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Catalan loops

Published online by Cambridge University Press:  19 July 2010

LING LONG  and
JONATHAN D. H. SMITH
LING LONG
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A. e-mail: [email protected], [email protected]
JONATHAN D. H. SMITH
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A. e-mail: [email protected], [email protected]
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 3 , November 2010 , pp. 445 - 453
Copyright
Copyright © Cambridge Philosophical Society 2010

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