongs to ⺪⌬⬘, so that 0 is a weight in W w8, Exercise 21.3x. Now, for any 0 / w g T0 l W, w Z⌬⬘ , w x : w T0 , T0 x s 0, since H q T0 is a Cartan subalgebra Žhence abelian . of the semisimple Lie algebra L s S [ T Žw15, Lemma 4.12x or w12, Lemma 8.1.b.x.. Since W is irreducible, it follows that w Z⌬⬘ , W x s 0, so that Z⌬⬘ is precisely the center of L⌬⬘. Besides, any abelian ideal of L⌬⬘ is easily shown to be contained in Z⌬⬘ , so that w L⌬⬘ , L⌬⬘ x is semisimple, as required. 3.5 Let T be a simple Lie triple system, let L s S [ T be its standard imbedding, and assume that T is an irreducible S-module. As before, we take H a Cartan subalgebra of S, so that the sum of H and the zero weight space T0 of T is a Cartan subalgebra of L. Let ⌽ be the root system of S and ⌳ the set of weights of T with respect to H. A look at the marked diagrams shows that: 3.5Ži. If S is simple of type A1 and 0 is a weight of T, then dim T0 s 1. Actually, either T is of adjoint type or the diagram is the one associated Ž . with AŽ2. 2 ; so L ( sl 3 and dim T0 s rank L y rank S s 2 y 1 s 1. 3.5Žii. If S is simple of type A 2 and 0 is a weight, then T is of adjoint type and dim T0 s 2. 3.5Žiii. If S is simple of type Bl Ž l G 2. and ⌽ : ⌳ then dim T0 s l. In fact ⌽ : ⌳ if and only if the difference between the highest weight and the longest root is a sum of positive roots. In this case either T is of adjoint type or its diagram is obtained from AŽ2. 2 l , hence the result Ž2 l y l s l ..
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3.5Živ. There is no simple Lie triple system where S is a direct sum of two simple ideals and all nonzero weights are roots; because for such S, T is a tensor product of an irreducible module for the first ideal times an irreducible module for the second, and the maximal weight cannot be a root. 3.5Žv. If S is simple of type Cl Ž l G 2. and ⌳ s short roots4 j 04 , then dim T0 s l y 1. This is because the only possibilities are given by the Ž2. Ž Ž . affine diagrams AŽ2. for l s 2.. 2 ly1 l G 3 and D 3 3.5Žvi. If S is simple of type D l Ž l G 3 with D 3 s A 3 . and 0 is a weight, then dim T0 s l y 1 Žthe only possibility is given by the affine . diagrams AŽ2. 2 ly1 . 3.6 With this in mind we can tackle the problem of bounding the dimension of T0 s t g T0 : S␣ t s 0 ᭙␣ H 4 Ž being the highest weight. for the triple systems with diagrams in 2.4ŽIV.. In each case we will find a subset ⌬⬘ of the base ⌬ such that all the roots in ⌬⬘ are orthogonal to , so that T0 : Z⌬⬘ and Lemma 3.4 will apply. Ž AŽ2. 2 . Here dim T0 s 1 and we already know that dim Hom S T mF T, T . G 1 by 3.2 and 2.5Žii.; hence dim Hom S ŽT mF T, T . s dim T0 s 1. 䢇
䢇
Ž . AŽ2. 2l l G 2
Here all the roots are weights. With ⌬⬘ s ␣ 2 , . . . , ␣ l 4 , S⌬⬘ is simple of type Bly1 if l G 3 or A1 if l s 2. From 3.5Žiii. Ž l G 3. or 3.5Ži. Ž l s 2. and Lemma 3.4 it follows that codim Z⌬⬘ s l y 1; so, since dim T0 s 2 l y l s l, it follows that dim T0F1 and we already know that dim Hom S ŽTmF T, T . G 1 in this case Ž2.5Žii.., so that again dim Hom S ŽT mF T, T . s 1.
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Ž . AŽ2. 2 ly1 l G 3 marked as
Here the nonzero weights are the short roots. We take ⌬⬘ s ␣ 1 , ␣ 3 , . . . , ␣ l 4 if l G 4; then because of 3.5Živ. w L⌬⬘ , L⌬⬘ x is the sum of two graded simple ideals and by 3.5Ži. and 3.5Žv. codim Z⌬⬘ s 1 q l y 3 s l y 2; so dim T0 F 1 Ždim T0 s l y 1.. Also, if l s 3 we take ⌬⬘ s ␣ 14 Žsince ␣ l is not a weight. and then S⌬⬘ is of type A1 and by 3.5Ži. codim Z⌬⬘ s 1 and dim T0 F 1. Again by 2.5 we get that dim Hom S ŽT mF T, T . s 1 in this case. 䢇
Ž . AŽ2. 2 ly1 l G 3 marked as
Here dim T0 s 2 l y 1 y l s l y 1. If l s 3 we take ⌬⬘ s ␣ 3 4 , and by 3.5Ži. codim Z⌬⬘ s 1; so dim T0 F 1. Also, for l ) 3 we take ⌬⬘ s ␣ 2 , . . . , ␣ l 4 and by 3.5Žvi. codim Z⌬⬘ s Ž l y 1. y 1 s l y 2; so again dim T0 F 1. Now by 2.5 we get dim Hom S ŽT mF T, T . s 1. 䢇
Ž2. Ž D lq1 l G 2. marked at one end
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Since dim T0 s rank D lq1 y rank Bl s 1, the short roots are weights and it is enough to take ⌬⬘ s ␣ l 4 and apply 3.5Ži. to conclude that Hom S ŽT mF T, T . s 0. 䢇
Ž2. Ž D lq1 l G 2. marked in the middle
Here L is simple of type D lq1 Ž D 3 s A 3 . and S is a sum of two simple ideals of type Br and Bs , respectively, with r, s G 1 Ž B1 s A1 . and r q s s l. Hence dim T0 s 1. If r s s s 1 we conclude by 2.5Žii. that dim Hom S ŽT mF T, T . s 1. If, for instance, r ) 1 we can take ⌬⬘ s ␣ r 4 and conclude from 3.5Ži. that Hom S ŽT mF T, T . s 0. 䢇
E6Ž2. marked as
Then S is of type F4 , so that dim T0 s 2. With ⌬⬘ s ␣ 34 Ž ␣ 3 is a short root and hence a weight. we get codim Z⌬⬘ s 1 by 3.5Ži.; so by 2.5Živ. we get dim Hom S ŽT mF T, T . s 1. 䢇
Finally, E6Ž2. marked as
With ⌬⬘ s ␣ 1 , ␣ 2 4 and because of 3.5Žii. we get codim Z⌬⬘ s 2 s dim T0 ; so Hom S ŽT mF T, T . s 0. 3.7 As a conclusion we get that in all cases dim Hom S ŽT mF T, T . s dim T0 and this is nonzero only in the cases contemplated in 3.2. So, summarizing all the work done, we get:
ALGEBRAS RELATED TO LIE TRIPLE SYSTEMS
247
THEOREM. Let T be a simple Lie triple system o¨ er an algebraically closed field F of characteristic zero and let L s S [ T be its standard imbedding. Then Hom S ŽT mF T, T . s 0 unless either: Ža. T is of adjoint type other than A n Ž n G 2.. In this case Hom S ŽT mF T, T . ( Hom S Ž S mF S, S ., which is spanned by the Lie multiplication in S or Žb. there exists a simple Jordan algebra J of degree n G 3 such that T is the Lie triple system J 0 . In this case either: Ži. J is not of type A Ž that is, J is not isomorphic to Mat nŽ F .q. ; then Hom S ŽT mF T, T . s Hom Der J Ž J 0 mF J 0 , J 0 . is spanned by the product in Ž).. Žii. J is of type A; then J 0 is of adjoint type S ( sl Ž n, F . and Hom S ŽT mF T, T . s Hom Der J Ž J 0 mF J 0 , J 0 . is spanned by the product in Ž). and the Lie multiplication in J 0 ( sl Ž n, F .. As mentioned in the Introduction, this theorem provides alternative proofs of w13, Theorem 8.1x and w14, Theorem 2.1x.
4. ARBITRARY FIELDS This section will be devoted to showing that the hypothesis in 3.7 that the field is algebraically closed can be avoided with only some minor modifications. 4.1 Given a Lie algebra L over a field K, two L-modules M and N and a subfield F of K, the F-descent of L is just L but considered as a Lie algebra over F. Then M and N are also modules for the F-descent of L. We can form M mF N, which is a module for the F-descent of L, and also M mK N, which is a module for L and, a fortiori, also a module for its F-descent. We have then a natural homomorphisms of modules Žfor the F-descent of L. p: M mF N ª M mK N mmn¬mmn Žwith different meanings of the same sign m.. LEMMA. Let L be a Lie algebra o¨ er a field K, let M, N, and P be three irreducible nontri¨ ial L-modules, and let F be a subfield of K such that KrF is a finite separable field extension. Then the linear map Ž of ¨ ector spaces
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o¨ er K . p*: Hom LŽ M mK N, P . ª Hom LŽ M mF N, P .
¬ ( p Ž p as abo¨ e . is a bijection. Proof. Notice that Hom LŽ M mF N, P . is a vector space over K because so is P. Let ⍀ be an algebraic closure of F and 1 , . . . , r : K ª ⍀ the r s w K : F x different F-homomorphisms. Then by separability ⍀ mF K s ⍀ e1 [ ⭈⭈⭈ [ ⍀ e r with e1 , . . . , e r orthogonal idempotents. Besides, as a right K-vector space, e i ␣ s i Ž ␣ . e i for any i s 1, . . . , r and any ␣ g K. Now, L ⍀ s ⍀ mF L ( Ž ⍀ mF K . mK L s Ž ⍀ e1 mK L . [ ⭈⭈⭈ [ Ž ⍀ e r mK L . s L1 [ ⭈⭈⭈ [ L r , where each L i s ⍀ e i mK L is the algebra obtained from L by extending scalars from K to ⍀ by means of i . Similarly M⍀ s ⍀ mF M ( Ž ⍀ e1 mK M . [ ⭈⭈⭈ [ Ž ⍀ e r mK M . s M1 [ ⭈⭈⭈ [ Mr , and the same for N⍀ and P⍀ . By orthogonality, L i M j s 0 if i / j. By scalar extension ⍀ mF Hom LŽ M mF N, P . ( Hom L ⍀Ž M⍀ m⍀ N⍀ , P⍀ . , but r
M⍀ m⍀ N⍀ s
[
i , js1
Mi m⍀ Nj .
For any g Hom L ⍀Ž Mi m⍀ Nj , Pk ., if i, j / k L k Ž Mi m⍀ Nj . : Ž L k Mi m⍀ Nj . q Ž Mi m⍀ L k Nj . s 0, so that
Ž Mi m⍀ Nj . s 0 since
x g Pk : L k x s 0 4 s ⍀ e k m x g P : Lx s 0 4 s 0;
249
ALGEBRAS RELATED TO LIE TRIPLE SYSTEMS
while if i s k / j
Ž Mi m⍀ Nj . s Ž Mi m⍀ L j Nj . s Ž L j Ž Mi m⍀ Nj . . s L j Ž Mi m⍀ Nj . : L j Pk s 0. Therefore r
Hom L ⍀Ž M⍀ m⍀ N⍀ , P⍀ . s
[ Hom L Ž Mi m⍀ is1
i
Ni , Pi .
r
s
[ ⍀ ei mK is1
Hom LŽ M mK N, P . .
Hence dim F Hom LŽ M mK N, P . s r dim K Hom LŽ M mK N, P . r
s
Ý dim ⍀ Hom L Ž Mi m⍀ i
Ni , Pi .
is1
s dim ⍀ Hom L ⍀Ž M⍀ m⍀ N⍀ , P⍀ . s dim F Hom LŽ M mF N, P . . But p: M mF N ª M mK N is onto; so p* is one-to-one and hence, because of the same dimension, it is a bijection. As a consequence, if T is a simple Lie triple system over a field F of characteristic zero, ⌫ is its centroid and S s w TT y x : gl ŽT ., to determine Hom S ŽT mF T, T . it is enough to determine Hom S ŽT m⌫ T, T .. That is, it is enough to deal with central simple Lie triple systems. Moreover, for these systems we can extend scalars up to an algebraically closed field and then use Theorem 3.7. 4.2 Let T be a central simple Lie triple system over a field F of characteristic zero with standard imbedding L s S [ T, and let ⍀ be an algebraic closure of F. Assume there is a simple Jordan algebra of degree n G 3 over ⍀ such that T⍀ s ⍀ mF T is the triple system J 0 constructed in 2.5. Then by Theorem 3.7, dim F Hom S Ž S 2 ŽT ., T . s 1, where S 2 ŽT . denotes the space of symmetric tensors in T mF T. Let : T = T ª T be symmetric, bilinear, and such that the map determined by u m u ¬ Ž u, u. Ž u g T . spans Hom S Ž S 2 ŽT ., T .. Then denoting by the same symbol its extension to a bilinear map T⍀ = T⍀ ª T⍀ , it follows that there is a 0 / g ⍀ such that x ⭈ y s Ž x, y . for
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BENITO, DRAPER, AND ELDUQUE
any x, y g T⍀ s J 0 Ž x ⭈ y as in Ž)... Thus the multiplication in the Jordan algebra J s ⍀1 [ J 0 is given by
Ž ␣ 1 q x . ( Ž  1 q y . s ␣ q
ž
1 n
t Ž x, y . 1 q Ž ␣ y q  x q x ⭈ y . , Ž 䉫 .
/
where t Ž x, y . s t Ž x ( y . defines an invariant symmetric bilinear form on J 0 . Since the algebra J is isomorphic Žby means of the map a ¬ a. to the algebra defined on the same J but with the new multiplication given by y1 a( b, we can assume that x ⭈ y s Ž x, y . for any x, y g J 0 . Now, the Jordan identity Ž x⬚2 ( y .( x s x⬚2 (Ž y ( x . gives for x, y g J 0 that 1 n
t Ž x ⭈ x, y . x q Ž Ž x ⭈ x . ⭈ y . ⭈ x s
1 n
t Ž y, x . x ⭈ x q Ž x ⭈ x . ⭈ Ž y ⭈ x . . Ž † .
On the other hand, t g Hom Der J Ž J 0 m⍀ J 0 , ⍀ . ( ⍀ mF Hom S ŽT mF T, F ., and by irreducibility of T and Schur’s lemma the dimension of these spaces is 1; hence there are a : T = T ª F bilinear, symmetric, and nondegenerate and a 0 / ␣ g ⍀ such that t Ž x, y . s ␣Ž x, y . for any x, y g T. If ␣ f F, for any x g T, take y g T with Ž x, y . s 1 and Ž†. allows us to conclude that x ⭈ x g ⍀ x for any x g T. But this implies that the degree of J is at most 2, a contradiction. Hence ␣ g F, t Ž x, y . g F for any x, y g T and A s F1 [ T is closed under the Jordan product in J. In other words, A s F1 [ T is a Jordan algebra over F which is a form of J. Moreover S s Der A since S⍀ s Der J. But we cannot conclude that T is the Lie triple system associated with the simple Jordan algebra A Žbecause of the scaling we did of the product in J .; we can only conclude that there is a nonzero scalar g F such that w xyz x s ŽŽ y ( z .( x y y (Ž z ( x .. for any x, y, z g A 0 s T. We will denote this triple system by Ž A 0 , .. Let us put all this another way. For any central simple Jordan algebra J over F of degree G 3 let LŽ J . s Der J [ J 0 be the ⺪ 2-graded simple Lie algebra which is the standard imbedding of the Lie triple system J 0 . Also, for any ⺪ 2-graded Lie algebra L s S [ T and any 0 / g F, let L be the new ⺪ 2-graded Lie algebra constructed on the same vector space as L with the same grading and the same product of elements in S and of elements in S times elements in T, but with the new product of elements in T given by
w t1 , t 2 x [ w t1 , t 2 x . Then the arguments above prove the first part of:
ALGEBRAS RELATED TO LIE TRIPLE SYSTEMS
251
PROPOSITION. Let L be a ⺪ 2-graded simple Lie algebra o¨ er a field F and let ⍀ be an algebraic closure of F. Assume that there is a simple Jordan algebra J o¨ er ⍀ of degree G 3 such that L ⍀ ( LŽ J .. Then: Ži. There exist a nonzero scalar g F and a Jordan algebra A o¨ er F, unique up to isomorphism, with A ⍀ ( J and L ( LŽ A. Ž as graded algebras.. Žii. LŽ A. ( LŽ A. Ž as graded algebras. if and only if y1 g F 2 . Proof. Assume that A and B are central simple Jordan algebras with A ⍀ ( J ( B⍀ and let 0 / , g F and : LŽ A. ª LŽ B . an isomorphism of graded algebras. Then it is immediate to check that is an isomorphism from LŽ A. to LŽ B .y1 , so that we may assume s 1. Let S s Der A, then the restriction < S : Der A ª Der B is an isomorphism which makes B0 an irreducible S-module. Let s < A 0 : A 0 ª B0 , then since Hom S Ž S 2 Ž A 0 ., A 0 . and Hom S Ž S 2 Ž B0 ., B0 . s Hom Der B Ž S 2 Ž B0 ., B0 . have dimension 1, there is a nonzero scalar 0/ gF such that Ž a1 ⭈ a2 . s Ž a1 . ⭈ Ž a2 . for any a1 , a2 g A 0 . Therefore Ž .Ž a1 ⭈ a2 . s Ž .Ž a1 . ⭈ Ž .Ž a2 . and is an isomorphism between the algebras Ž A 0 , ⭈ . and Ž B0 , ⭈ .. By Ž†. and since the degree of the Jordan algebras A and B is G 3 it follows that the traces of A and B are determined by the algebras Ž A 0 , ⭈ . and Ž B0 , ⭈ . respectively. From the expression in Ž䉫. extends to an isomorphism of Jordan algebras : A ª B Žwith Ž1. s 1 and Ž a. s Ž a. for any a g A 0 .. This shows the uniqueness in Ži.. Besides, induces an isomorphism ⌿: LŽ A. ª LŽ B . by means of ⌿ Ž s . s ( s( y1 for any s g S and ⌿ Ž a. s Ž a. for any a g A 0 . Now, ( ⌿y1 : LŽ B . ª LŽ B . is an isomorphism of simple graded Lie algebras with ( ⌿y1 Ž b . s y1 b for any b g B0 Žthe odd part.. But for any b1 , b 2 g B0
( ⌿y1 Ž w b1 , b 2 x . s ( y1 Ž b1 . , ( y1 Ž b 2 .
s y2 w b1 , b 2 x
and for any b1 , b 2 , b 3 g B0
y1 w b1 , b 2 x , b 3 s ( y1 Ž w b1 , b 2 x , b 3 y1
s (⌿
.
Ž w b1 , b2 x . , ( y1 Ž b3 .
s y2y1 w b1 , b 2 x , b 3 . Thus y2 s 1, giving the converse in Žii.. Conversely, if y1 s 2 for 0 / g F, the map : LŽ A. ª LŽ A. given by Ž s . s s for any s g S and Ž t . s t for any t g A 0 is easily seen to be an isomorphism. Remarks. Ži. The proposition above can be deduced also from w18, Theorem 4.5x. The proof above is more in the spirit of the rest of the paper.
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Žii. Over the real field, attached to any central simple Jordan algebra A of degree G 3 there appear two nonisomorphic triple systems with standard imbeddings LŽ A. and LŽ A.y1 . One is the dual of the other w7, p. 235x. The irreducible compact simply connected Riemannian symmetric spaces of type I with noncanonical affine connections Žsee w14, Theorem 2.1x. appear as the duals of the ones associated with the Jordan algebras of symmetric real matrices, symmetric matrices of quaternions with respect to the standard involution, and the simple exceptional Jordan algebra H3 Ž⺟.. 4.3 To state the result that summarizes most of the work done throughout the paper, we need one more thing. Let J be a central simple Jordan algebra of type A Žthat is, after extension of scalars it becomes isomorphic to Mat nŽ K .q. over a field ⌫ of characteristic zero. Then Žsee w9, Chaps. IV and Vx. either Ži. there is a central simple associative algebra B over ⌫ such that J is the Jordan algebra Bq or Žii. there is a central simple associative algebra A over a quadratic field extension P s ⌫w q x Ž q 2 g ⌫ . equipped with an involution of second kind j such that J is the Jordan algebra of symmetric elements H Ž A, j . s x g A : jŽ x . s x 4 . In the first case put P s ⌫ [ ⌫ s ⌫w q x with q 2 s 1, A s B [ B op Žwhere op denotes the opposite algebra., and j the involution of the second kind given by jŽ b1 , b 2 . s Ž b 2 , b1 .. So in both cases J s H Ž A, j . for a central simple involutorial algebra Ž A, j . of the second kind. Then, Der J s SŽ A, j . 0 s w SŽ A, j ., SŽ A, j .x acting on H Ž A, j . by means of the Lie bracket in A, where SŽ A, j . s x g A : jŽ x . s yx 4 Žsee w9, Theorem 6.9x.. Besides, H Ž A, j . s qSŽ A, j ., H Ž A, j . 0 is the adjoint module for SŽ A, j . 0 , and under the isomorphism of S Ž A, j . 0-modules given by H Ž A, j . 0 ª SŽ A, j . 0 , x ¬ qx, the Lie bracket on SŽ A, j . 0 becomes the SŽ A, j . 0-invariant product on H Ž A, j . 0 x 夹 y s q Ž xy y yx . .
Ž )).
Now, Theorem 3.7 plus the arguments in 4.1 and 4.2 prove the following: THEOREM. Let T be a simple Lie triple system o¨ er a field F of characteristic zero with centroid ⌫ and let L s S [ T be its standard imbedding. Then Hom S ŽT mF T, T . s 0 unless either: Ža. T is of adjoint type with S being a central simple Lie algebra o¨ er ⌫ of type different from A n Ž n G 2.. In this case Hom S ŽT mF T, T . ( Hom S Ž S mF S, S ., which is spanned o¨ er ⌫ by the Lie multiplication in S.
ALGEBRAS RELATED TO LIE TRIPLE SYSTEMS
253
Žb. There exists a central simple Jordan algebra J of degree n G 3 o¨ er ⌫ and a nonzero scalar g ⌫ Ž which can be taken modulo ⌫ 2 . such that T is the Lie triple system Ž J 0 , .. In this case either: Ži. J is not of type A o¨ er ⌫; then Hom S ŽT mF T, T . s Hom Der J Ž J 0 mF J 0 , J 0 . is spanned o¨ er ⌫ by the product in Ž).. Žii. J s H Ž A, j . for some central simple associati¨ e in¨ olutorial algebra Ž A, j . of the second kind o¨ er ⌫. Let P s ⌫w q x Ž0 / q 2 g ⌫ . be the center of A. Then Hom S Ž T mF T , T . s Hom Der J Ž J 0 mF J 0 , J 0 . s Hom SŽ A , j. 0Ž H Ž A, j . 0 m⌫ H Ž A, j . 0 , H Ž A, j . 0 . is spanned o¨ er ⌫ by the products in Ž). and Ž)).. This theorem may be considered as an extension of the classification of the flexible Lie-admissible algebras over a field of characteristic zero with an underlying simple Lie algebra Žsee w17, Chap. IIIx and the references therein ., since this classification is subsumed in cases Ža. and Žb.Žii. of the above result.
REFERENCES 1. G. M. Benkart and J. M. Osborn, An investigation of real division algebras using derivations, Pacific J. Math. 96 Ž1981., 265᎐300. 2. A. Elduque and J. M. Perez-Izquierdo, Composition algebras with large derivation ´ algebras, J. Algebra 190 Ž1997., 372᎐404. 3. A. Elduque and H. C. Myung, Color algebras and affine connections on S 6 , J. Algebra 149 Ž1992., 234᎐261. 4. A. Elduque and H. C. Myung, The reductive pair Ž B3 , G 2 . and affine connections on S 7, J. Pure Appl. Algebra 86 Ž1993., 155᎐171. 5. J. R. Faulkner, Dynkin diagrams for Lie triple systems, J. Algebra 62 Ž1980., 384᎐392. 6. J. R. Faulkner, Identity classification in triple systems, J. Algebra 94 Ž1985., 352᎐363. 7. S. Helgason, ‘‘Differential Geometry, Lie Groups and Symmetric Spaces,’’ Academic Press, San Diego, 1978. 8. J. E. Humphreys, ‘‘Introduction to Lie Algebras and Representation Theory,’’ SpringerVerlag, BerlinrHeidelbergrNew York, 1972. 9. N. Jacobson, ‘‘Structure and Representations of Jordan Algebras,’’ Am. Math. Soc., Providence, 1968. 10. V. G. Kac, Graduated Lie algebras and symmetric spaces, Funct. Anal. Appl. 2 Ž1968., 182᎐183. 11. V. G. Kac, Automorphisms of finite order of semisimple Lie algebras, Funct. Anal. Appl. 3 Ž1969., 252᎐254. 12. V. G. Kac, ‘‘Infinite Dimensional Lie Algebras,’’ 3rd ed., Cambridge Univ. Press, Cambridge, UK, 1990.
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13. H. T. Laquer, ‘‘Invariant affine connections on Lie groups,’’ Trans. Amer. Math. Soc. 331 Ž1992., 541᎐551. 14. H. T. Laquer, Invariant affine connections on symmetric spaces, Proc. Amer. Math. Soc. 115 Ž1992., 447᎐454. 15. W. G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 Ž1952., 217᎐242. 16. O. Loos, ‘‘Symmetric Spaces. II. Compact Spaces and Classification,’’ Benjamin, New York, 1969. 17. H. C. Myung, ‘‘Malcev᎐Admissible Algebras,’’ Progress in Mathematics, Vol. 64, Birkhauser, Boston, 1986. ¨ 18. E. Neher, ‘‘On the classification of Lie and Jordan triple systems,’’ Comm. Algebra 13 Ž1985., 2615᎐2667. 19. K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 Ž1954., 33᎐65. 20. J. M. Perez-Izquierdo, Division composition algebras through their derivation algebras, J. ´ Algebra, to appear.
On Some Algebras Related to Simple Lie Triple Systems Pilar Benito* and Cristina Draper † Departamento de Matematicas y Computacion, ´ ´ Uni¨ ersidad de La Rioja, 26004 Logrono, ˜ Spain
and Alberto Elduque ‡ Departamento de Matematicas, Uni¨ ersidad de Zaragoza, 50009 Zaragoza, Spain ´ Communicated by Georgia Benkart Received August 10, 1998
The set Hom De r T ŽT mF T, T . is determined for any simple finite dimensional Lie triple system T over a field F of characteristic zero. It turns out that it contains nontrivial elements if and only if T is related to a simple Jordan algebra. In particular this provides a new proof of the determination by Laquer of the invariant affine connections in the simply connected compact irreducible Riemannian symmetric spaces. 䊚 1999 Academic Press
1. INTRODUCTION Given a Lie group G acting smoothly and transitively on a manifold M, the isotropy subgroup H at any point p g M Žso that M , GrH ., the Lie algebra ᒄ of the group G and the Lie algebra ᒅ of H Ž ᒅ F ᒄ ., the manifold M is called a reducti¨ e homogeneous space in case ᒄsᒅ[ᒊ
Ž>.
*Supported by the Spanish DGICYT ŽPb 94-1311-C03-02. and by the Universidad de La Rioja ŽAPI-97rB05.. † Supported by a grant from the ‘‘Programa de Formacion ´ de Profesorado Universitario’’ ŽMEC. and by the Universidad de La Rioja ŽAPI-97rB05.. ‡ Supported by the Spanish DGICYT ŽPb 94-1311-C03-03. and by the Universidad de La Rioja ŽAPI-97rB05.. 234 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
ALGEBRAS RELATED TO LIE TRIPLE SYSTEMS
235
for a subspace ᒊ with ŽAd h.Ž ᒊ . ; ᒊ for any h g H, which implies that w ᒅ, ᒊ x : ᒊ Žand if H is connected, these are equivalent.. In this situation Nomizu w19, Theorem 8.1x established a bijection between the set of G-invariant affine connections on M and the set of bilinear maps
␣: ᒊ=ᒊªᒊ with the property that
␣ Ž Ž Ad h . X , Ž Ad h . Y . s Ž Ad h . ␣ Ž X , Y .
᭙h g H , ᭙ X , Y g ᒊ ,
that is, with Ad H < ᒊ : AutŽ ᒊ, ␣ . Žthe automorphism group of the nonassociative algebra Ž ᒊ, ␣ ... Again the condition Ad H < ᒊ : AutŽ ᒊ, ␣ . implies the condition ad ᒅ < ᒊ : DerŽ ᒊ, ␣ ., and they are equivalent if H is connected. The torsion, curvature, geodesics, holonomy, etc. of the connection associated to Ž ᒊ, ␣ . can then be expressed and studied in terms of this nonassociative algebra. Notice too that the set of the multiplications ␣ : ᒊ = ᒊ ª ᒊ such that ad ᒅ < ᒊ : DerŽ ᒊ, ␣ . is nothing else but the vector space of the homomorphisms of ᒅ-modules from ᒊ m⺢ ᒊ into ᒊ: Hom ᒅ Ž ᒊ m⺢ ᒊ, ᒊ ., which is a purely algebraic object. Thus, the problem of determining the invariant affine connections on reductive homogeneous spaces reduces to the problem of classifying algebras with a given subgroup of its automorphism group or a given subalgebra of its Lie algebra of derivations. The same problem has arisen in studying real division algebras w1x, nonunital composition algebras w2, 20x or in the classification of some flexible Lie-admissible algebras Žsee w17x and the references therein ., a subject we will come back to later on. In some cases, the algebras that appear related to some reductive homogeneous spaces are already known algebras. This is what happens for the six- and seven-dimensional spheres, viewed as S 6 s G 2rSUŽ3. and S 7 s SpinŽ7.rG 2 , where algebras related to a so-called color algebra w3x and the simple non-Lie Malcev algebras w4x appear. In 1992, Laquer w13, 14x computed the sets Hom ᒅ Ž ᒊ m⺢ ᒊ, ᒊ . associated with the simply connected compact irreducible Riemannian symmetric spaces Žsee w7x for definitions.. For these spaces, the decomposition Ž >. is a grading of ᒄ over ⺪ 2 s ⺪r2⺪, ᒊ thus being a Žsimple. Lie triple system. Therefore the problem was to compute Hom S ŽT m⺢ T, T . for some ⺪ 2-graded simple Žthat is, without nontrivial homogeneous ideals. finite dimensional real Lie algebra L s S [ T Ž S being the even part and T the odd one.. This was done by extending scalars to the complex field and by decomposing there the tensor product T⺓ m⺓ T⺓ as a direct sum of irreducible modules. The vector space Hom S ŽT m⺢ T, T . turned out to be
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BENITO, DRAPER, AND ELDUQUE
0 Žwhich amounts to saying that the only invariant affine connection is the canonical one. with a few exceptions. Our purpose in this paper is twofold: Ži. We will determine the set Hom S Ž T mF T , T . for any ⺪ 2-graded simple finite dimensional Lie algebra L s S [ T over any arbitrary field of characteristic zero, without actually decomposing T mF T Žwhen restricted to the real field, this determines the invariant affine connections on the irreducible affine symmetric spaces.. Žii. We will relate the resulting nontrivial algebras with the simple finite dimensional Jordan algebras. In a sense, we can say that the Jordan algebras are responsible for the existence of noncanonical invariant affine connections on the symmetric spaces. The close relationship of Jordan algebras with some symmetric spaces is well known Žsee, for instance, w16, Chapter VIIIx.. Our results give another aspect of this relationship. Actually, our determination of the sets Hom S ŽT mF T, T . over algebraically closed fields will be very quick, using information on the ⺪ 2-graded simple Lie algebras that is contained in the Dynkin diagrams associated with the corresponding Lie triple systems T by Faulkner w5x, with no need of a case-by-case decomposition of T mF T. Our approach is inspired by w6, Theorem 1x. This allows us to simplify some of the results in w13x and w14x. From the point of view of nonassociative algebras, a central role in what follows is played by the multiplication defined on the set J 0 of Žgeneric. trace zero elements in a Jordan algebra J, given by projecting the product in J onto J 0 Žsee Ž). in 3.2.. This procedure is not new; for example, when applied to Cayley᎐Dickson algebras it gives all the central simple non-Lie Malcev algebras; it appears too in the construction of the pseudo-octonion algebras Žsee w17x and the references therein ..
2. LIE TRIPLE SYSTEMS In this section, some facts concerning simple Lie triple systems and the Dynkin diagrams attached to them will be recalled.
ALGEBRAS RELATED TO LIE TRIPLE SYSTEMS
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2.1 A Lie triple system is a vector space T, over a ground field F, equipped with a trilinear product w xyz x satisfying
w xxy x s 0 w xyz x q w yzx x q w zxy x s 0 the map w xy y x : T ª T , z ¬ w xyz x is a derivation of T for any x, y, z g T. If S s span²w xy y x : x, y g T :, S is a Lie subalgebra of the Lie algebra gl ŽT . Žactually of DerŽT ..; the ⺪ 2-graded Lie algebra L s S [ T Žwhere the multiplication is given by the multiplication in S Ževen part. as a subalgebra of gl ŽT ., the natural action of S on T and w t 1 , t 2 x s w t 1 t 2 y x for any t 1 , t 2 g T . is called the standard imbedding of T. Thus any Lie triple system is nothing else but the odd part of a ⺪ 2-graded Lie algebra with product w xyz x s ww x, y x, z x. The Lie triple system is simple if and only if its standard imbedding is graded simple Žfor these facts see w15x.. 2.2 Now assume that T is a simple Lie triple system over the field F and let ⌫ s ␣ g End F ŽT . : ␣ w xyz x s wŽ ␣ x . yz x s w x Ž ␣ y . z x s w xy Ž ␣ z .x ᭙ x, y, z g T 4 be its centroid. ⌫ is a field extension of F and, for any x, y g T,w xy y x g End ⌫ ŽT .; therefore the standard imbedding L is a ⌫-algebra. Moreover, the centroid ⌳ of L is a commutative ⺪ 2-graded algebra: ⌳ s ⌳ 0 [ ⌳ 1 , with ⌳ 0 s ␣ g ⌳ : ␣ S : S, ␣ T : T 4 and ⌳ 1 s ␣ g ⌳ : ␣ S : T, ␣ T : S4 , with any nonzero homogeneous element being invertible Žby the graded simplicity of L., and it easily follows that ⌳ 0 s ⌫. There are two possibilities: 2.2Ža. If there is an element 0 / u g ⌳ 1 , then ⌳ 1 s ⌫u, so that ⌳ s ⌫w u x, with 0 / u 2 s g ⌫, T s uS, and L s ⌫w u x m⌫ S. In this case S is a central simple Lie algebra over ⌫ and T s uS is the adjoint module for S; so we will refer to this case as T being of adjoint type. For any x, y, z g S, wŽ ux .Ž uy .Ž uz .x s u 2 Ž uww x, y x, z x. s Ž uww x, y x, z x.; so T is isomorphic to the Lie triple system S under the product xyz 4 s ww x, y x, z x. Notice that if u 2 g ⌫ 2 , then can be taken to be 1. 2.2Žb. ⌳ s ⌫. Then L is a simple algebra Žungraded.: otherwise for any proper ideal I of L, by graded simplicity Ž S l I . [ ŽT l I . s 0 and S Ž I . [ T Ž I . s L Žwhere S and T denote the projections on S and T respectively.; therefore S l I s T l I s 0 and S Ž I . s S, T Ž I . s T. Hence for any s g S there is a unique t s g T with s q t s g I, and also for
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any t g T there is a unique, st g S with st q t g I. Then the odd map given by s ¬ t s and t ¬ st is easily checked to be an odd element in the centroid of L, a contradiction with ⌳ s ⌫. Therefore in this case, for any extension field Kr⌫, K m⌫ L is a simple algebra Žand ⺪ 2-graded.. From now on all the algebras and systems considered will be assumed to be finite dimensional. 2.3 Lister showed in w15, Theorem 4.5x Žsee also w5, Proposition 1x. that, given any simple Lie triple system T over an algebraically closed field of characteristic zero with standard imbedding L s S [ T, either 2.3Ži. S is semisimple and T is an irreducible self-dual S-module Žthis is always the situation for the adjoint type cases. or 2.3Žii. S s Fz [ w S, S x, where z is a nonzero central element of S and w S, S x is semisimple, and T s T1 [ T2 , where T1 and T2 are irreducible dual S-modules, so that z acts as a nonzero scalar on T1 and as its negative on T2 . This is the situation that occurs after complexifying the simple Lie triple systems associated to the hermitian symmetric spaces. 2.4 Faulkner w5x associated a diagram with any such simple Lie triple system, starting with the Dynkin diagram of the semisimple Lie algebra w S, S x and adding ‘‘marked’’ nodes representing the lowest weight of the representation of w S, S x on each irreducible summand of T. These diagrams thus contain the information on the structure of S and of T as an S-module Žwhich is enough to determine L.. The resulting diagrams are exactly the affine diagrams X NŽ1. and X NŽ2. Žwhich are equipped with some numerical labels in the nodes representing the linear dependence of the roots and weights involved; see w12, Chap. 4x. with one or two marked nodes. More precisely, the diagrams that appear are exactly: 2.4ŽI. The diagrams X NŽ1. with a node labeled by 1 marked Žby the symmetry of these diagrams, in this case it does not matter which is the marked node. ᎏthese correspond to the adjoint types. 2.4ŽII. The diagrams X NŽ1. with two nodes labeled by 1 markedᎏthese correspond to case 2.3Žii. above. Here the same diagram may give different possibilities according to which two nodes are marked. The same comment applies to the next cases.
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2.4ŽIII. The diagrams X NŽ1. with a node labeled by 2 markedᎏin this case the highest weight of T as an S-module Žwhich is the negative of the lowest weight since T is self-dual. is not in the root lattice, which is equivalent to 0 not being a weight of the representation of S on T Žsee w8, Exercise 21.3x.. 2.4ŽIV. The diagrams X NŽ2. Žso XN is A N Ž N G 2., DN , or E6 . with a node labeled by 1 markedᎏin this case 0 is a weight of the representation of S on T. This implies that there are roots which are weights and, therefore, at least the short roots of every simple ideal of S are weights Žnotice that the action of S on T is faithful, so that, for any simple ideal I of S and any 0 / ¨ g T in the zero weight space, I¨ / 0.. In case 2.4ŽI. L s S [ T is isomorphic Žas ungraded algebra. to the direct sum of two copies of the simple algebra of type X N . In the remaining cases L is a simple Žungraded. Lie algebra of type X N and the semisimple Lie algebra w S, S x Žwhich is the whole S if only one marked node appears. is the semisimple Lie algebra associated with the finite Dynkin diagram obtained by removing the marked nodes. Remark. Faulkner obtained these diagrams just for the Dynkin diagram of w S, S x plus the lowest weights of the irreducible subrepresentations of w S, S x on T. V. Kac had previously arrived at the same situation in his research on automorphisms of finite order in semisimple Lie algebras over an algebraically closed field of characteristic zero Žsee w10; 11; 12, Chapter 8; 7, Chap. Xx., since a ⺪ 2-graded Lie algebra is just a Lie algebra with an automorphism of order 2 Žcharacteristic not 2.. 2.5 Among the simple Lie triple systems, the ones obtained from Jordan algebras will be of particular interest to us. Given any simple Jordan algebra J of degree n G 3 with generic trace t w9, Chap. VIx and multiplication x ( y over a field of characteristic zero, its set of trace zero elements J 0 s x g J : t Ž x . s 04 becomes a simple Lie triple system under the trilinear multiplication w xyz x s Ž y ( z .( x y y (Ž z ( x . Žthe associator of y, z, and x .. Since w xy y x s w R x , R y x, where R u¨ s u( ¨ s ¨ ( u, it follows that in this case S s w R J , R J x s Der J is the Lie algebra of derivations of the Jordan algebra J and the standard imbedding is L s Der J [ J 0 ( Der J [ R J 0 , which is the derived subalgebra of the Lie multiplication algebra of J w9, Theorem 8.3x.
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Over an algebraically closed field F of characteristic zero there are the following possibilities for the simple Jordan algebras of degree n G 3: 2.5Ži. J s Mat nŽ F .q is the algebra of n = n matrices over F under the product x ( y s 12 Ž xy q yx .. In this case J 0 s sl Ž n, F . is a simple Lie triple system of adjoint type since Der J s ad J 0 , so that the associated diagram is
2.5Žii. J is the algebra of symmetric n = n matrices over F: J s Ž H Mat nŽ F ., t . Ž t the transposition.. In this case Der J is isomorphic to the set of skew-symmetric matrices Žsee w9, Theorem 6.9x. and the Lie algebra L s Der J [ J 0 is isomorphic to sl Ž n, F .. The associated diagram is
2.5Žiii. J is the algebra of symmetric 2 n = 2 n matrices with respect to the standard symplectic involution. Again in this case Der J is isomorphic to the skew-symmetric matrices with respect to this involution w9, Theorem 6.9x, L is isomorphic to sl Ž2 l, F ., and the associated diagram is
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2.5Živ. J is the exceptional simple Jordan algebra, whose derivation algebra is the Lie algebra of type F4 w9, Sect. 9.11x. The associated diagram is then
3. ALGEBRAICALLY CLOSED FIELDS We intend to determine in this section the vector space Hom S ŽT mF T, T . for any simple Lie triple system T with standard imbedding L s S [ T, over an algebraically closed field F of characteristic zero, an assumption we will keep throughout the section. 3.1 In case T s T1 [ T2 is a sum of two irreducible S-modules Ždiagrams 2.4ŽII.., the center of S is one-dimensional: ZŽ S . s Fz, with ad z < T1 s 1, ad z < T 2 s y1. Therefore T mF T is the sum of the eigenspaces 2, 0, y2 for ad z, which forces Hom S ŽT mF T, T . s 0. 3.2 In case T is of adjoint type, then Hom S ŽT mF T, T . ( Hom S Ž S mF S, S . and there is a distinguished Žskew-symmetric. element in this latter space: the Lie multiplication in S. So at least dim Hom S ŽT mF T, T . G 1. Actually, by w6, Theorem 1 and Corollary 2x dim Hom S Ž T mF T , T . s
½
2 1
if S is of type A n Ž n G 2 . otherwise.
On the other hand, if T s J 0 is the triple system associated with a simple Jordan algebra of degree n G 3 with multiplication x ( y, then S s Der J and we obtain a nonzero Žsymmetric. element of Hom S ŽT mF T, T . by means of J 0 mF J 0 ª J 0 x m y ¬ x ⭈ y s x( y y
1 n
tŽ x( y.1
Ž ).
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Žthe projection of the element x ( y onto J 0 .. Notice that if the degree were 2, this would be zero. Besides, if J s Mat nŽ F .q, then the triple system J 0 is both of adjoint type Ž A ny1 . and of ‘‘Jordan type.’’ Therefore, for the adjoint type Ždiagrams 2.4ŽI.., Hom S ŽT mF T, T . is spanned by the ‘‘Lie bracket in S’’ for types other than A n Ž n G 2. and by the ‘‘Lie bracket in S’’ and the ‘‘Jordan product Ž).’’ for types A n Ž n G 2.. 3.3 The situation for the simple Lie triple systems with diagrams 2.4ŽIII. can be settled immediately with the following lemma Žinspired by w6, Theorem 1x.: LEMMA. Let S be a semisimple Lie algebra, H a Cartan subalgebra of S, V an irreducible self-dual module with highest weight relati¨ e to H, and ¨ and ¨ y nonzero weight ¨ ectors for and y. For any root ␣ of S relati¨ e to H let S␣ be the corresponding root space. If V0 is the weight 0-space in V, then the linear map
: Hom S Ž V mF V , V . ª ¨ g V0 : S␣ ¨ s 0 ᭙␣ H 4 ¬ Ž ¨ m ¨ y . is well defined and one-to-one. Proof. Since V is self-dual, y is the lowest weight of V and ¨ m ¨ y generates V mF V. Hence any g Hom S Ž V mF V, V . is determined by Ž ¨ m ¨ y . g V0 . But, for any root ␣ orthogonal to the weight and any x␣ g S␣ , x␣ ¨ s 0 s x␣ ¨ y , so x␣ Ž ¨ m ¨ y . s 0 and x␣ Ž ¨ m ¨ y . s 0. As a direct consequence, Hom S ŽT mF T, T . s 0 for the simple Lie triple systems with diagrams 2.4ŽIII., for which 0 is not a weight of the representation of S on T. 3.4 From the paragraphs above, only the simple Lie triple systems with diagrams 2.4ŽIV. remain to be studied. According to Lemma 3.3, we will try to bound the dimension of t g T0 : S␣ t s 0 ᭙␣ H 4 for a triple system T which is an irreducible S-module of highest weight . LEMMA. Let T be a simple Lie triple system which is an irreducible module for S s w TT᎐ x. Let H be a Cartan subalgebra of the semisimple Lie algebra S, ⌽ the associated root system, ⌬ a base of ⌽, and ⌳ the set of weights of T relati¨ e to H. For any ⭋ / ⌬⬘ : ⌬, let S⌬⬘ be the subalgebra of S generated by the root spaces S␣ , ␣ g ⌬⬘ j y ⌬⬘, let T⌬⬘ be the sum of the weight spaces T for g ⌳ l ⺪⌬⬘, and let L⌬⬘ s S⌬⬘ [ T⌬⬘.
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Then L⌬⬘ is a graded subalgebra of L s S [ T that decomposes as L⌬⬘ s Z⌬⬘ [ w L⌬⬘ , L⌬⬘ x , where Z⌬⬘ s ¨ g T0 : S␣ ¨ s 0 ᭙␣ g ⌬⬘ j y ⌬⬘4 is its center and w L⌬⬘ , L⌬⬘ x is semisimple Ž hence a direct sum of graded simple algebras.. In particular, the codimension in T0 of Z⌬⬘ equals the dimension of the zero weight space in w S⌬⬘ , T⌬⬘ x Ž the odd part of w L⌬⬘ , L⌬⬘ x.. Proof. Since S⌬⬘ s Ý ␣ g ⌬⬘ Fh␣ q Ý  g ⌽ l ⺪ ⌬⬘ S Žthe h␣ ’s as in w8, p. 37x., S⌬⬘ is a semisimple subalgebra of S with Cartan subalgebra H⌬⬘ s H l S⌬⬘ s Ý ␣ g ⌬⬘ Fh␣ . Besides, ⌬⬘ is a base of the root system of S⌬⬘: ⌽ l ⺪⌬⬘. Moreover, L⌬⬘ is clearly a graded subalgebra of L. By complete reducibility T⌬⬘ s Z⌬⬘ [ w S⌬⬘ , T⌬⬘ x, and if 0 / W is an irreducible S⌬⬘-module in w S⌬⬘ ,T⌬⬘ x, its highest weight belongs to ⺪⌬⬘, so that 0 is a weight in W w8, Exercise 21.3x. Now, for any 0 / w g T0 l W, w Z⌬⬘ , w x : w T0 , T0 x s 0, since H q T0 is a Cartan subalgebra Žhence abelian . of the semisimple Lie algebra L s S [ T Žw15, Lemma 4.12x or w12, Lemma 8.1.b.x.. Since W is irreducible, it follows that w Z⌬⬘ , W x s 0, so that Z⌬⬘ is precisely the center of L⌬⬘. Besides, any abelian ideal of L⌬⬘ is easily shown to be contained in Z⌬⬘ , so that w L⌬⬘ , L⌬⬘ x is semisimple, as required. 3.5 Let T be a simple Lie triple system, let L s S [ T be its standard imbedding, and assume that T is an irreducible S-module. As before, we take H a Cartan subalgebra of S, so that the sum of H and the zero weight space T0 of T is a Cartan subalgebra of L. Let ⌽ be the root system of S and ⌳ the set of weights of T with respect to H. A look at the marked diagrams shows that: 3.5Ži. If S is simple of type A1 and 0 is a weight of T, then dim T0 s 1. Actually, either T is of adjoint type or the diagram is the one associated Ž . with AŽ2. 2 ; so L ( sl 3 and dim T0 s rank L y rank S s 2 y 1 s 1. 3.5Žii. If S is simple of type A 2 and 0 is a weight, then T is of adjoint type and dim T0 s 2. 3.5Žiii. If S is simple of type Bl Ž l G 2. and ⌽ : ⌳ then dim T0 s l. In fact ⌽ : ⌳ if and only if the difference between the highest weight and the longest root is a sum of positive roots. In this case either T is of adjoint type or its diagram is obtained from AŽ2. 2 l , hence the result Ž2 l y l s l ..
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3.5Živ. There is no simple Lie triple system where S is a direct sum of two simple ideals and all nonzero weights are roots; because for such S, T is a tensor product of an irreducible module for the first ideal times an irreducible module for the second, and the maximal weight cannot be a root. 3.5Žv. If S is simple of type Cl Ž l G 2. and ⌳ s short roots4 j 04 , then dim T0 s l y 1. This is because the only possibilities are given by the Ž2. Ž Ž . affine diagrams AŽ2. for l s 2.. 2 ly1 l G 3 and D 3 3.5Žvi. If S is simple of type D l Ž l G 3 with D 3 s A 3 . and 0 is a weight, then dim T0 s l y 1 Žthe only possibility is given by the affine . diagrams AŽ2. 2 ly1 . 3.6 With this in mind we can tackle the problem of bounding the dimension of T0 s t g T0 : S␣ t s 0 ᭙␣ H 4 Ž being the highest weight. for the triple systems with diagrams in 2.4ŽIV.. In each case we will find a subset ⌬⬘ of the base ⌬ such that all the roots in ⌬⬘ are orthogonal to , so that T0 : Z⌬⬘ and Lemma 3.4 will apply. Ž AŽ2. 2 . Here dim T0 s 1 and we already know that dim Hom S T mF T, T . G 1 by 3.2 and 2.5Žii.; hence dim Hom S ŽT mF T, T . s dim T0 s 1. 䢇
䢇
Ž . AŽ2. 2l l G 2
Here all the roots are weights. With ⌬⬘ s ␣ 2 , . . . , ␣ l 4 , S⌬⬘ is simple of type Bly1 if l G 3 or A1 if l s 2. From 3.5Žiii. Ž l G 3. or 3.5Ži. Ž l s 2. and Lemma 3.4 it follows that codim Z⌬⬘ s l y 1; so, since dim T0 s 2 l y l s l, it follows that dim T0F1 and we already know that dim Hom S ŽTmF T, T . G 1 in this case Ž2.5Žii.., so that again dim Hom S ŽT mF T, T . s 1.
ALGEBRAS RELATED TO LIE TRIPLE SYSTEMS 䢇
245
Ž . AŽ2. 2 ly1 l G 3 marked as
Here the nonzero weights are the short roots. We take ⌬⬘ s ␣ 1 , ␣ 3 , . . . , ␣ l 4 if l G 4; then because of 3.5Živ. w L⌬⬘ , L⌬⬘ x is the sum of two graded simple ideals and by 3.5Ži. and 3.5Žv. codim Z⌬⬘ s 1 q l y 3 s l y 2; so dim T0 F 1 Ždim T0 s l y 1.. Also, if l s 3 we take ⌬⬘ s ␣ 14 Žsince ␣ l is not a weight. and then S⌬⬘ is of type A1 and by 3.5Ži. codim Z⌬⬘ s 1 and dim T0 F 1. Again by 2.5 we get that dim Hom S ŽT mF T, T . s 1 in this case. 䢇
Ž . AŽ2. 2 ly1 l G 3 marked as
Here dim T0 s 2 l y 1 y l s l y 1. If l s 3 we take ⌬⬘ s ␣ 3 4 , and by 3.5Ži. codim Z⌬⬘ s 1; so dim T0 F 1. Also, for l ) 3 we take ⌬⬘ s ␣ 2 , . . . , ␣ l 4 and by 3.5Žvi. codim Z⌬⬘ s Ž l y 1. y 1 s l y 2; so again dim T0 F 1. Now by 2.5 we get dim Hom S ŽT mF T, T . s 1. 䢇
Ž2. Ž D lq1 l G 2. marked at one end
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BENITO, DRAPER, AND ELDUQUE
Since dim T0 s rank D lq1 y rank Bl s 1, the short roots are weights and it is enough to take ⌬⬘ s ␣ l 4 and apply 3.5Ži. to conclude that Hom S ŽT mF T, T . s 0. 䢇
Ž2. Ž D lq1 l G 2. marked in the middle
Here L is simple of type D lq1 Ž D 3 s A 3 . and S is a sum of two simple ideals of type Br and Bs , respectively, with r, s G 1 Ž B1 s A1 . and r q s s l. Hence dim T0 s 1. If r s s s 1 we conclude by 2.5Žii. that dim Hom S ŽT mF T, T . s 1. If, for instance, r ) 1 we can take ⌬⬘ s ␣ r 4 and conclude from 3.5Ži. that Hom S ŽT mF T, T . s 0. 䢇
E6Ž2. marked as
Then S is of type F4 , so that dim T0 s 2. With ⌬⬘ s ␣ 34 Ž ␣ 3 is a short root and hence a weight. we get codim Z⌬⬘ s 1 by 3.5Ži.; so by 2.5Živ. we get dim Hom S ŽT mF T, T . s 1. 䢇
Finally, E6Ž2. marked as
With ⌬⬘ s ␣ 1 , ␣ 2 4 and because of 3.5Žii. we get codim Z⌬⬘ s 2 s dim T0 ; so Hom S ŽT mF T, T . s 0. 3.7 As a conclusion we get that in all cases dim Hom S ŽT mF T, T . s dim T0 and this is nonzero only in the cases contemplated in 3.2. So, summarizing all the work done, we get:
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247
THEOREM. Let T be a simple Lie triple system o¨ er an algebraically closed field F of characteristic zero and let L s S [ T be its standard imbedding. Then Hom S ŽT mF T, T . s 0 unless either: Ža. T is of adjoint type other than A n Ž n G 2.. In this case Hom S ŽT mF T, T . ( Hom S Ž S mF S, S ., which is spanned by the Lie multiplication in S or Žb. there exists a simple Jordan algebra J of degree n G 3 such that T is the Lie triple system J 0 . In this case either: Ži. J is not of type A Ž that is, J is not isomorphic to Mat nŽ F .q. ; then Hom S ŽT mF T, T . s Hom Der J Ž J 0 mF J 0 , J 0 . is spanned by the product in Ž).. Žii. J is of type A; then J 0 is of adjoint type S ( sl Ž n, F . and Hom S ŽT mF T, T . s Hom Der J Ž J 0 mF J 0 , J 0 . is spanned by the product in Ž). and the Lie multiplication in J 0 ( sl Ž n, F .. As mentioned in the Introduction, this theorem provides alternative proofs of w13, Theorem 8.1x and w14, Theorem 2.1x.
4. ARBITRARY FIELDS This section will be devoted to showing that the hypothesis in 3.7 that the field is algebraically closed can be avoided with only some minor modifications. 4.1 Given a Lie algebra L over a field K, two L-modules M and N and a subfield F of K, the F-descent of L is just L but considered as a Lie algebra over F. Then M and N are also modules for the F-descent of L. We can form M mF N, which is a module for the F-descent of L, and also M mK N, which is a module for L and, a fortiori, also a module for its F-descent. We have then a natural homomorphisms of modules Žfor the F-descent of L. p: M mF N ª M mK N mmn¬mmn Žwith different meanings of the same sign m.. LEMMA. Let L be a Lie algebra o¨ er a field K, let M, N, and P be three irreducible nontri¨ ial L-modules, and let F be a subfield of K such that KrF is a finite separable field extension. Then the linear map Ž of ¨ ector spaces
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o¨ er K . p*: Hom LŽ M mK N, P . ª Hom LŽ M mF N, P .
¬ ( p Ž p as abo¨ e . is a bijection. Proof. Notice that Hom LŽ M mF N, P . is a vector space over K because so is P. Let ⍀ be an algebraic closure of F and 1 , . . . , r : K ª ⍀ the r s w K : F x different F-homomorphisms. Then by separability ⍀ mF K s ⍀ e1 [ ⭈⭈⭈ [ ⍀ e r with e1 , . . . , e r orthogonal idempotents. Besides, as a right K-vector space, e i ␣ s i Ž ␣ . e i for any i s 1, . . . , r and any ␣ g K. Now, L ⍀ s ⍀ mF L ( Ž ⍀ mF K . mK L s Ž ⍀ e1 mK L . [ ⭈⭈⭈ [ Ž ⍀ e r mK L . s L1 [ ⭈⭈⭈ [ L r , where each L i s ⍀ e i mK L is the algebra obtained from L by extending scalars from K to ⍀ by means of i . Similarly M⍀ s ⍀ mF M ( Ž ⍀ e1 mK M . [ ⭈⭈⭈ [ Ž ⍀ e r mK M . s M1 [ ⭈⭈⭈ [ Mr , and the same for N⍀ and P⍀ . By orthogonality, L i M j s 0 if i / j. By scalar extension ⍀ mF Hom LŽ M mF N, P . ( Hom L ⍀Ž M⍀ m⍀ N⍀ , P⍀ . , but r
M⍀ m⍀ N⍀ s
[
i , js1
Mi m⍀ Nj .
For any g Hom L ⍀Ž Mi m⍀ Nj , Pk ., if i, j / k L k Ž Mi m⍀ Nj . : Ž L k Mi m⍀ Nj . q Ž Mi m⍀ L k Nj . s 0, so that
Ž Mi m⍀ Nj . s 0 since
x g Pk : L k x s 0 4 s ⍀ e k m x g P : Lx s 0 4 s 0;
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while if i s k / j
Ž Mi m⍀ Nj . s Ž Mi m⍀ L j Nj . s Ž L j Ž Mi m⍀ Nj . . s L j Ž Mi m⍀ Nj . : L j Pk s 0. Therefore r
Hom L ⍀Ž M⍀ m⍀ N⍀ , P⍀ . s
[ Hom L Ž Mi m⍀ is1
i
Ni , Pi .
r
s
[ ⍀ ei mK is1
Hom LŽ M mK N, P . .
Hence dim F Hom LŽ M mK N, P . s r dim K Hom LŽ M mK N, P . r
s
Ý dim ⍀ Hom L Ž Mi m⍀ i
Ni , Pi .
is1
s dim ⍀ Hom L ⍀Ž M⍀ m⍀ N⍀ , P⍀ . s dim F Hom LŽ M mF N, P . . But p: M mF N ª M mK N is onto; so p* is one-to-one and hence, because of the same dimension, it is a bijection. As a consequence, if T is a simple Lie triple system over a field F of characteristic zero, ⌫ is its centroid and S s w TT y x : gl ŽT ., to determine Hom S ŽT mF T, T . it is enough to determine Hom S ŽT m⌫ T, T .. That is, it is enough to deal with central simple Lie triple systems. Moreover, for these systems we can extend scalars up to an algebraically closed field and then use Theorem 3.7. 4.2 Let T be a central simple Lie triple system over a field F of characteristic zero with standard imbedding L s S [ T, and let ⍀ be an algebraic closure of F. Assume there is a simple Jordan algebra of degree n G 3 over ⍀ such that T⍀ s ⍀ mF T is the triple system J 0 constructed in 2.5. Then by Theorem 3.7, dim F Hom S Ž S 2 ŽT ., T . s 1, where S 2 ŽT . denotes the space of symmetric tensors in T mF T. Let : T = T ª T be symmetric, bilinear, and such that the map determined by u m u ¬ Ž u, u. Ž u g T . spans Hom S Ž S 2 ŽT ., T .. Then denoting by the same symbol its extension to a bilinear map T⍀ = T⍀ ª T⍀ , it follows that there is a 0 / g ⍀ such that x ⭈ y s Ž x, y . for
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any x, y g T⍀ s J 0 Ž x ⭈ y as in Ž)... Thus the multiplication in the Jordan algebra J s ⍀1 [ J 0 is given by
Ž ␣ 1 q x . ( Ž  1 q y . s ␣ q
ž
1 n
t Ž x, y . 1 q Ž ␣ y q  x q x ⭈ y . , Ž 䉫 .
/
where t Ž x, y . s t Ž x ( y . defines an invariant symmetric bilinear form on J 0 . Since the algebra J is isomorphic Žby means of the map a ¬ a. to the algebra defined on the same J but with the new multiplication given by y1 a( b, we can assume that x ⭈ y s Ž x, y . for any x, y g J 0 . Now, the Jordan identity Ž x⬚2 ( y .( x s x⬚2 (Ž y ( x . gives for x, y g J 0 that 1 n
t Ž x ⭈ x, y . x q Ž Ž x ⭈ x . ⭈ y . ⭈ x s
1 n
t Ž y, x . x ⭈ x q Ž x ⭈ x . ⭈ Ž y ⭈ x . . Ž † .
On the other hand, t g Hom Der J Ž J 0 m⍀ J 0 , ⍀ . ( ⍀ mF Hom S ŽT mF T, F ., and by irreducibility of T and Schur’s lemma the dimension of these spaces is 1; hence there are a : T = T ª F bilinear, symmetric, and nondegenerate and a 0 / ␣ g ⍀ such that t Ž x, y . s ␣Ž x, y . for any x, y g T. If ␣ f F, for any x g T, take y g T with Ž x, y . s 1 and Ž†. allows us to conclude that x ⭈ x g ⍀ x for any x g T. But this implies that the degree of J is at most 2, a contradiction. Hence ␣ g F, t Ž x, y . g F for any x, y g T and A s F1 [ T is closed under the Jordan product in J. In other words, A s F1 [ T is a Jordan algebra over F which is a form of J. Moreover S s Der A since S⍀ s Der J. But we cannot conclude that T is the Lie triple system associated with the simple Jordan algebra A Žbecause of the scaling we did of the product in J .; we can only conclude that there is a nonzero scalar g F such that w xyz x s ŽŽ y ( z .( x y y (Ž z ( x .. for any x, y, z g A 0 s T. We will denote this triple system by Ž A 0 , .. Let us put all this another way. For any central simple Jordan algebra J over F of degree G 3 let LŽ J . s Der J [ J 0 be the ⺪ 2-graded simple Lie algebra which is the standard imbedding of the Lie triple system J 0 . Also, for any ⺪ 2-graded Lie algebra L s S [ T and any 0 / g F, let L be the new ⺪ 2-graded Lie algebra constructed on the same vector space as L with the same grading and the same product of elements in S and of elements in S times elements in T, but with the new product of elements in T given by
w t1 , t 2 x [ w t1 , t 2 x . Then the arguments above prove the first part of:
ALGEBRAS RELATED TO LIE TRIPLE SYSTEMS
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PROPOSITION. Let L be a ⺪ 2-graded simple Lie algebra o¨ er a field F and let ⍀ be an algebraic closure of F. Assume that there is a simple Jordan algebra J o¨ er ⍀ of degree G 3 such that L ⍀ ( LŽ J .. Then: Ži. There exist a nonzero scalar g F and a Jordan algebra A o¨ er F, unique up to isomorphism, with A ⍀ ( J and L ( LŽ A. Ž as graded algebras.. Žii. LŽ A. ( LŽ A. Ž as graded algebras. if and only if y1 g F 2 . Proof. Assume that A and B are central simple Jordan algebras with A ⍀ ( J ( B⍀ and let 0 / , g F and : LŽ A. ª LŽ B . an isomorphism of graded algebras. Then it is immediate to check that is an isomorphism from LŽ A. to LŽ B .y1 , so that we may assume s 1. Let S s Der A, then the restriction < S : Der A ª Der B is an isomorphism which makes B0 an irreducible S-module. Let s < A 0 : A 0 ª B0 , then since Hom S Ž S 2 Ž A 0 ., A 0 . and Hom S Ž S 2 Ž B0 ., B0 . s Hom Der B Ž S 2 Ž B0 ., B0 . have dimension 1, there is a nonzero scalar 0/ gF such that Ž a1 ⭈ a2 . s Ž a1 . ⭈ Ž a2 . for any a1 , a2 g A 0 . Therefore Ž .Ž a1 ⭈ a2 . s Ž .Ž a1 . ⭈ Ž .Ž a2 . and is an isomorphism between the algebras Ž A 0 , ⭈ . and Ž B0 , ⭈ .. By Ž†. and since the degree of the Jordan algebras A and B is G 3 it follows that the traces of A and B are determined by the algebras Ž A 0 , ⭈ . and Ž B0 , ⭈ . respectively. From the expression in Ž䉫. extends to an isomorphism of Jordan algebras : A ª B Žwith Ž1. s 1 and Ž a. s Ž a. for any a g A 0 .. This shows the uniqueness in Ži.. Besides, induces an isomorphism ⌿: LŽ A. ª LŽ B . by means of ⌿ Ž s . s ( s( y1 for any s g S and ⌿ Ž a. s Ž a. for any a g A 0 . Now, ( ⌿y1 : LŽ B . ª LŽ B . is an isomorphism of simple graded Lie algebras with ( ⌿y1 Ž b . s y1 b for any b g B0 Žthe odd part.. But for any b1 , b 2 g B0
( ⌿y1 Ž w b1 , b 2 x . s ( y1 Ž b1 . , ( y1 Ž b 2 .
s y2 w b1 , b 2 x
and for any b1 , b 2 , b 3 g B0
y1 w b1 , b 2 x , b 3 s ( y1 Ž w b1 , b 2 x , b 3 y1
s (⌿
.
Ž w b1 , b2 x . , ( y1 Ž b3 .
s y2y1 w b1 , b 2 x , b 3 . Thus y2 s 1, giving the converse in Žii.. Conversely, if y1 s 2 for 0 / g F, the map : LŽ A. ª LŽ A. given by Ž s . s s for any s g S and Ž t . s t for any t g A 0 is easily seen to be an isomorphism. Remarks. Ži. The proposition above can be deduced also from w18, Theorem 4.5x. The proof above is more in the spirit of the rest of the paper.
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Žii. Over the real field, attached to any central simple Jordan algebra A of degree G 3 there appear two nonisomorphic triple systems with standard imbeddings LŽ A. and LŽ A.y1 . One is the dual of the other w7, p. 235x. The irreducible compact simply connected Riemannian symmetric spaces of type I with noncanonical affine connections Žsee w14, Theorem 2.1x. appear as the duals of the ones associated with the Jordan algebras of symmetric real matrices, symmetric matrices of quaternions with respect to the standard involution, and the simple exceptional Jordan algebra H3 Ž⺟.. 4.3 To state the result that summarizes most of the work done throughout the paper, we need one more thing. Let J be a central simple Jordan algebra of type A Žthat is, after extension of scalars it becomes isomorphic to Mat nŽ K .q. over a field ⌫ of characteristic zero. Then Žsee w9, Chaps. IV and Vx. either Ži. there is a central simple associative algebra B over ⌫ such that J is the Jordan algebra Bq or Žii. there is a central simple associative algebra A over a quadratic field extension P s ⌫w q x Ž q 2 g ⌫ . equipped with an involution of second kind j such that J is the Jordan algebra of symmetric elements H Ž A, j . s x g A : jŽ x . s x 4 . In the first case put P s ⌫ [ ⌫ s ⌫w q x with q 2 s 1, A s B [ B op Žwhere op denotes the opposite algebra., and j the involution of the second kind given by jŽ b1 , b 2 . s Ž b 2 , b1 .. So in both cases J s H Ž A, j . for a central simple involutorial algebra Ž A, j . of the second kind. Then, Der J s SŽ A, j . 0 s w SŽ A, j ., SŽ A, j .x acting on H Ž A, j . by means of the Lie bracket in A, where SŽ A, j . s x g A : jŽ x . s yx 4 Žsee w9, Theorem 6.9x.. Besides, H Ž A, j . s qSŽ A, j ., H Ž A, j . 0 is the adjoint module for SŽ A, j . 0 , and under the isomorphism of S Ž A, j . 0-modules given by H Ž A, j . 0 ª SŽ A, j . 0 , x ¬ qx, the Lie bracket on SŽ A, j . 0 becomes the SŽ A, j . 0-invariant product on H Ž A, j . 0 x 夹 y s q Ž xy y yx . .
Ž )).
Now, Theorem 3.7 plus the arguments in 4.1 and 4.2 prove the following: THEOREM. Let T be a simple Lie triple system o¨ er a field F of characteristic zero with centroid ⌫ and let L s S [ T be its standard imbedding. Then Hom S ŽT mF T, T . s 0 unless either: Ža. T is of adjoint type with S being a central simple Lie algebra o¨ er ⌫ of type different from A n Ž n G 2.. In this case Hom S ŽT mF T, T . ( Hom S Ž S mF S, S ., which is spanned o¨ er ⌫ by the Lie multiplication in S.
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Žb. There exists a central simple Jordan algebra J of degree n G 3 o¨ er ⌫ and a nonzero scalar g ⌫ Ž which can be taken modulo ⌫ 2 . such that T is the Lie triple system Ž J 0 , .. In this case either: Ži. J is not of type A o¨ er ⌫; then Hom S ŽT mF T, T . s Hom Der J Ž J 0 mF J 0 , J 0 . is spanned o¨ er ⌫ by the product in Ž).. Žii. J s H Ž A, j . for some central simple associati¨ e in¨ olutorial algebra Ž A, j . of the second kind o¨ er ⌫. Let P s ⌫w q x Ž0 / q 2 g ⌫ . be the center of A. Then Hom S Ž T mF T , T . s Hom Der J Ž J 0 mF J 0 , J 0 . s Hom SŽ A , j. 0Ž H Ž A, j . 0 m⌫ H Ž A, j . 0 , H Ž A, j . 0 . is spanned o¨ er ⌫ by the products in Ž). and Ž)).. This theorem may be considered as an extension of the classification of the flexible Lie-admissible algebras over a field of characteristic zero with an underlying simple Lie algebra Žsee w17, Chap. IIIx and the references therein ., since this classification is subsumed in cases Ža. and Žb.Žii. of the above result.
REFERENCES 1. G. M. Benkart and J. M. Osborn, An investigation of real division algebras using derivations, Pacific J. Math. 96 Ž1981., 265᎐300. 2. A. Elduque and J. M. Perez-Izquierdo, Composition algebras with large derivation ´ algebras, J. Algebra 190 Ž1997., 372᎐404. 3. A. Elduque and H. C. Myung, Color algebras and affine connections on S 6 , J. Algebra 149 Ž1992., 234᎐261. 4. A. Elduque and H. C. Myung, The reductive pair Ž B3 , G 2 . and affine connections on S 7, J. Pure Appl. Algebra 86 Ž1993., 155᎐171. 5. J. R. Faulkner, Dynkin diagrams for Lie triple systems, J. Algebra 62 Ž1980., 384᎐392. 6. J. R. Faulkner, Identity classification in triple systems, J. Algebra 94 Ž1985., 352᎐363. 7. S. Helgason, ‘‘Differential Geometry, Lie Groups and Symmetric Spaces,’’ Academic Press, San Diego, 1978. 8. J. E. Humphreys, ‘‘Introduction to Lie Algebras and Representation Theory,’’ SpringerVerlag, BerlinrHeidelbergrNew York, 1972. 9. N. Jacobson, ‘‘Structure and Representations of Jordan Algebras,’’ Am. Math. Soc., Providence, 1968. 10. V. G. Kac, Graduated Lie algebras and symmetric spaces, Funct. Anal. Appl. 2 Ž1968., 182᎐183. 11. V. G. Kac, Automorphisms of finite order of semisimple Lie algebras, Funct. Anal. Appl. 3 Ž1969., 252᎐254. 12. V. G. Kac, ‘‘Infinite Dimensional Lie Algebras,’’ 3rd ed., Cambridge Univ. Press, Cambridge, UK, 1990.
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13. H. T. Laquer, ‘‘Invariant affine connections on Lie groups,’’ Trans. Amer. Math. Soc. 331 Ž1992., 541᎐551. 14. H. T. Laquer, Invariant affine connections on symmetric spaces, Proc. Amer. Math. Soc. 115 Ž1992., 447᎐454. 15. W. G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 Ž1952., 217᎐242. 16. O. Loos, ‘‘Symmetric Spaces. II. Compact Spaces and Classification,’’ Benjamin, New York, 1969. 17. H. C. Myung, ‘‘Malcev᎐Admissible Algebras,’’ Progress in Mathematics, Vol. 64, Birkhauser, Boston, 1986. ¨ 18. E. Neher, ‘‘On the classification of Lie and Jordan triple systems,’’ Comm. Algebra 13 Ž1985., 2615᎐2667. 19. K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 Ž1954., 33᎐65. 20. J. M. Perez-Izquierdo, Division composition algebras through their derivation algebras, J. ´ Algebra, to appear.