Quantum groups are mathematical structures that arise in the context of quantum algebra and have applications in various fields, including representation theory, theoretical physics, and algebraic geometry. They generalize the notion of groups and play a crucial role in the study of quantum symmetries, particularly in the context of quantum mechanics and quantum field theory. Quantum groups can be thought of as certain deformations of classical Lie groups (or Lie algebras).
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A quantum group is a type of algebraic symmetry commonly encoded by a Hopf algebra deforming a classical symmetry algebra. The term covers several related constructions; it does not specify a single additional Hopf algebra axiom. A standard example over , for , is with generators and relationsIts coproduct can be taken as , , and .[ref]
These formulas make the symmetry act on tensor products of representations. In particular, the coproduct is generally not unchanged by swapping its two tensor factors. Formally writing and letting approach one recovers the familiar Lie bracket relations for . Making that limit rigorous requires choosing a suitable algebra over a parameter ring.