Group cohomology and gauge equivalence of some twisted quantum doubles

Abstract

The twisted quantum double D\sp ω(G) of a finite groupG is a quasitriangular quasi Hopf algebra obtained by twistingthe ordinary Drinfeld double by a suitable 3 cocycleω\in H\sp 3(G, {\bf C}\sp *). If G is abelian,D\sp ω(G) has a nontrivial associator Φ but it isalso a genuine Hopf algebra.\par The study of twisted quantumdoubles is motivated by an equivalence between the category ofmodules of D\sp ω(G) and the category of modules of someparticular vertex operator algebras, the so called holomorphicorbifold models [see R. Dijkgraaf, V. Pasquier and P. Roche,Nuclear Phys. B Proc. Suppl. 18B (1990), 60 72 (1991); MR\Cite{Dijkgraaf90:Quasi:1991}[92m:81238]].\par The authors giveconditions on G abelian and on the 3 cocycle ω underwhich D\sp ω(G) becomes a group algebra,{\bf C}[Γ\sp {ω}(G)]. In this case, the cocycle iscalled ``abelian'' and it turns out that the groupΓ\sp {ω}(G) is an abelian group. There is then acentral abelian extension1\longrightarrow {\widehat G}\longrightarrow Γ\sp {ω}(G)\longrightarrow G\longrightarrow1where {\widehat G} is the dual group. This defines a map fromthe set of abelian 3 cocycles to the group of ordinary 2cocycles with values in \widehat G. This map yields a grouphomomorphism Λ\sb G at cohomology level. The kernel ofΛ\sb G is described using the bar complex and it isisomorphic to the 2 torsion part of the group G; the image isdescribed in terms of the dual central abelianextension.\par Equivalence between D\sp ω(G) mod andD\sp { ω'}(G') mod (with nontrivial associativeconstraint) as monoidal categories is treated when G andω are abelian. In particular, it is proved that if\vert G\vert is odd, the categories are equivalent if and onlyif Γ\sp {ω'}(G')\simeq Γ\sp {ω}(G). Thisresult can be related to the results in [P. I. Etingof and S.Gelaki, Internat. Math. Res. Notices 2001, no. 2, 59 76; MR\Cite{Etingof01:Isocategorical:59--76}[2001j:20011]] where thecase of group algebras viewed as genuine Hopf algebras istreated.\par Gauge equivalence classes of twisted quantum doublesare put in correspondence with equivalence classes of quadraticfunctions on Γ\sp {ω}(G) with a metabolizerisomorphic to \widehat G and with pairs of rational latticesM\subset L with M even and self dual andL/M\simeq G.\par In the language of forms with a metabolizer,more detailed descriptions of D\sp ω(G) are obtained whenG is a p group