Topology
《Topology》 · 第 9 页 · Chapter 1，Section 1.1
Fix a ground field $k$ and an element $q \in k$ with $q \neq 0$ and $q^2 \neq 1$. Then $U_q(\mathfrak{sl}_2)$ is defined as the (associative) algebra (with 1 and over $k$) with generators $E, F, K$, and $K^{-1}$ and relations (R1) $K K^{-1} = 1 = K^{-1} K$, (R2) $K E K^{-1} = q^2 E$, (R3) $K F K^{-1} = q^{-2} F$, (R4) $E F - F E = \frac{K - K^{-1}}{q - q^{-1}}$. The algebra $U_q(\mathfrak{sl}_2)$ just defined is supposed to be a quantum analogue of the enveloping algebra $U(\mathfrak{sl}_2)$.