. R A ] 2 4 Se p 20 09 ASSOCIATIVE GEOMETRIES . II : INVOLUTIONS , THE CLASSICAL GROUDS , AND THEIR HOMOTOPES

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called homotopes. The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion. Introduction: The classical groups revisited The purpose of this work is to explain two remarkable features of classical groups: (1) every classical group is a member of a “continuous” family interpolating between the group and its “flat” Lie algebra; put differently, there is a geometric construction of “contractions” (in this context also called homotopes), (2) every classical group and all of its homotopes admit a canonical completion to a semigroup; the underlying (compact) space of all of these “semigroup hulls” is the same for all homotopes. In fact, these results hold much more generally. The key property of classical groups is that they are closely related to associative algebras: either they are (quotients of) unit groups of such algebras, or they are (quotients of) ∗-unitary groups (0.1) U(A, ∗) := {u ∈ A| uu = 1} for some involutive associative algebra (A, ∗). This way of characterizing classical groups suggests to consider as “classical” also all other groups given by these constructions, including infinite-dimensional groups and groups over general base fields or rings K, obtained from general involutive associative algebras (A, ∗) over K. On an algebraic, or “infinitesimal”, level, features (1) and (2) are supported by simple observations on associative algebras: as to (1), associative algebras really are families of products (x, y) 7→ xay (the homotopes, see below), and as to (2), it is obvious that an associative algebra forms a semigroup and not a group with respect to multiplication. Our task is, then, to “globalize” these simple observations, and at the same time to put them into the form of a geometric theory: we have to free them from choices of base points (such as 0 and the unit 1 in an associative algebra). Just as in classical geometry, this means to proceed from a “linear” to a “projective” formulation, with an “affine” formulation as intermediate piece. For classical groups of the “general linear type” (An), this has already been achieved in Part I of this 2000 Mathematics Subject Classification. 20N10, 17C37, 16W10.