The Kashiwara operators ${\tilde{e}}_i$ and ${\tilde{f}}_i$ of an integrable module

Linearity and weights. On a string position $F_i^{(n)}{u}_n$ with $u_n\in H_{\lambda +n{\alpha }_i}$ put ${\tilde{e}}_i(F_i^{(n)}{u}_n):=F_i^{(n-1)}{u}_n$ for $n\ge 1$ and $0$ for $n=0$, and ${\tilde{f}}_i(F_i^{(n)}{u}_n):=F_i^{(n+1)}{u}_n$. By [F2] every $u$ is a unique sum of such positions inside its decomposition, so these prescriptions define linear maps on $M$; the shifted vectors lie in $M_{\lambda +{\alpha }_i}$ and $M_{\lambda -{\alpha }_i}$ respectively, since $F_i^{(n-1)}{u}_n\in M_{\lambda +{\alpha }_i}$ and $F_i^{(n+1)}{u}_n\in M_{\lambda -{\alpha }_i}$ by [F1].

The top projection. Writing $p_{\mathrm{t}\mathrm{o}\mathrm{p}}u:=u_0$ gives a well-defined linear idempotent by [F2]. For $u={\sum }_n{F}_i^{(n)}{u}_n$ one has ${\tilde{f}}_i{\tilde{e}}_{i}u={\sum }_{n\ge 1}{F}_i^{(n)}{u}_n$ by step 1.1, hence $u-{\tilde{f}}_i{\tilde{e}}_{i}u=u_0=p_{\mathrm{t}\mathrm{o}\mathrm{p}}u$; equivalently ${\tilde{f}}_i{\tilde{e}}_i=\mathrm{i}\mathrm{d}-p_{\mathrm{t}\mathrm{o}\mathrm{p}}$, and on each position with $n\ge 1$ the composition ${\tilde{f}}_i{\tilde{e}}_i$ is the identity.

The bottom projection. Put $a_n:=⟨\lambda +n{\alpha }_i,h_i⟩$. A nonzero admissible coefficient has $n\le a_n$, and when $n=a_n$ its shifted term $F_i^{(n+1)}{u}_n$ is zero by [F1]. Thus ${\tilde{f}}_{i}u={\sum }_{n<a_n}{F}_i^{(n+1)}{u}_n$. Its admissible top coefficients are $v_0=0$ and, for $m\ge 1$, $v_m=u_{m-1}$ if $m\le a_{m-1}$, and $v_m=0$ otherwise. Indeed $v_m\in H_{\lambda -{\alpha }_i+m{\alpha }_i}$ and the surviving positions satisfy $m\le a_{m-1}$, so [F2] identifies this as the unique admissible decomposition; bottom coefficients are discarded. Applying step 1.1 gives ${\tilde{e}}_i{\tilde{f}}_{i}u={\sum }_{n<a_n}{F}_i^{(n)}{u}_n$, hence $u-{\tilde{e}}_i{\tilde{f}}_{i}u={\sum }_{n=a_n}{F}_i^{(n)}{u}_n=p_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{m}}u$; on every position that is not a bottom this composition is the identity.

Kernels. For $u={\sum }_n{F}_i^{(n)}{u}_n$, step 1.1 gives ${\tilde{e}}_{i}u=0$ exactly when $u_n=0$ for all $n\ge 1$, that is $u=u_0\in H_{\lambda }$. On the other hand [F3] gives $E_{i}u={\sum }_{n\ge 1}[a_n-n+1{]}_i{F}_i^{(n-1)}{u}_n$ with $a_n=⟨\lambda +n{\alpha }_i,h_i⟩$; here $a_n-n+1\ge 1$ for $n\le a_n$ by admissibility and the term vanishes for $n>a_n$, so $E_{i}u=0$ exactly when $u_n=0$ for all $n\ge 1$ as well; hence $\ker {\tilde{e}}_i=\ker E_i$. Likewise ${\tilde{f}}_{i}u={\sum }_{n<a_n}{F}_i^{(n+1)}{u}_n=0$ exactly when $u_n=0$ for all $n<a_n$, and $F_{i}u={\sum }_{n<a_n}[n+1{]}_i{F}_i^{(n+1)}{u}_n$ vanishes exactly then because $[n+1{]}_i\mathrm{\ne }0$; hence $\ker {\tilde{f}}_i=\ker F_i$.

Agreement on the kernels. If $u\in \ker E_i$ then $u=u_0$ by [F2], so ${\tilde{f}}_{i}u=F_i{u}_0=F_{i}u$; if $u\in \ker F_i$ then every nonzero coefficient $u_n$ sits at a bottom, $n=a_n$, so $E_{i}u={\sum }_{n\ge 1}[a_n-n+1{]}_i{F}_i^{(n-1)}{u}_n={\sum }_{n\ge 1}{F}_i^{(n-1)}{u}_n={\tilde{e}}_{i}u$; the $n=a_n=0$ strings contribute zero to both raising operators by [F3].

Naturality. Let $\phi :M\to N$ be a homomorphism of weight-graded $A_i$-modules, so $\phi$ commutes with $E_i$, $F_i$ and $K_i$ and preserves each $H_{\mu }$ and each weight space. Applying $\phi$ to the admissible decomposition of $u$ gives the admissible decomposition of $\phi (u)$ by uniqueness [F2], because $\phi$ commutes with divided powers; hence $\phi {\tilde{e}}_{i}u={\tilde{e}}_i\phi (u)$, $\phi {\tilde{f}}_{i}u={\tilde{f}}_i\phi (u)$, and the same holds for the projections.

Interior comparison. In $L_i(2)$ take the highest vector $v=v_0$ and $u=F_{i}v=[1{]}_i{v}_1$; by [F3] $F_{i}u=[2{]}_i{F}_i^{(2)}v$, whereas $u$ is the position-$1$ vector of the string of $v$ of length $2$, so ${\tilde{f}}_{i}u=F_i^{(2)}v$. Since $[2{]}_i=q_i+q_i^{-1}$ and $q_i$ is transcendental over $\mathbb{Q}$, $[2{]}_i\mathrm{\ne }1$, so ${\tilde{f}}_i$ differs from $F_i$ at this interior position.