quantum symmetry QuantumSymmetry2 2011-01-15 13:36:43 2011-01-15 13:40:58 Definition QuantumTopologicalOrder quantum categories Poisson transformation Lie group Lie Algebra Hamilton algebroid graded Lie superalgebra extended quantum symmetry topological order symmetry group \section{Quantum Symmetry} Often quantum symmetry is understood in terms of properties of symmetry groups, their representations and related algebras. Quantum groups also possess quantum symmetries which are distinct from those exhibited by classical Lie groups, groups of rotations and Poisson or Lorentz transformation groups. Extended quantum symmetries are also encountered for quantum groupoids, quantum categories, Hamilton algebroids, graded Lie super-algebras, Lie algebroids and quantum systems with topological order.