# Why the coproduct of quantum groups are defined in this way?

Let $U_q(g)$ be a quantum group generated by $e_i, f_i, k_{\lambda}$, $\lambda \in Q$, $Q$ is the weight lattice of the Lie algebra $g$. The coproduct of $U_q(g)$ is defined as follows (I only write the formulas for $e_i, f_i$): \begin{align} & \Delta(e_i) = e_i \otimes k_i + 1 \otimes e_i, \\ & \Delta(f_i) = f_i \otimes 1 + k_i^{-1} \otimes f_i. \end{align} Here $k_i = k_{\alpha_i}$.

Why the coproduct of quantum groups are defined in this way? Can the coproduct defined using other formulas? Thank you very much.

• They mimick the coproduct of the universal enveloping algebra $U(\mathfrak{g})$. For example by using $k_i=1$ here you would get the usual coproduct rule. This was to be expected, because the idea is to deform $U(\mathfrak{g})$ somehow. – Jyrki Lahtonen Apr 20 '16 at 13:05