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Generalized projective geometries: General theory and equivalance with Jordan structures. (English) Zbl 1035.17043

The classical Jordan pairs belong in a natural way to geometries like projective spaces or Grassmannian geometries. Therefore, it is natural to ask if there exists a general method to assign a geometry to an arbitrary Jordan pair and, moreover, if there is an axiomatic description of the resulting geometries. In his fundamental paper, the author gives a extensive answer to both questions.
In the sequel we let \(K\) be a commutative unital ring such that \(2\) is invertible in \(K\). An affine pair geometry over \(K\) is given by two sets \(X\) and \(X'\) and a subset \(M \subseteq X \times X'\) (whose elements are called “remote” or “distant” pairs) such that (i) for every \(o'\in X'\), the set \(V_ {o'} := \{ x \in X \,| \, (x,o')\in M\}\) is a non-empty affine space over \(K\), and (ii) the same property holds for the analogously defined sets \(V'_ o \subseteq X'\), \(o\in X\). Let \(D\) be the set of \((o,o',x) \in X\times X'\times X\) such that \((o,o'),(x,o') \in M\). For \(r \in K\) we define a ternary “multiplication” \(\mu_r : D \rightarrow X; (o,o',x) \mapsto \mu_r(o,o',x):=rx\), where \(rx\) is the usual multiplication by scalars in the \(K\)-module \(V_ {o'}\) with origin \(o\). (It is worth remarking that the maps \(\mu_ r(o,o',\cdot)\), \(r\in K\), comprise all information on the \(K\)-module \(V_ {o'}\).) Exchanging the rôles of \(X\) and \(X'\) yields the dual multiplication maps \(\mu'_ r\).
A generalized projective geometry is an affine pair geometry \((X,X')\) whose multiplication maps \(\mu_ r\) and \(\mu'_ r\) satisfy certain “fundamental identities” (whose archetypes are identities for the maps of a projective space \(X\) and its dual \(X'\)). A homomorphism of generalized projective geometries is a pair \((f,f'):(X,X')\rightarrow (Y,Y')\) which preserves remoteness as well as the multiplication maps \(\mu_ r\), \(\mu'_ r\).
The scalar extension of a generalized projective geometry \((X,X')\) over \(K\) by the ring \(K[\varepsilon]\) of dual numbers over \(K\) can be considered as a kind of tangent bundle of \((X,X')\). Choose a base point \((o,o')\in M\) and put \(V:=T_ o X\), \(V':=T'_ {o'}X'\). Similar to the construction of a tangent Lie triple system of a symmetric space, derivations of the ternary maps \(\mu_ r\), \(\mu'_ r\) at \((o,o')\) give rise to a pair of trilinear maps \(T : V \times V' \times V \rightarrow V\) and \(T' : V'\times V\times V' \rightarrow V'\). If \(3\) is invertible in \(K\), then \((V,V',T,T')\) is a linear Jordan pair over \(K\). The assignment \((X,X') \mapsto (V,V')\) (which is called Jordan functor) is functorial. Conversely, if \((V,V')\) is a Jordan pair over \(K\), then there exists a generalized projective geometry with base point whose associated Jordan pair is isomorphic to \((V,V')\). The link between \((X,X')\) and \((V,V')\) is the Kantor-Koecher-Tits algebra of \((V,V')\), which can be considered as the Lie algebra of “infinitesimal automorphisms” of \((X,X')\).
Moreover, the Jordan functor also yields an equivalence between Jordan triple systems (which are the same as Jordan pairs with involution) and so-called generalized polar geometries. Here, a generalized polar geometry is a generalized projective geometry \((X,X')\), together with a bijection \(p : X \rightarrow X'\) such that (i) \((p,p^ {-1}) :(X,X') \rightarrow (X',X)\) is an isomorphism and (ii) there exists at least one “non-isotropic” point \(x\in X\) (i.e.\((x,p(x)) \in M\)). The set \(M^ {(p)}\) of all non-isotropic points of \(p\) is closed under the binary map \(m(x,y):=\mu^ {-1} (x,p(x),y)\). In fact, the pair \((M ^ {(p)}, m)\) satisfies all axioms of a symmetric space in the sense of Loos (provided that one uses an appropriate substitute of the term “isolated fixed point”): similar to the finite-dimensional real and complex case, every Jordan triple system defines a symmetric space.

MSC:

17C37 Associated geometries of Jordan algebras
17C30 Associated groups, automorphisms of Jordan algebras
17C36 Associated manifolds of Jordan algebras
17B70 Graded Lie (super)algebras
51A05 General theory of linear incidence geometry and projective geometries
51A50 Polar geometry, symplectic spaces, orthogonal spaces
53C35 Differential geometry of symmetric spaces
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References:

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