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**Associative geometries. II: Involutions, the classical torsors, and their homotopes.**
*(English)*
Zbl 1206.20075

This work is the second part of [J. Lie Theory 20, No. 2, 215-252 (2010; Zbl 1206.20074)]. There, the key notion was that of an associative geometry (an object combining aspects of Lie groups and of generalized projective geometries). These associative geometries are also related to associative pairs. Now, using involutions of associative geometries as an ingredient, the authors construct homotopes for all classical groups and for their analogs in infinite dimension or over general base fields.

They also prove that under certain conditions, the groups and their homotopes have a canonical semigroup completion.

If we analyze the two key points in the above paragraphs: (1) construct homotopes, and (2) semigroup completion, one realizes that at an infinitesimal level (in words of the authors), that is, in an algebraic setting, both points can be observed on associative algebras. Indeed, associative algebras can be equipped always with a family of products \((x,y)\mapsto xay\) (homotopes), and an associative algebra forms a semigroup with respect to multiplication.

The paper deals with the globalization problem: that is passing from the algebraic (or infinitesimal) level to a geometric theory (in which we can get rid of the predominant role of \(0\) and \(1\) proceeding from a linear theory to an projective one).

For classical groups of type \(A\) this program has been achieved in part I of the present paper. The remaining classical groups (types B-D) and their generalizations, are studied in this work.

They also prove that under certain conditions, the groups and their homotopes have a canonical semigroup completion.

If we analyze the two key points in the above paragraphs: (1) construct homotopes, and (2) semigroup completion, one realizes that at an infinitesimal level (in words of the authors), that is, in an algebraic setting, both points can be observed on associative algebras. Indeed, associative algebras can be equipped always with a family of products \((x,y)\mapsto xay\) (homotopes), and an associative algebra forms a semigroup with respect to multiplication.

The paper deals with the globalization problem: that is passing from the algebraic (or infinitesimal) level to a geometric theory (in which we can get rid of the predominant role of \(0\) and \(1\) proceeding from a linear theory to an projective one).

For classical groups of type \(A\) this program has been achieved in part I of the present paper. The remaining classical groups (types B-D) and their generalizations, are studied in this work.

Reviewer: Candido Martín González (Málaga)

### 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 |

20G15 | Linear algebraic groups over arbitrary fields |