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Associative geometries. I: Torsors, linear relations and Grassmannians. (English) Zbl 1206.20074

The authors define and investigate the notion of “associative geometry”. The axiomatic approach to this concept given in Section 3 of the paper is more than convenient to give an idea of the definition. An associative geometry over a commutative ring \(K\) is given by a set \(\mathcal X\) which is a complete lattice (relative to \(\wedge\) and \(\vee\)) and maps \(\Gamma\colon\mathcal X^5\to\mathcal X\) and \(\Pi_s\colon\mathcal X^3\to\mathcal X\) (with \(s\in K\)). Defining \(L_{xaby}(z):=M_{xabz}(y)=R_{aybz}(x)=\Gamma(x,a,y,b,z)\) and \(x\top y\) iff \(x\wedge y=0\), \(x\vee y=1\) and \[ C_a:=a^\top:=\{x\in\mathcal X:x\top a\},\quad C_{a,b}:=C_a\cap C_b, \] then the following properties must be satisfied:
(1) \(\Gamma(\Gamma(x,a,y,b,z),a,u,b,v)=\Gamma(x,a,\Gamma(u,a,z,b,y),b,v)= \Gamma(x,a,y,b,\Gamma(z,a,u,b,v))\).
(2) (i) \(\Gamma(x,a,y,b,z)=\Gamma(z,b,y,a,x)\)(ii) \(\Gamma(x,a,y,b,z)=\Gamma(a,x,y,z,b)\).
(3) The following couples of maps are structural transformations: \((L_{xayb},L_{yaxb})\), \((M_{xabz},M_{zabx})\) and \((R_{aybz},R_{azby})\).
(4) (i) \(\forall a,b,y\in{\mathcal X},\;\Gamma(a,a,y,b,b)=a\vee b\),
(ii) \(\forall a,b,y\in{\mathcal X},\;\Gamma(a,b,y,a,b)=a\wedge b\),
(iii) if \(x\in C_{ab}\) then \(\Gamma(x,a,x,b,z)=\Gamma(z,b,x,a,x)\),
(iv) if \(a\top x\) and \(y\top b\) then \(\Gamma(x,a,y,b,b)=b\),
(v) if \(a\top x\) and \(y\top b\) then \(\Gamma(x,a,y,b,a)=a\).
(5) for all \(a\in\mathcal X\) and \(r\in K\) the set \(C_a\) is stable under \(\Pi_r\) and \(C_a\) becomes an affine space (see details in the paper).
(6) for all \(a,b\in\mathcal X\) we have \(\Gamma(U_a,a,U_b,b,U_a)\subset U_a\) and \(\Gamma(U_b,a,U_a,b,U_b)\subset U_b\) (definition of \(U_a\) in p. 224 of the paper).
In words of the authors, associative geometries combine aspects of Lie groups and of generalized projective geometries (where the former corresponds to the Lie product of an associative algebra, and the latter to its Jordan product).
One can associate an associative pair structure to any associative geometry with base point. And reciprocally: it is proved that for any associative pair there exists an associative geometry with base point whose associated associative pair is the given one.
Associative geometries are also related to Jordan geometries. Any associative pair gives rise to a Jordan pair with quadratic operator \(Q(x)y:=\langle xyz\rangle\). But also an associative geometry gives rise to a Jordan geometry. Again following the authors: the new feature is that there are two diagonal restrictions \(\Gamma(x,a,y,b,x)\) and \(\Gamma(x,a,y,a,z)\) which are equivalent. This can be used to give a new axiomatic foundation of Jordan geometries (also valid in characteristic two in contrast to previous versions).

MSC:

20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
17C37 Associated geometries of Jordan algebras
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
51A05 General theory of linear incidence geometry and projective geometries
17C50 Jordan structures associated with other structures
53C35 Differential geometry of symmetric spaces
51A50 Polar geometry, symplectic spaces, orthogonal spaces
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