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Extensions of racks and quandles

Extensions of racks and quandles

Nicholas Jackson

A \defn{rack} is a set equipped with a bijective, self-right-distributivebinary operation, and a \defn{quandle} is a rack which satisfies anidempotency condition.

In this paper, we introduce a new definition of modules over a rack orquandle, and show that this definition includes the one studied by Etingofand Gra\~na \cite{etingof/grana:orc} and the more general one given byAndruskiewitsch and Gra\~na \cite{andr/grana:pointed-hopf}. We furthershow that this definition coincides with the appropriate specialisationof the definition developed by Beck \cite{beck:thesis}, and hence thatthese objects form a suitable category of coefficient objects in whichto develop homology and cohomology theories for racks and quandles.

We then develop an Abelian extension theory for racks and quandles whichcontains the variants developed by Carter, Elhamdadi, Kamada andSaito \cite{carter/elhamdadi/saito:twisted,carter/kamada/saito:diag} asspecial cases.

Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 151-167

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