The algebra of discrete torsion

TL;DR

This paper explores the algebraic structures of G--Frobenius algebras associated with finite group quotients, revealing their module structure over the Drinfeld double and the universal role of discrete torsion as a group action.

Contribution

It provides an algebraic framework for discrete torsion as a universal group action on G--Frobenius algebras, with explicit constructions and classifications.

Findings

G--Frobenius algebras are modules over the Drinfeld double of k[G]

Discrete torsion acts as a universal group action via tensoring with twisted group rings

Classification of super-structure transformations by Hom(G,Z/2Z)

Abstract

We analyze the algebraic structures of G--Frobenius algebras which are the algebras associated to global group quotient objects. Here G is any finite group. These algebras turn out to be modules over the Drinfeld double of the group ring k[G]. We furthermore prove that discrete torsion is a universal group action of H^2(G,k^*) on G--Frobenius algebras by isomorphisms of the underlying linear structure. These morphisms are realized explicitly by taking the tensor product with twisted group rings. This gives an algebraic realization of discrete torsion and allows for a treatment analogous to the theory of projective representations of groups, group extensions and twisted group ring modules. Lastly, we identify another set of discrete universal transformations among G--Frobenius algebras pertaining to their super--structure and classified by Hom(G,Z/2Z).