[Paper Review] The algebra of discrete torsion

Ralph M. Kaufmann|ArXiv.org|Aug 9, 2002

Finite Group Theory Research8 references5 citations

TL;DR

This paper establishes that discrete torsion in $G$-Frobenius algebras arises from a universal group action of $H^2(G, k^*)$ via tensoring with twisted group rings, providing an algebraic realization analogous to projective representations. The key contribution is showing that all definitions of discrete torsion are unified through this action, which preserves the underlying linear structure and correctly transforms partition functions via a 2-cocycle-derived bi-character.

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).

Motivation & Objective

• To unify disparate definitions of discrete torsion in $G$-Frobenius algebras under a single algebraic framework.
• To show that discrete torsion acts universally on $G$-Frobenius algebras via tensor product with twisted group rings.
• To establish an analogy between discrete torsion and the theory of projective group representations and twisted group algebras.
• To identify a second class of discrete deformations related to $\mathbb{Z}/2\mathbb{Z}$-graded structures, classified by $\mathrm{Hom}(G, \mathbb{Z}/2\mathbb{Z})$.

Proposed method

• Define $G$-Frobenius algebras as $k[G]$-module algebras satisfying the Yetter–Drinfeld condition, making them modules over the Drinfeld double $D(k[G])$.
• Construct the action of $H^2(G, k^*)$ on $G$-Frobenius algebras by tensoring with twisted group rings $k^ au[G]$, where $\tau \in Z^2(G, k^*)$.
• Show that this tensor product action preserves the algebraic structure and realizes the bi-character $\epsilon(g,h)$ as derived from the 2-cocycle $\alpha$, with $\epsilon \in H^1(G, k^*[G])$.
• Introduce a generalized construction using central extensions $G^\alpha$ of $G$ by an Abelian group $H$, and define $A^\alpha = A \#_\alpha k[H]$, a $G^\alpha$-Frobenius algebra.
• Demonstrate that the twist $A_\alpha$ (via $\alpha \in H^2(G, k^*)$) lifts to $A^\alpha$ via a push-down map induced by $\chi \in \mathrm{Hom}(H, k^*)$, with $A^{\alpha'} \simeq (A_\alpha)^\chi$.
• Use the universal extension $G^*$ with cocycle $\beta \in H^2(G, H^2(G, k^*))$ to show that any twist $A_\alpha$ lifts to a component of $A^\beta$.

Experimental results

Research questions

• RQ1 How can discrete torsion be universally realized as an algebraic action on $G$-Frobenius algebras?
• RQ2 What is the precise algebraic mechanism by which $H^2(G, k^*)$ acts on $G$-Frobenius algebras via tensor products with twisted group rings?
• RQ3 How does this action reproduce the expected behavior of the bi-character $\epsilon(g,h)$ in partition functions?
• RQ4 Can the theory of discrete torsion be placed on the same footing as the theory of projective representations and twisted group algebras?
• RQ5 What is the role of $\mathbb{Z}/2\mathbb{Z}$-graded structures in the deformation theory of $G$-Frobenius algebras?

Key findings

• Discrete torsion is realized as a universal action of $H^2(G, k^*)$ on $G$-Frobenius algebras via tensor product with twisted group rings $k^\tau[G]$, preserving the underlying linear structure.
• The action correctly reproduces the bi-character $\epsilon(g,h)$ as a 1-cocycle in $H^1(G, k^*[G])$, derived from a 2-cocycle $\alpha \in Z^2(G, k^*)$, and appears as a factor in partition functions.
• The construction establishes a complete analogy with projective representations: just as twisted group algebras classify projective representations, twisted $G$-Frobenius algebras classify discrete torsion deformations.
• For any central extension $G^\alpha$ of $G$ by $H$ and $\chi \in \mathrm{Hom}(H, k^*)$, the $G^\alpha$-Frobenius algebra $A^\alpha = A \#_\alpha k[H]$ lifts the twist $A_\alpha$, and $(A^\alpha)^\chi \simeq A_\alpha$.
• The universal extension $G^*$ with cocycle $\beta \in H^2(G, H^2(G, k^*))$ ensures that every twist $A_\alpha$ lifts to a component of $A^\beta$, with $A^\beta \simeq \bigoplus_{\alpha \in T} A^\alpha$ for a transversal $T$ of $B^2(G, k^*)$ in $Z^2(G, k^*)$.