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Jordan structures and non-associative

geometry

Wolfgang Bertram

Institut Elie Cartan, Université Nancy I, Faculté des Sciences, B.P. 239, 54506 Vandœuvre-lès-Nancy, Cedex, France bertram@iecn.u-nancy.fr

Abstract. We give an overview of constructions of geometries associated to Jor-

dan structures (algebras, triple systems and pairs), featuring analogs of these con-

structions with the Lie functor on the one hand and with the approach of non-

commutative geometry on the other hand.

Keywords: Jordan pair, Lie triple system, graded Lie algebra, filtered Lie alge-

bra, generalized projective geometry, flag geometries, (non-) associative algebra and

geometry

AMS subject classification: 17C37, 17B70, 53C15

Introduction

Let us compare two aspects of the vast mathematical topic “links between ge- ometry and algebra”: on the one hand, the Lie functor establishes a close rela- tion between Lie groups (geometric side) and Lie algebras (algebraic side); this is generalized by a correspondence between symmetric spaces and Lie triple systems (see [Lo69]). On the other hand, the philosophy of Non-Commutative Geometry generalizes the relation between usual, geometric point-spaces M (e.g., manifolds) and the commutative and associative algebra Reg(M, K) of “regular” (e.g., bounded, smooth, algebraic,..., according to the context) K- valued functions on M , where K = R or C, by replacing the algebra Reg(M, K) by more general, possibly non-commutative algebras A. The interaction be- tween these two aspects seems to be rather weak; indeed, in some sense they are “orthogonal” to each other: first, the classical setting of Lie’s third the- orem is finite-dimensional, whereas the algebras A of commutative or non- commutative geometry are typically infinite dimensional; second, and more importantly, taking a commutative and associative algebra as input, we ob- tain as Lie bracket [x, y] = xy−yx = 0, and hence (if we forget the associative structure and retain only the Lie bracket) we are left with constructing a Lie

2 Wolfgang Bertram

group with zero Lie bracket. But this is not very interesting: it does not even capture the specific information encoded in “commutative geometries”.1

This remark suggests that, if one looks for a link between the two aspects just mentioned, it should be interesting to ask for an analog of the Lie func- tor for a class of algebras that contains faithfully the class of commutative associative algebras, but rather sacrifices associativity than commutativity. At this point let us note that the algebras A of non-commutative geometry are of course always supposed to be associative, so that the term “associative geometry” might be more appropriate than “non-commutative geometry”. Indeed, behind the garden of associative algebras starts the realm of general algebras, generically neither associative nor commutative, and, for the time being, nobody has an idea of what their “geometric interpretation” might be. Fortunately, two offsprings of associative algebras grow not too far away behind the garden walls: Lie algebras right on the other side (the branches reach over the wall so abundantly that some people even consider them as still belonging to the garden), and Jordan algebras a bit further. Let us just recall that the former are typically obtained by skew-symmetrizing an asso- ciative algebra, [x, y] = xy − yx, whereas the latter are typically obtained by symmetrizing them, x • y = 12 (xy + yx) (the factor

1 2 being conventional, in

order to obtain the same powers xk as in the associative algebra).

So let us look at Jordan algebras – their advantage being that the class of commutative associative algebras is faithfully embedded. In this survey paper we will explain their geometric interpretation via certain generalized projec- tive geometries, emphasizing that this interpretation really combines both aspects mentioned above. Of course, it is then important to include the case of infinite-dimensional algebras, and to treat them in essentially the same way as finite-dimensional ones. This is best done in a purely algebraic framework, leaving aside all questions of topologizing our algebras and geometric spaces. Of course, such questions form an interesting topic for the further development of the theory: in a very general setting (topological algebraic structures over general topological fields or rings) basic results are given in [BeNe05], and cer- tainly many results from the more specific setting of Jordan operator algebras (Banach-Jordan structures over K = R or C; see [HOS84]) admit interesting geometric interpretations in our framework. Beyond the Banach-setting, it should be interesting to develop a theory of locally convex topological Jordan structures and their geometries, taking up the associative theory (cf. [Bil04]).

The general construction for Jordan algebras (and for other members of the family of Jordan algebraic structures, namely Jordan pairs and Jordan triple systems, which in some sense are easier to understand than Jordan algebras; cf. Section 1) being explained in the main text (Section 2 and Section 3), let us

1 Of course, this does not exclude that other ways of associating Lie groups to function algebras are interesting, for example by looking at their derivations and automorphisms; but our point is precisely that such constructions need more input than the trivial Lie bracket on the algebra of functions.

Jordan structures and non-associative geometry 3

here just look at the special case of “commutative geometry”, i.e., the case of the commutative and associative algebra A = Reg(M, K) of “regular K-valued functions” on a geometric space M (here, K is a commutative field with unit 1; in the main text we will allow also commutative rings with 1). Looking at A as a Jordan algebra, we associate to it the space X := Reg(M, KP1) of “regular functions from M into the projective line KP1”. Of course, X is no longer an algebra or a vector space; however, we can recover all these structures if we want: we call two functions f, g ∈ X “transversal”, and write f>g, if f(p) 6= g(p) for all points p ∈ M . Now choose three functions f0, f1, f∞ ∈ X that are mutually transversal. For each point p ∈ M , the value f∞(p) singles out an “affine gauge” (by taking f∞(p) as point at infinity in the projective line KP1 and looking at the affine line KP1 \ {f∞(p)}); then f0(p) singles out a “linear gauge” (origin in the affine line) and f1(p) a “unit gauge” (unit element in the vector line). Obviously, any such choice of f0, f1, f∞ leads to an identification of the set (f∞)

> of all functions that are transversal to f∞ with the usual algebra A of regular functions on M . In other words, A and (X ; f0, f1, f∞) carry the same information, and thus we see that the encoding via A depends on various choices, from which we are freed by looking instead at the “geometric space” X . It thus becomes evident that A really is a sort of “tangent algebra” of the geometric (“non-flat”) space X at the point f0, in a similar way as the Lie algebra of a Lie group reflects its tangent structure at the origin.

This way of looking at ordinary “commutative geometry” leads to gener- alizations that are different from non-commutative geometry (in the sense of A. Connes) but still have much in common with it. For instance, both the- ories have concepts of “states”, i.e., a notion that, in the commutative case, amounts to recover the point space M from the algebra of regular functions. Whereas in case of non-commutative geometry this concept relies heavily on positivity (and hence on the ordered structure of the base field R), the cor- responding concept for generalized projective geometries is purely geometric and closely related to the notion of inner ideals in Jordan theory (Section 4). There are many interesting open problems related to these items, some of them mentioned in Section 4.

Summing up, it seems that the topic of geometrizing Jordan structures, incorporating both ideas from classical Lie theory and basic ideas from as- sociative geometry, is well suited for opening the problem of “general non- associative geometry”, i.e., the problem of finding geometries corresponding to more general non-associative algebraic structures.

Acknowledgements. This paper is partly based on notes of a series of lec- tures given at the university of Metz in April 2007, and I would like to thank Said Benayadi for inviting me at this occasion.

Notation: Throughout, K is a commutative base ring with unit 1 and such that 2 and 3 are invertible in K.

4 Wolfgang Bertram

1 Jordan pairs and graded Lie algebras

1.1 Z/(2)-graded Lie algebras and Lie triple systems

Let (Γ, +) be an abelian group. A Lie algebra g over K is called Γ -graded if it is of the form

g = ⊕

n∈Γ

gn, with [gm, gn] ⊂ gn+m.

Let us consider the example Γ = Z/(2). Here, g = g0 ⊕ g1, with a subalgebra g0 and a subspace g1 which is stable under g0 and such that [g1, g1] ⊂ g0. It is then easily seen that the linear map σ : g → g which is 1 on g0 and −1 on g1 is an involution of g (automorphism of order 2). Conversely, every involution gives rise to Z/(2)-grading of g, and hence such gradings correspond bijectively to symmetric Lie algebras (g, σ), i.e., to Lie algebras g together with an automorphism σ of order 2. Then g1 (the −1-eigenspace of σ) is stable under taking triple Lie brackets [[x, y], z]. This leads to the notion of Lie triple system:

Definition 1.1. A Lie triple system (LT