The *-operation And Duality In Twisted Drinfeld Double Algebra

Given a finite group G and a three-cocycle ??H3(G,U(1))the twisted Drinfeld double D?(G)can be defined to be the algebra over the complex numbers.F is the field algebra of G-spin models.There is a*-operation so thatD?(G)is a Hopf C*-algebra when G is abelian.At this point,there is a C*-representation of D?(G)so that D?(G)and AG are commutants of each other where AG is the D?(G)-invariant space.