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Smooth quasigroups and loops. Recent achievements and open problems. (English) Zbl 0969.20043
Costa, Roberto (ed.) et al., Nonassociative algebra and its applications. Proceedings of the fourth international conference, São Paulo, Brazil. New York, NY: Marcel Dekker (ISBN 978-0-8247-0406-3/pbk; 978-1-138-40176-1/hbk; 978-0-429-18767-4/ebook). Lect. Notes Pure Appl. Math. 211, 337-344 (2000).
A short up-to-date survey on the matter is given. Recent achievements in the smooth loop theory and its applications are outlined, and some open problems are discussed. The author considers some new classes of loops (smooth $$M$$-loops, $$\nu PL$$-loops, binary-Bol loops) and gives some preliminary account how the structural theory of them may be elaborated. The problem of smooth loop actions is outlined, and related differential equations are introduced. In addition, the problem of the structure theory of Bol algebras, not elaborated until now, is analyzed. In conclusion, the loop-theoretic foundations of differential geometry (generalized geometric algebra) are discussed.
For the entire collection see [Zbl 0940.00027].
##### MSC:
 20N05 Loops, quasigroups 22A22 Topological groupoids (including differentiable and Lie groupoids) 53A60 Differential geometry of webs
##### Keywords:
smooth quasigroups; smooth loops; Bol loops; Bol algebras