Associative Geometries. I: Torsors, linear relations and Grassmannians

Abstract

We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in subsequent work.

v2: new results on relation with lattice theory added (Th. 2.4) v3: title and terminology changed: "torsor" instead of "groud"; to appear in Journal of Lie Theory

References in corpus (3)

• Associative Geometries. II: Involutions, the classical torsors, and their homotopes
• Homotopes and Conformal Deformations of Symmetric Spaces
• Inner Ideals and Intrinsic Subspaces of Linear Pair Geometries

Cited by in corpus (12)

• Associative Geometries. II: Involutions, the classical torsors, and their homotopes
• The projective geometry of a group
• Weil Spaces and Weil-Lie Groups
• Universal Associative Geometry
• Homotopes of Symmetric Spaces I. Construction by Algebras with Two Involutions
• An Essay on the Completion of Quantum Theory. I: General Setting
• Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach
• Generalized heaps, inverse semigroups and Morita equivalence
• Torsors and ternary Moufang loops arising in projective geometry
• Distributive lattices, associative geometries: the arithmetic case
• An Essay on the Completion of Quantum Theory. II: Unitary Time Evolution
• Jordan Geometries - an Approach by Inversions