= Quantum group
{title2=$U_q(\mathfrak g)$}
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 $q\ne0,\pm1$, is $U_q(\mathfrak{sl}_2)$ with generators $E,F,K,K^{-1}$ and relations
$$
KEK^{-1}=q^2E,\qquad KFK^{-1}=q^{-2}F,\qquad EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.
$$
Its <coproduct> can be taken as $\Delta K=K\otimes K$, $\Delta E=E\otimes K+1\otimes E$, and $\Delta F=F\otimes1+K^{-1}\otimes F$.<Tensor Categories by Pavel Etingof and others>[section 5.6, Definition 5.6.1 and Theorem 5.6.2]{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.
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