Extensions of racks and quandles

A rack is a set equipped with a bijective, self-right-distributive binary operation, and a quandle is a rack which satisfies an idempotency condition. In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Grana [9] and the more general one given by Andruskiewitsch and Grana [1]. We further show that this definition coincides with the appropriate specialisation of the definition developed by Beck [3], and hence that these objects form a suitable category of coefficient objects in which to develop homology and cohomology theories for racks and quandles. We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, El-hamdadi, Kamada and Saito [6, 7] as special cases.