Quasi Hopf quantum symmetry in quantum theory

In quantum theory, internal symmetries more general than groups are possible. We show that quasi-triangular quasi Hopf algebras G∗ (“quasi quantum groups”) as introduced by Drinfeld [1] permit a consistent formulation of a transformation law of states in the physical Hilbert space H, of invariance of the ground state, and of a transformation law of field operators which is consistent with local braid relations of field operators which generalise those proposed by Fröhlich [2]. All this remains true when Drinfeld's axioms are suitably weakened in order to build in truncated tensor products. Conversely, all the axioms of a weak quasi-triangular quasi Hopf algebra are motivated from what physics demands of a symmetry. Unitarity requires in addition that G∗ admits a ∗-operation with certain properties. Invariance properties of Green functions follow from invariance of the ground state and covariance of field operators as usual. Covariant adjoints and covariant products of field operators can be defined. The R-matrix elements in the local braid relations are in general operators in H. They are determined by the symmetry up to a phase factor. Quantum group algebras like $U_q(sl_2)$ with |q| = 1 are examples of symmetries with special properties. We show that a weak quasi-triangular quasi Hopf algebra G∗ is canonically associated with $U_q(sl_2)$ if $q^p$ = 1. We argue that these weak quasi Hopf algebras are the true symmetries of minimal conformal models. Their dual algebras G (“functions on the group”) are neither commutative nor associative