I am reading: "On crystal bases of the $q$-analogue of universal enveloping algebras" by Kashiwara.

On page 481 - Lemma 3.4.1:

For any $P\in U_q(\mathfrak{g})^-$ there exist unique $Q,R\in U_q(\mathfrak{g})^-$ such that: $$[e_i,P]=\frac{t_iQ-t_{i}^{-1}R}{q_i-q_i^{-1}}$$

Where $U_q(\mathfrak{g})$ is the $\Bbb Q(q)$-algebra generated by $e_i,f_i,q^h, h\in P^*$ with all the normal relations, and (not that it is likely to matter much here) $q_i=q^{(\alpha_i,\alpha_i)}$, $t_i=q^{(\alpha_i,\alpha_i)h_i}$.

My question: Why are $e_i'':P\mapsto Q$ and $e_i':P\mapsto R$ both endomorphisms of algebras?


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