Quantum group ($U_q(\mathfrak{g})$)

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 $\mathbb{C}$, for $q\ne 0,\pm 1$, is $U_q({\mathfrak{sl}}_2)$ with generators $E,F,K,K^{-1}$ and relationsIts coproduct can be taken as $\Delta K=K\otimes K$, $\Delta E=E\otimes K+1\otimes E$, and $\Delta F=F\otimes 1+K^{-1}\otimes F$.^[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 $K=q^h$ and letting $q$ approach one recovers the familiar Lie bracket relations for ${\mathfrak{sl}}_2$. Making that limit rigorous requires choosing a suitable algebra over a parameter ring.