Linear deformations of discrete groups and constructions of multivalued groups (Q2710720)

This interesting paper begins by fixing the needed definitions: multivalued groups, linear deformations, the manifold of associative algebra structures on a given linear space, etc. Then deformations of discrete multivalued groups are constructed.NEWLINENEWLINENEWLINEFrom the abstract: ``We show that the deformations of ordinary groups producing multivalued groups are defined by cocycles with coefficients in the group algebra of the original group and obtain classification theorems on these deformations. We indicate a connection between the linear deformations of discrete groups introduced in this paper and the well-known constructions of multivalued groups. We describe the manifold of \(3\)-dimensional associative commutative algebras with identity, fixed basis, and a constant number of values. The group algebras of \(n\)-valued groups of order \(3\) form a discrete set in the manifold.''NEWLINENEWLINENEWLINEIn Section 1, the author mentions that the orbits of the manifold of \(5\)-dimensional associative algebras with unit under the action of the general linear group are known in dimension up to 5; the reviewer whishes to add a reference to \textit{D. Happel}'s paper [Lect. Notes Pure Appl. Math. 51, 459-494 (1979; Zbl 0439.16014)]. Moreover these orbits are known in dimension 6 [\textit{Th. Dana-Picard, M. Schaps}, Houston J. Math. 22, No. 4, 749-773 (1996; Zbl 0877.16015)] and partially known in dimension 7 [\textit{Th. Dana-Picard}, CSM Conf. Proc. 11, 123-158 (1991; Zbl 0772.16013)] and 8 [\textit{Th. Dana-Picard}, Pac. J. Math. 164, No. 2, 229-261 (1994; Zbl 0799.16021)].