I am reading Kashiwara's paper [GLOBAL CRYSTAL BASES OF QUANTUM GROUPS](https://projecteuclid.org/euclid.dmj/1077293577). On page 462, the action of $\tilde{e}\_i$ on an integrable $U\_q(g)$-module $M$ is defined as follows. For $u \in \ker e\_i \cap M\_{\lambda}$ and $0 \leq n \leq \langle h\_i, \lambda \rangle$, define 
\begin{align}
& \tilde{e}\_i( f^{(n)} u ) = f^{(n-1)} u, \\
& \tilde{f}\_i( f^{(n)} u ) = f^{(n+1)} u.
\end{align}
Can we define the action of $\tilde{e}\_i$ on $M$ using by defining $\tilde{e}\_i( e^{(n)} v )$, where $v \in \ker f\_i \cap M\_{\lambda}$? What are the formulas for $\tilde{e}\_i( e^{(n)} v )$. Thank you very much.



$\tilde{e}_i$ action on $e^{(n)} u$
The actions are defined in the Kashiwara's paper: [On crystal bases of the Q-analogue of universal enveloping algebras](https://projecteuclid.org/euclid.dmj/1077295931) in (2.2.5).


Let $M$ be an integrable $U\_q(\mathfrak{g})$-module. Then
\begin{align\*}
M = \oplus\_{0 \leq n \leq -\langle h\_i, \mu \rangle} e\_i^{(n)} (\ker f\_i \cap M\_{\mu}),
\end{align\*}
where $M\_{\mu} = \{u \in M: h\cdot u = \langle h, \mu \rangle u, \forall \ h \in P^\* \}$, $P$ is the weight lattice of $\mathfrak{g}$. 


The actions of $\tilde{e}\_i$ and $\tilde{f}\_i$ on $M$ are given by 
\begin{align\*}
& \tilde{e}\_i(e\_i^{(n)} v) = e\_i^{(n+1)}v, \\
& \tilde{f}\_i(e\_i^{(n)} v) = e\_i^{(n-1)}v,
\end{align\*}
$v \in \ker f\_i \cap M\_{\mu}$, $0 \leq n \leq - \langle h\_i, \mu \rangle$.


