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Topological and analytic loops. (English) Zbl 0747.22004
Quasigroups and loops: theory and applications, Sigma Ser. Pure Math. 8, 205-262 (1990).
[For the entire collection see Zbl 0704.00017.]
This chapter is a survey of the major developments in the theory of topological and analytical loops over the past 40 years. It stands relatively independent from the other chapters in the book.
The first section focuses on the general topological theory of topological loops and quasigroups (both are non-associative analogs of groups). The results are guided primarily by the search for appropriate generalizations from the theory of topological groups. A vital link between topological loops and transformation groups arises by considering the groups of left, right, and bilateral translations; this link is the subject of the second section. Weak associative laws are considered in the third section. Most of the substantial results apply only to quite restrictive classes of loops, and some version of weak associativity is a common assumption.
Much of the motivation for the study of topological and analytical loops comes from their natural occurrence in geometry. Some of these relationships are introduced in latter parts of the paper. But the major topic of the latter sections is the Lie theory of loops. A generalization of a Lie algebra called an Akivis algebra is introduced and the extent to which one has a Lie theory for various classes of loops is explored.
The article contains substantial historical information concerning the development of the subject and numerous references. It successfully ties together the results that relate to the area from a number of quite diverse sources.

MSC:
22A30 Other topological algebraic systems and their representations
22E60 Lie algebras of Lie groups
53A60 Differential geometry of webs
22A22 Topological groupoids (including differentiable and Lie groupoids)
20N05 Loops, quasigroups
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