Catalan loops

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Motivated by a problem from number theory about the relationship between Fermat curves and modular curves, a new class of loops is introduced, the Catalan loops. In the number-theoretic context, these loops turn out to be abelian precisely when the Fermat curves and modular curves coincide. General Catalan loops arise on certain transversals to diagonal subgroups in special linear groups over rings with a topologically nilpotent element. The transversals consist of products of certain affine shears. In a Catalan loop, the multiplication and right division are given by rational functions. The left division is algebraic, corresponding to a quadratic irrationality. The left division embodies generating functions for the Catalan numbers. Structurally, Catalan loops are shown to be residually nilpotent. © Cambridge Philosophical Society 2010.

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Mathematical Proceedings of the Cambridge Philosophical Society

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