Extensions of racks and quandles. (English) Zbl 1077.18010

Jackson, Nicholas

Homology Homotopy Appl. 7, No. 1, 151-167 (2005).

A rack is a set \(X\) with a binary operation \(f\) so that for any \(a,b\in X\) there is a unique \(c\in X\) with \(f(c,b)=a\) and so that for any \(a,b,c\in X\) one has \(f(f(a,b),c)=f(f(a,c),f(b,c))\). A rack is a quandle if \(f(a,a)=a\) for every \(a\in X\). Racks proved to be useful in Hopf algebra theory by work of Andruskiewitch, Graña and others. Given a rack \(X\) one defines a ‘trunk’ \(T(X)\) associated to the rack as a certain representation in a certain generalisation of the notion of a category. Given a rack \(X\) a ‘rack-module’ is then a ‘trunk-map’ of the trunk \(T(X)\) to the category of abelian groups satisfying compatibility conditions.
The first main theorem of the paper is the statement that the category of \(X\)-modules for a given rack \(X\) is equivalent to the category of abelian group objects over \(X\). Similar statements are proven for quandles. Finally the author defines a theory of extension of racks by means of \(2\)-cocylces and \(2\)-coboundaries, defined appropriately.

Reviewer: Alexander Zimmermann (Amiens)

MSC:

18G15	Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
18E30	Derived categories, triangulated categories (MSC2010)
16W30	Hopf algebras (associative rings and algebras) (MSC2000)

Keywords:

racks; wracks; quandles; representations of racks