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Journal of Generalized Lie Theory and Applications
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Symmetric bundles and representations of Lie triple systems

Wolfgang BERTRAM and Manon DIDRY

Institut Élie Cartan de Nancy, Universite Nancy, CNRS, INRIA, Boulevard des Aiguillettes, B.P. 239, F-54506 Vanduvre-les-Nancy, France

*Corresponding Author:
Institut Élie Cartan de Nancy
UniversitÉe Nancy, CNRS, INRIA
Boulevard des Aiguillettes
B.P. 239, F-54506 Vandeauvre-lÉes-Nancy, France

Received Date: January 9, 2009; Revised Date: May 07, 2009

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Abstract

We de ne symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying re ection space, and we investigate the corresponding forgetful functor both from the point of view of di erential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing \unusual" symmetric bundle structures on the tangent bundles of certain symmetric spaces.

Keywords

Although this is not common, linear representations of Lie groups may be defined as vector bun- dles in the category of Lie groups: if image is a (say, finite-dimensional) representation of a Lie group in the usual sense, then the semidirect product image of G and V is a Lie group and at the same time a vector bundle over G such that both structures are compatible in the following sense:

(R1) the projection image is a Lie group homomorphism,

(R2) the group law image is a morphism of vector bundles, i.e., it preserves fibers and, fiberwise, group multiplication image is linear.

Conversely, given a vector bundle F over G with total space a Lie group and having such properties, the representation of G can be recovered as the fiber Fe over the unit element e on which G acts by conjugation. For instance, the tangent bundle TG corresponds to the adjoint representation, and the cotangent bundle T*G to the coadjoint representation of G.

In this work we wish to promote the idea that this way of viewing representations is the good point of view when looking for a notion of "representation" for other categories of spaces which, like Lie groups, are defined by one or several "multiplication maps": somewhat simplified, a rep- resentation of a given object M of such a category is a vector bundle over M in the given category, where "vector bundle in the given category" essentially means that the analogs of (R1) and (R2) hold. In fact, this simple notion came out as a result of our attempts to find a "global" or "geometric" analog of the notion of representation of general n-ary algebraic structures: given a multilinear algebraic structure defined by identities, such as Lie, Jordan or other algebras or triple systems, Eilenberg [8] introduced a natural notion of (general) representation (which is widely used in Jordan theory, see [12,17,18]).1 Essentially, a representation V of such an n-ary algebra m is equivalent to defining on the direct sumimagem an n-ary algebraic structure satisfying the same defining identities as m and such that some natural properties hold, which turn out to be exactly the "infinitesimal analogs" of (R1) and (R2): for instance, V will be an "abelian" ideal in image m corresponding to the role of the fiber in a vector bundle. The archetypical example is given by the adjoint representation which is simply image with ε2 = 0, the scalar extension of m by dual numbers, which will of course correspond to the tangent bundle in the geometric picture. However, nothing guarantees in principle that there be a "coadjoint representation" and a "cotangent bundle in the given category"!

This approach is very general and has a wide range of possible applications: at least locally, any ane connection on a manifold gives rise to a smooth "multiplication map" (a local loop, see [20]), which by deriving gives rise to n-ary algebras, and hence may be "represented" by vector bundles. Concretely, we will show how all these ideas work for the most proeminent example of such structures, namely for symmetric spaces (here, the approach to symmetric spaces by Loos [16] turns out to be best suited; we recall some basic facts and the relation with homogeneous spaces image in Chapter 2). Symmetric bundles are defined as vector bundles in the category of symmetric spaces (Section 2.3) and their infinitesimal analogs, representations of Lie triple systems are introduced (Chapter 3). These have already been studied from a purely algebraic point of view by Hodge and Parshall [11]. Another algebraic point of view (see [10]) features the aspect of representations of Lie algebras with involution (cf. Section 4.1), which we use to prove that, in the real finite-dimensional case, such representations are in one-to-one correspondence with symmetric bundles (Theorem 4.2). This result implies that the cotangent bundle TM of a real finite-dimensional symmetric space is again a symmetric bundle { this is much less obvious than the corresponding fact for the tangent bundle, and it owes its validity to the fact that, on the level of representations of Lie triples systems, every representation admits a dual representation (Section 5.2). Whereas, among the algebraic constructions of new vector bundles from old ones, the dual and the direct sum constructions survive in the category of symmetric bundles, this is not the case for tensor products and hom-bundles: they have to replaced by other, more complicated, constructions (Chapter 5).

Compared to the case of group representations, a new feature of symmetric bundles is that they are "composed objects": for a Lie group, the group structure on image and its structure of a homogeneous vector bundle are entirely equivalent. For symmetric bundles, the structure of a homogeneous vector bundle carries strictly less information than that of the symmetric bundle: let us assume F is a symmetric bundle over a homogeneous symmetric space M = G/H; then F carries two structures: it is a homogeneous symmetric space F = L/K, and, under the action of the smaller group G, it is a homogeneous vector bundle image , with V = Fo being the fiber over the base point o = eH. Basically, seeing F as a homogeneous vector bundle only retains the representation of H on Fo, whereas seeing F as symmetric space L/K takes into account the whole isotropy representation of the bigger group K. In other words, there is a forgetful functor from symmetric bundles to homogeneous vector bundles. Conversely, the following "extension problem" arises: given a homogeneous symmetric space M = G/H, which homogeneous vector bundles (i.e., which H-representations) admit a compatible structure of a symmetric bundle? On the infinitesimal level of general representations of Lie triple systems, the forgetful functor appears as follows: a general representation of a Lie triple system m consists of two trilinear maps (r;m), and we simply forget the second component m (Section 3.4). The extension problem is then: when does r admit a compatible trilinear map m such that (r;m) is a representation of Lie triple systems? For a geometric interpretation of this problem, one notes that the trilinear map r is of the type of a curvature tensor, and indeed one can prove that every symmetric bundle admits a canonical connection (Theorem 6.1) such that r becomes its curvature tensor (Theorem 6.2). It seems thus that the representations of H that admit an extension to a symmetric bundle are those that can themselves be interpreted as holonomy representation of a connection on a vector bundle. However, the situation is complicated by the fact that the canonical connection on F does not determine completely the symmetric bundle structure on F.

We do not attack in this work the problem of classifying representations of, say, finite dimensional simple symmetric spaces; but we give a large class of examples of "unusual" symmetric bundle structures on tangent bundles (Chapter 7), thus showing that the above mentioned forgetful functor is not injective. In fact, as observed in [3], many (but not all) symmetric spaces image admit, besides their "usual" complexification image another, "twisted" or "hermitian" one image We show here that a similar construction works when one replaces "complexification" by "scalar extension by dual numbers" (replace the condition image byimage and that in this way we obtain two difierent symmetric structures on the tangent bundle TM. A particularly pleasant example is the case of the general linear groupimage in this case, the usual tangent bundle TM is the group image (scalar extension by dual numbers), whereas the "unusual" symmetric structure on the tangent bundle is obtained by realizing TM as the homogeneous space image whereimage is some degenerate version of the quaternions (Theorem 7.1). We conjecture that, for real simple symmetric spaces, there are no other symmetric bundle structures on the tangent space than the ones just mentioned. In other words, we conjecture that the extension problem as formulated here is closely related to the "extension problem for the Jordan-Lie functor" from [3]; however, this remains a topic for future research.

The results presented in this paper partially extend results from the thesis [6], where a slightly di erent axiomatic definition of symmetric bundles was proposed in a purely algebraic setting, permitting to state the analog of Theorem 4.2 (equivalence of symmetric bundles and representations of Lie triple systems) in an algebraic framework (arbitrary dimension and arbitrary base field; see Theorem 2.1.2 in loc. cit.), based on results published in [7]. The present paper is independent from the results of [7], but nevertheless the framework still is quite general: our symmetric spaces are of arbitrary dimension and defined over very general topological base fields or rings K { for instance, the setting includes real or complex infinite dimensional (say, Banach) symmetric spaces or p-adic symmetric spaces (Section 2.1). We hope the reader will agree that, in the present case, this degree of generality does not complicate the theory, but rather simplifies it by forcing one to search for the very basic concepts.

After this work had been finished, we learned from Michael Kinyon that the question of defining "modules" for an object in a category had already been investigated by Beck in his thesis (see [2]; see also [1]): he defines a module to be an abelian group object in the slice category over the given object. It seems reasonable to conjecture that, in the cases considered here, this notion should agree with ours, but by lack of competence in category theory we have not been able to check this.

Symmetric bundles

Notation and general framework

This work can be read on two di erent levels: the reader may tak image to be the real base field and understand by "manifold" finite-dimensional real manifolds in the usual sense; then our symmetric spaces and Lie groups are the same as in [16] or [13], or one may consider a commutative topological field or ring image having dense unit groupimage and such that 2 is invertible in image;then we refer to [5] for the definition of manifolds and Lie groups over image. Readers interested in the general case should just keep in mind that, in general,

 symmetric spaces need no longer be homogeneous (cf. Example 2.2 below),

 there is no exponential map and hence no general tool to "integrate" infinitesimal structures to local ones.

If we use such tools, it will be specifically mentioned that we are in the real (or complex) finite-dimensional case. In the sequel, the word linear space means "(topological) image-module"

Symmetric spaces and re ection spaces

A re ection space ("Spiegelungsraum", introduced by Loos in [15]) is a smooth manifold M together with a smooth "product map"image satisfying, for all image

image

image

image

The reflection space image is called a symmetric space if in addition

(S4) for all x ε M, the differential image of the "symmetry" σx at x is the negative of the identity of the tangent space image.

In the real finite-dimensional case this is (via the implicit function theorem) equivalent to

(S4') for all x ε M, the fixed point x of σx is isolated.

Homomorphisms of such structures are smooth maps which commute with product maps. According to (S3), all maps of the formimage are automorphisms; the subgroup G(M) of Aut(M) generated by these elements is called the transvection group of M. Often one considers the category of re ection spaces, respectively, symmetric spaces with base point: a base point is just a distinguished point, often denoted by x0 or o, and homomorphisms are then required to preserve base points. If o ε M is a base point, one defines the quadratic map by

image

and the powers by image

Example 2.1. The group case. Every Lie group with the new multiplication image is a symmetric space.

Example 2.2. Homogeneous symmetric spaces. We say that a symmetric space is homogeneous if the group G := G(M) acts transitively on it and carries a Lie group structure such that this action is smooth. Let o be a base point and H its stabilizer, so that image Then the mapimage is an involution of G, and the multiplication map on G/H is given by

image

In finite dimension over image every connected symmetric space is of this form, for a suitable involution σ of a Lie group G (see [15,16]).

Example 2.3. Linear symmetric spaces. Assume V is a linear space over image ; we consider image as a linear space and thus writeimage Assume that V carries a symmetric space structureimage which is a image-linear map. Because of (S4), the symmetry image being a linear map, must agree with its tangent map -idV . Then it follows that

image

image

Conversely, every linear space equipped with the multiplication map image is a symmetric space (group case G = V ). With respect to the zero vector as base point, image is translation by 2x, and the powers areimage

Example 2.4. Polynomial symmetric spaces. In the same way as in the preceding example, we can consider linear spaces together with a symmetric structure which is a polynomial map image See [6] for a theory of such spaces.

Symmetric bundles

A symmetric bundle (or, longer but more precise, symmetric vector bundle) is a vector bundle image

image and image. are symmetric spaces such thatimage is a homomorphism of symmetric spaces,

(SB2) for all image the map induced byimage fiberwise,

image

(which is well defined according to (SB1)), is linear.

Homomorphisms of symmetric bundles are vector bundle homomorphisms that are also homomorphisms of the symmetric spaces in question. Clearly, the concept of symmetric bundle could be adapted to other classes of bundles whenever the fibers belong to a category that admits direct products (e.g., multilinear bundles in the sense of [5]): it suffices to replace (SB2) by the requirement that the map image be a morphism in that category. Also, it is clear that such concepts exist for any category of manifolds equipped with binary, ternary or other "multiplication maps", such as generalized projective geometries (cf. [4]). However, in the sequel, we will stick to the case of vector bundles and symmetric spaces.

A symmetric bundle is called trivial if it is trivial as a bundle, and if, as a symmetric space, it is simply the direct product of M with a vector space. The first nontrivial example of a symmetric bundle is the tangent bundle F := TM of a symmetric space image: as to (SB1), it is well known that TM with product mapimage is a symmetric space such that the canonical projection is a homomorphism and the fibers are at subspaces (see [16] for the real finite-dimensional case and [5] for the general case). Property (SB2) follows immediately from the linearity of the tangent mapimage

Some elementary properties of symmetric bundles

For a symmetric bundle F over M, the following holds:

(SB3) The symmetric space structure on the fiber image over x ε M coincides with the canonical symmetric space structure of the vector spaceimage

(SB4) The zero-section image is a homomorphism of symmetric spaces. (Hence in the sequel we may identifyM with z(M), and the use of the same letter μ for the multiplication maps of M and F does not lead to confusion.)

(SB5) For all image the fiberwise dilation map

image

is an endomorphism of the symmetric space F; for image it is an automorphism.

In fact, for p = q, (SB2) says that the fiber Fp is a symmetric subspace of F such that its structure map image is linear, and (SB3) now follows in view of Example 2.3. Since a linear map sends zero vector to zero vector, we haveimage proving (SB4), and to prove (SB5), just note that forimage

image

In particular, note that (0)F is the projection onto the zero section, and that (-1)F can be seen as a "horizontal re ection with respect to the zero section".

Horizontal and vertical symmetries

Let image a symmetric bundle andimage We define the horizontal (resp., vertical) symmetry (with respect to u) by

image

For image the maps image commute with each other because of (SB5):

image

Therefore vu then is also is of order 2. Conjugating by image we see that for all image we get three pairwise commuting automorphismsimage of order 2 and fixing the point image

Lemma 2.5. The vertical symmetry depends only on the fiber Fp, that is, for all image we have image

Proof. Let us show that image with image i.e.,

image

But this follows from image and the fact thatimage is an automorphism.

Proposition 2.6. The space F together with the binary map image is a reflection space.

Proof. The defining properties (S1) and (S2) say that vv is of order 2 and fixes v, and this has already been proved above. In order to establish (S3), let image and image Then, using the preceding lemma,

image

image

We say that (F;μ) is the reflection space associated to the symmetric bundle (F; μ). Thus image from symmetric vector bundles to re ection spaces; it will be a recurrent theme in this work to interpret this functor as a forgetful functor. Note that the di erential of vu has 1-eigenspace tangent to the fiber through u and -1-eigenspace complementary to it; thus the distribution of the "vertical" 1-eigenspaces is integrable, whereas the distribution of the "horizontal" -1-eigenspaces is in general not (see Chapter 6: the curvature of the corresponding Ehresmann connection does in general not vanish).

Automorphisms downstairs and upstairs

The canonical projection image does in general not admit a cross-section; we cannot even guarantee that it is surjective. However, it is easily seen that the projection of transvection groups image is surjective: write image as a composition of symmetries at points of M; identifying M with the zero section in F we see that g gives rise to an element image with image In particular, if M is homogeneous, then so is F: in fact, if image then there existsimage with g:o = x; thenimage In the real finite-dimensional case, we may replace G(M) by its universal covering; then the zero section z : M ! F induces a homomorphism of this universal covering into G(F), having discrete kernel. Hence, if we write F = L/K, it is not misleading to think of G as a subgroup of L and of H as a subgroup of K (possibly up to a discrete subgroup).

Homogeneous bundles over symmetric spaces

Assume that M = G/H is a homogeneous symmetric space (Example 2.2). To any smooth action image on a manifold U one can associate the homogeneous bundle

image

When the base M = G/H is a symmetric space, we define image

image

and one can show that image becomes a reflection space such that the projection onto M becomes a homomorphism of reflection spaces (cf. [15, Theorem 1.5]). Let us say that then F is a reflection space over the symmetric space M. The preceding formula shows that the reflection vv does not depend on the choice of v ε U, i.e., it depends only on the base. Loos has shown [15] that, conversely, every real finite-dimensional and connected reflection space can be written in this way as a homogeneous bundle over a symmetric space. In particular, linear representations of H and reflection spaces over M with linear fibers ("reflection vector bundles over M") correspond to each other.

Having this in mind, we now consider a symmetric bundle image over a homogeneous symmetric space M = G/H. As we have just seen, F is then also homogeneous, say, F = L/K. Looking at H as a subgroup of K (see Section 2.6), we get a linear representation of H on the fiber Fo, and we can write image as a homogeneous bundle over the base M. In this way, the functor from symmetric bundles to reflection spaces corresponds in the homogeneous case to the functor from symmetric bundles F = L/K overM = G/H to the associated homogeneous bundle image Conversely, we can formulate an extension problem: For which representationsimage does the homogeneous bundleimage carry a symmetric bundle structure? If it does, how many such structures are there?

Derivations of symmetric bundles and vertical automorphisms

A derivation of a symmetric bundle F is a homomorphism of symmetric spaces image which at the same time is a smooth section of π (see [16] for this terminology in case of the tangent bundle). A vertical automorphism of a symmetric bundle is an automorphism f of the symmetric bundle F preserving fibers, i.e., image then is a derivation of F. Conversely, if X is a derivation, define

image

Then f is a vertical automorphism: it is clearly smooth, preserves fibers, and is bijective. It is an automorphism: using (SB2),

image

image

Summing up, vertical automorphisms are the same as derivations. Moreover, they clearly form a normal subgroup Vaut(F) in the group Aut(F), where composition corresponds to addition of sections. It follows that the space of derivations is stable under addition; it is also stable under multiplication by scalars, hence forms a vector group. The same kind of arguments shows that in fact we have an exact sequence

image

(which essentially splits if we take transvection groups; cf. the discussion in Section 2.6). Now fix a base point o ε M; then the involution given by conjugation with σ0o restricts to Vaut(F) and thus defines a linear map. Let us write

image

for the corresponding eigenspace decomposition.

Lemma 2.7. We have image and the map

image

is a bijection with inverse image

Proof. See [16] or [5, Proposition 5.9] for the proof in the case of the tangent bundle; the same arguments apply here.

General representations of Lie triple systems

Definition 3.1. A Lie triple system (Lts) is a linear space m over image together with a trilinear mapimage such that, writing also R(X; Y ) for the endomorphismimage

image (skew-symmetry),

image(the Jacobi identity),

image is a derivation of the trilinear product on m, i.e.,

image

For instance, if image is a Lie algebra with involution, then the -1-eigenspace m of σ with image is an Lts. Every Lts arises in this way (see Section 4.1 below). For later use we introduce also the "middle multiplication operators"image then, in presence of (LT1), property (LT2) can be written in operator form

image

and similarly, reading (LT3) as an identity of operators, applied to the variable W; V or Y , we get the following equivalent conditions, respectively:

image

image

image

The Lie triple system of a symmetric space

Let (M;μ) be a symmetric space with base point o. Consider the tangent bundle TM and write image for the derivations of the symmetric bundle TM, and image for the eigenspace decomposition from Lemma 2.7. One shows that g, seen as a space of vector fields on M, is stable under the Lie bracket and that σ0 induces an involution of this Lie algebra structure. Hence the -1-eigenspace m is an Lts. Via the bijection image from Lemma 2.7 this Lts structure can be transferred to the tangent space ToM, which, by definition, is the Lts associated to the pointed symmetric space (M,o) (cf. [16] or [5, Chap 5]). The Lts depends functorially on M and plays a similar role for symmetric spaces as the Lie algebra for a Lie group. (In particular, in the real finite-dimensional case there is an equivalence of categories between Lts and connected simply connected spaces with base point, cf. [16].)

The Lts of a symmetric bundle

Now assume that image is a symmetric bundle over M and fix a base point o ε M and let image Since F is a symmetric space, f is a Lie triple system. We wish to describe its structure in more detail. The di erentials of the three involutionsimage and image from Section 2.5 act by automorphisms on the Lts f (where we write 0 instead of 0o). The +1-eigenspace of # is the tangent space image of the fiberimage which we identify with V (in the notation from [5] we could also write "εV for this "vertical space").

Lemma 3.2. The decomposition image has the following properties:

(1) V is an ideal of image as soon as one of the x; y; z belongs to V ,

image

image

image

Proof. (1) is clear since V is the kernel of a homomorphism (the differential image of image at 0); (2) follows from the fact that m is the fixed point space of the horizontal automorphism image (moreover, we see that m is isomorphically mapped by image onto the Lts of M), and (3) holds sinceimage is an automorphism and hence

image

hence image belongs to m and thus toimage since V is an ideal. Finally, the fiber Fo carries the "flat" symmetric space structure of a linear space and hence [V; V; V ] = 0.

A side-remark: if we adapt the whole set-up to the case of bilinear bundles (in the sense of [5]) instead of vector bundles, we get essentially the same properties, the only di erence being that (3) does no longer hold: e.g., for the bilinear bundle TTM over M, the tangent model is image where the term in brackets is still an abelian Lts, but (3) does no longer hold: in fact, image is nonzero in general.

Representations and modules

Let m be an Lts. An m-module is a vector space V such that the direct sum image carries the structure of an Lts satisfying the properties from the preceding lemma. More explicitly, this means, by decomposing

image

that we are given two trilinear maps r and m:

image

image

satisfying the properties given by the following lemma.

Lemma 3.3. For any Lts image the spaceimage with a triple bracket given by (3.1) is an Lts if and only if r and m satisfy the following relations:

image

image

image

image

image

Proof. We have to show that (LT1){(LT3) for image are equivalent to (R1)-{(R4): first of all, we note that a bracket is zero if more than one of the three arguments belongs to V . Now, (LT1) is equivalent to (R1) if both arguments belong to m and holds by (3.1) if one is in V and the other in m. Next, (LT2) is an identity in three variables. We may assume that two variables, say X and Z, belong to m, and write (LT2) in its operator form (LT2a). Thus we see that (LT2) is equivalent to (R2). Finally, (LT3) is an identity in 5 variables. In order to get a nontrivial identity, we can assume that at least four of them belong to m. We then write (LT3) in operator form (identities (LT3a,b,c) from Definition 3.1), and see that (LT3) is equivalent to (R3) and (R4), thus proving our claim.

Note that, in view of (R3), identity (R4) is equivalent to the following identity:

image

Condition (R4) can be rephrased by saying that the operator image defined by imageimage belongs to the space of derivations from m into V ,

image

image

Definition 3.4. A general representation of a Lie triple system m in a unital associative algebra A is given by two bilinear maps

image

image

such that (R1)-{(R4) hold (where o has to be interpreted as the product in A and the bracket is the Lie bracket in A). If A = End(V ) is the endomorphism algebra of a vector space, we say that V is an m-module. Homomorphisms of m-modules are defined in the obvious way, thus turning m-modules into a category. Given an m-module V , the Lts image with bracket defined by (3.1) is called the split null extension of m by the module V . It is fairly obvious that the split null extension depends functorially on the m-module V .

Example 3.5 (regular representation). For any Lts m, consider its "extension by dual numbers", i.e., let image (ring of dual numbers over image ), and

image

with the ε-trilinear extension of the bracket from m:

image

This is nothing but the split null extension of m by the regular representation, which by definition is given by V = m and

image

If M is a symmetric space, then the Lts of the tangent bundle TM is precisely image (cf. [5]).Hence the regular representation corresponds to the tangent bundle of M.

The extension problem revisited

The forgetful functor associating to a symmetric vector bundle its underlying re ection space corresponds to the forgetful functor image . Namely, if h is the image of the skew-symmetric map

image

then (LT3) implies that image is a Lie algebra, and for any representationimagewe may define image then the first relation of (R3) is equivalent to p being a representation. Thus we get the infinitesimal version of a homogeneous vector bundle. Now the problem of finding a compatible symmetric vector bundle structure corresponds to finding the second component m such that (r;m) defines a representation of m.

Reconstruction

We have shown that a representation of an Lts is the derived version of a symmetric bundle. Conversely, can one reconstruct a symmetric bundle from a representation of an Lts? As a first step, it is always possible to recover Lie algebras from Lie triple systems, and certain Lie algebra representations from Lie triple representations. The second step is then to lift these constructions to the space level: here we have to make assumptions on the base field and on the topological nature of M.

From Lie triple systems to Lie algebras with involution

Lie triple systems and imagegraded Lie algebras: the standard imbedding. Every Lie algebra image together with an involution σ gives rise to an Ltsimage equipped with the triple Lie bracket [[X; Y ];Z]. Conversely, every Lts m can be obtained in this way: let h be the subalgebra of the algebra of derivations of m generated by the endomorphisms R(x; y), x; y ε m. Then the space image carries a Lie bracket given by

image

This Lie algebra, called the standard imbedding of the Lts m, does in general not depend functorially on m; see [11,21] for a detailed study of functorial properties related to this and other constructions. Note that, in terms of the Lie algebra g, we can write

image

image-modules with involution. Assume image is a Lie algebra with involution. A representationimage is called aimagemodule with involution if W is equipped with a direct sum decomposition image which is compatible with σ in the sense that

image

commutes, where image is the identity on W+ and -1 on W-, and image

Lemma 4.1. Let m be an Lts and g its standard imbedding. There exists a bijection between -image-modules with involution and m-modules.

Proof. Given a image-module with involution (W; T ), we first form the semidirect product imageimage This is a Lie algebra carrying an involution given byimage Its -1-eigenspace image is an Lts satisfying the relations from Lemma 3.2, and hence is the split null extension corresponding to an m-module W-.

Given an m-module V , we construct first the split null extension image and then its standard imbedding image Then

image

is a g-module with involution.

Again, the correspondence set up by the lemma is functorial in one direction but not in the other; see [11] for this issue.

From modules to bundles

Theorem 4.2. Let image and M be a finite-dimensional connected simply connected symmetric space with base point o. Let image be its associated Lts and g its standard imbedding, with involution σ. Then the following objects are in one-to-one correspondence:

(1) (finite-dimensional) symmetric vector bundles over M,

(2) (finite-dimensional) image-modules with involution,

(3) (finite-dimensional) image-modules.

The bijection between (1) and (3) is an equivalence of categories.

Proof. We have already seen how to go from (1) to (3), and that (2) and (3) are in bijection. Let us give a construction from (2) and (3) to (1): let image be the split null extension coming from an image-module V andimage the standard imbedding ofimage and let W be the corresponding g-module with involution. Let G be the simply connected covering of the transvection group G(M) and write M = G/H. Then the representation of g on W integrates to a representation of G on W. Let

image

We claim that

(i) F is a vector bundle over M, isomorphic to the homogeneous bundle image

(ii) F carries the structure of a symmetric bundle over M.

Proof of (i): first of all,

image

is a well-defined bijection. Since ,this proves the first claim.

Proof of (ii): the Lie algebra image is the fixed imageed point space of an involution of b, and hence image is a symmetric space. Its Lts is image. The projection map image has as di erential the projection from image to m and hence is a homomorphism of symmetric spaces.

Let us show that the structure map image is linear. Since we already know that F is a homogeneous G-bundle, we may assume that p = o is the base point. Now we proceed in two steps:

(a) we show that image is linear. In fact, here we use that W is a G-module with involution, i.e.,image

image

image

Thus this map is described by image and thus is linear.

(b) Since image is a linear bijection,image is linear (and well defined) if and only if so is the map image But the last map is the same asimage (the point stands for the action of W- on F; recall that in every symmetric spaceimage Summing up, it suffices now to show that the map

image

is well defined and linear. Proof of this: let image and imageimageThen

image

image

Thus our map is described by image with a linear mapimage that depends on g, and hence is linear, proving claim (ii).

Finally, the fact that homomorphisms in the categories defined by (1) and (3) correspond to each other follows from the corresponding fact for (connected simply connected) symmetric spaces and Lie triple systems, see [16].

Linear algebra and representations

So far we do not know any representations other than the regular one and the trivial ones. In the following we discuss the standard linear algebra constructions producing new representations from old ones.

Direct sums

Clearly, if image and image are m-modules, thenimage is again a general representation. Correspondingly, the direct sum of symmetric bundles can be turned into a symmetric bundle.

The dual representation

If (V; r;m) is an m-module, then the dual space V* can be turned into an m-module by putting

image (5.1)

where image is the dual operator of an operatorimage In fact, the properties (R1)-{(R3) for image are easily verified; for (R4) note that (R4') written out for the dual is precisely (R4).

Equivalently: if image is aimage)-module with involution, one verifies that the dual module, image also is a module with involutionimage It follows that

image

leading to Formula (5.1).

In particular, in the finite-dimensional real case, invoking Theorem 3.4, the dual of the regular representation corresponds to the cotangent bundle image which thus again carries a symmetric bundle structure. (If M = G/H, then we may also write image whereimage It remains intriguing that there seems to be no really intrinsic construction of this symmetric space structure on image For this reason we cannot affirm that (in cases where a reasonable topological dual m* of m exists), in the infinite-dimensional case or over other base fields than image is again a symmetric space.

A duality principle

Note that, for finite-dimensional modules over a field, V is the dual of its dual module V*. More generally, we can define for any general representation of q in an algebra A its opposite representation in the algebra Aopp by putting

image

as above it is seen that this is again a representation. As an application of these remarks we get a duality principle similar as the o ne for Jordan pairs formulated by Loos (cf. [18]):

Proposition 5.1. If I is an identity in R(X; Y ) and M(U; V ) valid for all Lie triples over image then its dual identity I*, obtained by replacing R(X; Y ) by R(Y;X) and M(U; V ) by M(V;U) and reversing the order of all factors, is also valid for all Lie triples over image

Proof. If I is valid for all Lts, then it is also valid for all split null extensions obtained from representations and hence the corresponding identity, with R(X; Y ) replaced by r(X; Y ) and M(X; Y ) by m(X; Y ), is valid for all representations. Since the set of all representations is the same as the set of all opposite representations, and since the opposite functor changes order of factors and order of arguments, we see that I* is valid for all representations. In particular, it is valid for the regular representation and hence holds in q.

For instance, identities (R4) and (R4') (cf. Lemma 3.3 and remark following it) are dual in the sense of the proposition. We don't know about any application of Proposition 5.1; however, one may note that the original definition of Lie triple systems by Jacobson [12] as well as the exposition by Lister [14] are based on a set of five identities, among which two identities are equivalent to each other by the duality principle; they correspond to (LT3c) and its dual identity.

Tensor products

The tensor product image of two symmetric vector bundles is in general no longer a symmetric vector bundle: let image be the two fibers in question, regarded as m-modules. Extends A and B to image modules with involution,imageimage It is easily verified that thenimage is again a image module with involution. Now, the minus-part in

image

is

image

which therefore is another m-module, replacing the ordinary tensor product image

image

and there is a similar expression for the m-components and for image It is obvious that the operation fi is compatible with direct sums, and it also associative (in the same sense as the usual tensor product): the minus-part both in imageand in imageis

image

where image and with imageetc.

Hom-bundles

If A and B are m-modules, then Hom(A;B) is in general not an m-module, but we can use the same construction as in Section 5.4 to see that

image

is again an m-module.

Universal bundles

Various definitions of universal or enveloping algebras (resp., universal representations), attached to triple systems or algebras with involution, can be given; see [10,17,18,19]. These objects should correspond to certain "universal bundles" over a given symmetric space M. We intend to investigate such questions elsewhere.

The canonical connection of a symmetric bundle

We now study in more detail the di erential geometric aspects of symmetric bundles. It is immediately clear from Section 2.5 that a symmetric bundle F carries a fiber bundle connection in the general sense of Ehresmann, i.e., there is a distribution of horizontal subspaces, complementary to the vertical subspaces image (tangent spaces of the fiber); namely, as horizontal subspace take the fixed point spaces of the di erentials of the horizontal symmetry vu

image

In the sequel, we show that this Ehresmann connection is indeed a linear connection (in the general sense defined in [5], which in the real case amounts to the usual definitions), and that in general it has nonvanishing curvature, so that the distribution image is in general not integrable.

Theorem 6.1. Let image be a symmetric bundle over M. Then there exists a unique linear connection on the vector bundle F which is invariant under all symmetriesimage

Proof. The proof is similar to the one of [5, Theorem 26.3], and therefore will not be spelled out here in full detail. The uniqueness statement is proved by observing that the di erence image of two linear connections on F is a tensor field such thatimage is bilinear; if both L1 and L2 are invariant under σp, then image and henceimage It follows thatimage The main argument for the proof of existence consists in proving that the fibers of the bundle TF over M are abelian symmetric spaces (for this one has to analyze the map image in the fiber over a point image in the same way as the corresponding map image was analyzed in [5, Lemma 26.4]); then one concludes by general arguments that the fibers of TF over M carry canonically the structure of a linear space which is bilinearly related to all linear structures induced by bundle charts. By definition, this is what we call a linear connection on F.

Extension problem: on uniqueness

When image is the tangent bundle, with its canonical symmetric bundle structure, the connection defined by the preceding theorem is precisely the canonical connection of the symmetric space M (cf. [5, 16]). We will see in the next chapter that the abstract bundle image may carry several di erent (nonequivalent) symmetric bundle structures over M. The uniqueness statement of the theorem shows that they all lead to the same linear connection on image over M. Therefore the symmetric bundle structure is not uniquely determined by the linear connection from Theorem 6.1. Only at second order, by considering connections on image over image one is able to distinguish two symmetric bundle structures on image

Theorem 6.2. Let be the curvature tensor of the linear connection on the symmetric bun- dle F over M defined in the preceding theorem. Then is given by the r-component of the corresponding Lts representation: for all sections X; Y of TM and sections image

image

(i.e., with respect to an arbitrary base point image for all imageimage

Proof. In case image where Ω is the curvature of the canonical connection of M, it is well known that is (possibly up to a sign, which is a matter of convention) given by the Lie triple system of M, i.e.,image see [16,13,3,5] for three di erent proofs. It is inevitable to go into third-order calculations, and therefore none of these proofs is really short. Theorem 6.2 generalizes this result, and it can also be proved in several di erent ways. We will here just present the basic ideas and refer the reader to the above references for details. (a) Approaches using sections. Recall that the curvature may be defined by

image

where image is the horizontal lift of a vector fieldimage According to the definition of the horizontal space Hu given above, the horizontal lift of a vector field X is given,forimage by

image

By homogenity, it suces to calculate Ωo, the value at the base point, and then it is enough to plug in the vector fields image having valueimage for image This already implies thatimage and so we are left with calculatingimage This can be done by a calculation in a chart, corresponding essentially to [5, Lemma 26.4]; the outcome is [v;w; u], as expected.

(b) Approaches using higher-order tangent bundles. We analyze the structure of the bundles TF and TTF in exactly the same way as done in [5, Chapter 27], for the case of the tangent bundle: as mentioned in the proof of the preceding theorem, TF has abelian fibers defining the canonical connection, and TTF is non-abelian, leading to an intrinsic description of the curvature in terms of the symmetric space structure of the fibers. The conclusion is the same as with image for u; v;w belonging to the three "axes"image of TTF.

Extension problem: on existence

Via Theorem 5.3, we can relate the extension problem to a problem on holonomy representations: assume M = G/H is a homogeneous symmetric space and assume given a representation imageimage then the associated bundleimage carries a tensor field of curvature typeimage coming from the derived representationimage Can we find a connection (coming from a symmetric bundle structure on F) such that r is its curvature, i.e., such that this representation becomes a holonomy representation? In case of the tangent bundle, F = TM, the answer clearly is positive, since H is the holonomy group of the canonical connection on the tangent bundle.

Symmetric structures on the tangent bundle

In this chapter we will show that, for a rather big class of symmetric spacesM, the tangent bundle TM carries (at least) two di erent symmetric structures with the same underlying re ection space structure. For instance, this is the case for the general linear group, seen as symmetric space.

Example: the general linear group

The tangent bundle of the group image can be identified with the general linear group over the dual numbers image

image

with ε-bilinear multiplication image The canonical symmetric space structures on image and on its tangent bundle are given by the product map image (see Example 2.1).

Theorem 7.1. The vector bundle image admits a second symmetric bundle structure iso- morphic to the homogeneous symmetric space

image

where image is the noncommutative ring of "degenerate quaternions over image",

image

with group involution of L induced by conjugation of image with respect to its subalgebra of diagonal matrices.

Proof. Before coming to details of the calculation, let us give a heuristic argument: the "Hermitian complexification" of the group image is the symmetric space image of complex structures onimage (see [3, Chapter IV]). In principle, in the present context we should replace complex structuresimage by "infinitesimal structures"image but this attempt fails since E is not invertible. However, changing the point of view by considering a complex structure on image rather as aimage-form of the algebraimage and viewingimage rather asimage the suitably reformulated arguments carry over from the Cayley- Dickson extension image to the "degenerate Cayley-Dickson extension"image (and in fact to any extension Á la Cayley-Dickson of a commutative ring with nontrivial involution, see below). In the following calculations we need the matrices

image

Let us abbreviate image Then

image

where imagewe call map image an image this is justified by the fact that the fixed ring image is the ring of diagonal matrices in D, which is isomorphic to A, and that T anticommutes with the "structure map"imageFor the corresponding K-linear maps image we will again writeimage and f instead ofimage and image

The group

image

image

acts on the space image by conjugation. The stabilizer of image is given by b = 0, i.e., it is the group image Thus the image orbit of image is a homogeneous symmetric space:

image

We will show now that image is a vector bundle overimage and that this vector bundle is isomorphic to the homogeneous bundlimage over image To this end, observe that the groupimage contains a subgroup isomorphic toimage namely the group of matrices of the form image

with image Now let this subgroup act onimage

image

The stabilizer of T is gotten by taking g = h, and so the orbit of  under this group is isomorphic to the symmetric space image (group case). On the other hand,image contains the abelian normal subgroup of matrices of the form

image

with image and image is a semidirect product of these two subgroups. It follows that O is a vector bundle over the orbitimage Let us determine fiber over the base point Fn. Since

image

the stabilizer is gotten by X = 0, whereas for Y = 0 we get the fiber of image over the base point:

image

Thus the fiber is isomorphic to image and hence O is isomorphic as a homogeneous bundle toimage The remaining task of calculating the Lts of the symmetric space O becomes easier by transforming everything with the "Cayley transform" Rn: the Cayley transformed version of image

image

The Lie algebra of image is imbedded as the subalgebra given by the conditions a = d and b = c. The tangent spaces of our two special orbits inside image are complementary to this subalgebra, namely

image

so that image is the Lts of O, where the triple product is the usual triple Lie bracket of matrices since the group action is by ordinary conjugation of matrices. The formula shows that m clearly is isomorphic to the usual Lts of image that V is abelian and thatimage acts on V by conjugation, i.e., the r-component of the corresponding Lts representation of m is the usual one (corresponding to the fact that O is isomorphic to image as a homogeneous bundle). However, the whole Lts representation just defined is not equivalent to the adjoint Lts representation of image In fact, the corresponding involutive Lie algebrasimage are not isomorphic: already for n = 1, they are not isomorphic since image is commutative, whereasimage is not.

As mentioned above, instead of dual numbers we could have taken for A another ring extension of the formimage with arbitraryimage instead ofimage Then the symmetric spaceimage has two di erent scalar extensions fromimage the "straight one", simply gotten by taking the group case image and another, "twisted" one, given by the homogeneous symmetric spaceimagewhere image is the split Cayley-Dickson extension ofimage (see [6] for details). One could even replace here the algebra of square matrices by any other associative K-algebra. For image we are back in the example of the "twisted complexification ofimage It is interesting that the interpretation of O as the "space of complex structures" (i.e., endomorphisms with E2 = ) works only for invertible scalars , whereas the interpretation given here works uniformly for all scalars

ordan-extensions

Besides general linear groups, for all other "classical" symmetric spaces, there exist similar descriptions of symmetric bundle structures on the tangent bundle, see [6] for the case of Grassmannians, Lagrangians and orthogonal groups. The latter example gives rise to the "D-unitary groups", analogs of Sp(n) with H replaced by D. The general construction principle behind these examples uses Jordan theory:

Definition 7.2. A Jordan-extension of an Lts (m;R) is given by a Jordan triple product T : image such that

image

Recall that a Jordan triple system (Jts) is a linear space m together with a trilinear map image such that

image is symmetric in the outer variables:image

image

For any Jts T, the trilinear map defined by image is an Lts: we call the correspondenceimage the Jordan-Lie functor (cf. [3]).

Theorem 7.3 (the twisted regular representation defined by a Jordan extension). Assume image is an Lts having a Jordan extension T. Letimage be the regular representation of m (recall that the corresponding split null extensionimage is just the scalar extension by dual numbers). Then there exists another representationimage in general, not isomorphic to the regular representation, but having the same r-component .

Proof. We follow the lines of the proof of the corresponding statement for complexifications in [3, Chapter III] LET image and image be the ε-trilinear scalar extension of T by dual numbers. Then the conjugation

image

is a K-automorphism of T[ε]. But for any involutive K-automorphism, the "-isotope"

image

is again a Jts (cf. [3, Lemma III.4.5] for the easy proof). Moreover, this new Jts is "-linear in the outer variables and "-antilinear in the inner variable, and since  acts trivially on m, restriction of this new Jts to m3 gives us back T again. Now we let

image

This is an Lts which coincides with R on m since T was chosen to be a Jordan extension of R. Next, "m is an ideal of image by (anti-)linearity it is an ideal ofimage and hence it is one of image Finally, if two terms belong to image then application ofimage gives zero, and therefore also application of image gives zero. Thusimage is an Lts having the properties from Lemma 3.2, and hence is the split null extension corresponding to a representation image on εm.

Now we show that image for image

image

In order to prove that (r;m) and image are in general not isomorphic, observe that the split null extension of (r;m), being just scalar extension byimage has the property thatimageimage On the other hand,

image

Thus R and image cannot belong to isomorphic representations unless they vanish both. (This does not exclude that, as Lie triple systems over K, they may be isomorphic in special cases.)

Final comments

Essentially, all classical Lie triple systems (and about half of the exceptional ones) admit Jordan extensions (cf. [3, Chapter XII]). For instance, it is easily checked that image with the triple productimage is a Jts, and then

image

so that we have a Jordan extension of image Correspondingly,image and essentially all classical symmetric spaces admit on their tangent bundle a symmetric bundle structure that is di erent from the usual one. We conjecture that (at least for simple finite-dimensional Lts over image all symmetric bundle structures on the tangent bundle are exactly of two types:

(1) "straight": given by the canonical symmetric structure on TM, corresponding to the regular representation of the Lts m,

(2) "twisted": given by the construction from the preceding theorem.

This conjecture is of course supported by the corresponding fact for complexifications of symmetric spaces, which (for simple finite-dimensional Lts over R) are either straight or twisted (see [3, Corollary V.1.12]). However, the proof given in loc. cit. for the complex and para-complex cases does not carry over to the tangent case (the invertibility of i, resp. j, in K[i], resp. K[j], is used at a crucial point, and " clearly is not invertible in K["ε"]). A proof covering all three cases at the same time would be of great value for a better understanding of the "Jordan-Lie functor" (see [3]), and it should relate the "extension problem for the Jordan-Lie functor" with the extension problem for Lts representations as discussed here. One might conjecture that an interpretation in terms of the Cayley-Dickson process, which turned out to be useful in the special case of GL(n,K), could be the key for proving the conjecture, but this is not clear at present.

Acknowledgement

The first author would like to thank the Hausdor Institute (Bonn) for hospitality when part of this work was carried out.

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