Uniqueness of the $i$-string decomposition of an integrable module

Finite-dimensional reduction. Let $u\in M_{\lambda }$. Choose $N\ge 0$ with $E_i^{N}u=0$ and set $W:={\mathrm{span}}_{\mathbb{Q}(q)}\{F_i^{(a)}{E}_i^{b}u:0\le b<N,a\ge 0\}$. Then $W$ is finite-dimensional: for each of the finitely many $b<N$ the local nilpotence of $F_i$ gives an integer $A_b$ with $F_i^{A_b}{E}_i^{b}u=0$, so the span of the vectors $F_i^{(a)}{E}_i^{b}u$, $a\ge 0$, is already spanned by the finitely many vectors with $a<A_b$.

The commutation scalar. Let $v$ be a weight vector with $E_{i}v=0$ and $K_{i}v=q_i^{a}v$, where $a=⟨\mathrm{w}\mathrm{t}(v),h_i⟩$. For $b\ge 1$ the formula of [F4] gives $E_i{F}_i^{(b)}v=F_i^{(b-1)}{\mathrm{\Theta }}_{i,b}v$, and ${\mathrm{\Theta }}_{i,b}$ acts on $v$ by the scalar $(q_i^b-q_i^{-b}{)}^{-1}{\sum }_{s=0}^{b-1}(q_i^{-2s}{q}_i^a-q_i^{2s}{q}_i^{-a})=[a-b+1{]}_i$ by the finite geometric-sum identities; hence $E_i{F}_i^{(b)}v=[a-b+1{]}_i{F}_i^{(b-1)}v$.

The Casimir. Put $\phi (K_i):=(q_i{K}_i+q_i^{-1}{K}_i^{-1})\mathrm{/}(q_i-q_i^{-1}{)}^2$ and $C:=F_i{E}_i+\phi (K_i)\in A_i$. Then $C$ is central in $A_i$: from $K_i{E}_i{K}_i^{-1}=q_i^2{E}_i$ and $K_i{F}_i{K}_i^{-1}=q_i^{-2}{F}_i$ one computes $\phi (q_i^2{K}_i)-\phi (K_i)=(q_i^2{K}_i-q_i^{-2}{K}_i^{-1})\mathrm{/}(q_i-q_i^{-1})$ and hence $[\phi (K_i),E_i]=E_i(\phi (q_i^2{K}_i)-\phi (K_i))=(K_i-K_i^{-1})E_i\mathrm{/}(q_i-q_i^{-1})=-[F_i{E}_i,E_i]$, while $\phi (q_i^{-2}{K}_i)-\phi (K_i)=-(K_i-K_i^{-1})\mathrm{/}(q_i-q_i^{-1})$ gives $[\phi (K_i),F_i]=-F_i(K_i-K_i^{-1})\mathrm{/}(q_i-q_i^{-1})=-[F_i{E}_i,F_i]$; so $[C,E_i]=[C,F_i]=0$, and $C$ commutes with $K_i^{\pm 1}$ visibly. Moreover $C$ acts on every weight vector $v$ with $E_{i}v=0$ and $K_{i}v=q_i^{a}v$ by the scalar $c_a:=\phi (q_i^a)=(q_i^{a+1}+q_i^{-a-1})\mathrm{/}(q_i-q_i^{-1}{)}^2$.

The scalars $c_a$ for $a\ge 0$ are pairwise distinct: if $c_a=c_b$ with $a,b\ge 0$, then $q_i^{a+1}+q_i^{-a-1}=q_i^{b+1}+q_i^{-b-1}$, that is $(q_i^{a+1}-q_i^{b+1})(1-q_i^{-(a+b+2)})=0$; as $q_i$ is transcendental over $\mathbb{Q}$, $q_i^m=1$ forces $m=0$, so the second factor is nonzero and $q_i^{a+1}=q_i^{b+1}$ gives $a=b$.

$W$ is a finite-dimensional $A_i$-submodule on which $E_i$ and $F_i$ are nilpotent and $K_i$ is diagonalisable with eigenvalues in $q_i^{\mathbb{Z}}$. Indeed $F_i{F}_i^{(a)}=[a+1{]}_i{F}_i^{(a+1)}$ and $K_i{F}_i^{(a)}{K}_i^{-1}=q_i^{-2a}{F}_i^{(a)}$ show that $F_i$ and $K_i$ preserve $W$, and for $a\ge 1$ the commutation formula of [F4] gives $E_i{F}_i^{(a)}{E}_i^{b}u=F_i^{(a)}{E}_i^{b+1}u+F_i^{(a-1)}{\mathrm{\Theta }}_{i,a}{E}_i^{b}u$, where ${\mathrm{\Theta }}_{i,a}$ acts on the weight vector $E_i^{b}u$ by a scalar; for $b+1<N$ both terms lie in $W$, and for $b+1=N$ the first term is $F_i^{(a)}{E}_i^{N}u=0$, so $E_i$ preserves $W$. Nilpotence is inherited from local nilpotence on the finite-dimensional $W$, and diagonalisability with the eigenvalue form from [F1] and [F2].

Claims (i) and (ii). Let $0\mathrm{\ne }v\in H_{\mu }$. By steps 1.1 and 2.2 applied with $N=1$, the span $W_v$ of $F_i^{(a)}v$, $a\ge 0$, is a finite-dimensional module with $F_i$ nilpotent, so there is a minimal $t\ge 1$ with $F_i^{(t)}v=0$, and $F_i^{(t-1)}v\mathrm{\ne }0$. Step 1.2 gives $0=E_i{F}_i^{(t)}v=[a_{\mu }-t+1{]}_i{F}_i^{(t-1)}v$, hence $[a_{\mu }-t+1{]}_i=0$, which by [F3] means $a_{\mu }-t+1=0$, that is $t=a_{\mu }+1\ge 1$. Thus $a_{\mu }\ge 0$, and $F_i^{(n)}v\mathrm{\ne }0$ for $0\le n\le a_{\mu }$ while $F_i^{(n)}v=([a_{\mu }+1{]}_i!\mathrm{/}[n{]}_i!)F_i^{n-a_{\mu }-1}{F}_i^{(a_{\mu }+1)}v=0$ for $n>a_{\mu }$. As $v$ was an arbitrary nonzero element of $H_{\mu }$, this proves $H_{\mu }=0$ for $a_{\mu }<0$ and the injectivity and vanishing clauses for $F_i^{(n)}$ on $H_{\mu }$.

Simple factors. Let $S$ be a simple subquotient of a finite-dimensional type-1 $A_i$-module on which $E_i$ is nilpotent. Then $S$ contains a nonzero weight vector $v$ with $E_{i}v=0$: choose $0\mathrm{\ne }w\in S$, let $m\ge 1$ be minimal with $E_i^{m}w=0$ and put $v_0:=E_i^{m-1}w\mathrm{\ne }0$; decomposing $v_0$ into weight components and using that $E_i$ preserves weights, some nonzero weight component $v$ satisfies $E_{i}v=0$. With $a:=⟨\mathrm{w}\mathrm{t}(v),h_i⟩$, step 3.1 gives $a\ge 0$, $F_i^{(a)}v\mathrm{\ne }0$ and $F_i^{(a+1)}v=0$. The span of $v,F_{i}v,\dots ,F_i^{(a)}v$ is a nonzero $A_i$-submodule by the formulas of steps 1.2 and 2.2 together with $[0{]}_i=0$, so it equals the simple module $S$; these $a+1$ vectors have distinct weights, whence $\dim S=a+1$ and $C$ acts on $S$ by the scalar $c_a$ of step 1.3.

Generalized-eigenspace splitting. Let $W$ be as in steps 1.1-2.2. A finite-dimensional module has a composition series: take a proper submodule of largest possible dimension, whose quotient is simple, and repeat inside that submodule; dimensions strictly decrease. Since $C$ is central, it preserves this finite filtration and acts on each simple factor by some $c_a$ from step 4.1. The product of the corresponding linear operators $(C-c_a)$ annihilates $W$, since each factor lowers the filtration by one. Its minimal polynomial therefore has only these linear factors. Applying [F5], the generalized eigenspaces $W[a]:=\ker (C-c_a{)}^m\cap W$ are $A_i$-submodules with $W={⨁}_{a}W[a]$. Every composition factor of $W[a]$ has highest label $a$: a factor of label $b\mathrm{\ne }a$, $b\ge 0$, would make $C-c_b$ non-invertible on $W[a]$, whereas $C-c_b$ is invertible on the generalized eigenspace of the distinct scalar $c_a$.

Blocks. Fix $a\ge 0$ with $W[a]\mathrm{\ne }0$ and let $r:=\dim W[a]\mathrm{/}(a+1)$, the number of composition factors of $W[a]$ counted with multiplicity. Each factor has by step 4.1 a weight-space decomposition into one-dimensional spaces of $K_i$-eigenvalues $q_i^a,q_i^{a-2},\dots ,q_i^{-a}$; Weight multiplicities add along this filtration: a lift of an eigenvector in a quotient can be projected to its eigenspace by the finite interpolation polynomial in the diagonal operator $K_i$, so the eigenspace sequence is exact. Hence the weight space of eigenvalue $q_i^{a-2n}$ has dimension $r$ for $0\le n\le a$, and $W[a]$ has no weight of eigenvalue $q_i^m$ with $m>a$. In particular $k:=\dim W[a{]}_a=r$, $\dim W[a]=k(a+1)$, and $E_i$ kills the top weight space $W[a{]}_a$ because it would raise the eigenvalue to $q_i^{a+2}$.

String splitting of a block. For $0\mathrm{\ne }w\in W[a{]}_a$ and $0\le n\le a$ one has $F_i^{(n)}w\mathrm{\ne }0$: if $F_i^{(n)}w=0$ for some minimal $n\ge 1$, then $F_i^{(n-1)}w\mathrm{\ne }0$ and step 1.2 gives $0=E_i{F}_i^{(n)}w=[a-n+1{]}_i{F}_i^{(n-1)}w$, forcing $n=a+1>a$ by [F3]. Hence each $N_n:=F_i^{(n)}W[a{]}_a$, $0\le n\le a$, has dimension $k$, and the $N_n$ lie in the distinct weight spaces of eigenvalues $q_i^{a-2n}$, so they are independent; their sum is $A_i$-stable because $E_i{F}_i^{(n)}w=[a-n+1{]}_i{F}_i^{(n-1)}w$ and $F_i{F}_i^{(n)}w=[n+1{]}_i{F}_i^{(n+1)}w$ for $w\in W[a{]}_a$; being a submodule of dimension $k(a+1)=\dim W[a]$, the sum equals $W[a]$.

Existence. Let $u\in M_{\lambda }$ be homogeneous and let $W$ be the finite-dimensional reduction of steps 1.1-2.2, which contains $u$ and splits as $W={⨁}_{a}W[a]={⨁}_a{⨁}_{n=0}^a{F}_i^{(n)}W[a{]}_a$ by steps 5.1-7.1. All generators of $W$ have labels in $\lambda +\mathbb{Z}{\alpha }_i$; within this coset the integer $⟨\cdot ,h_i⟩$ distinguishes labels because $⟨{\alpha }_i,h_i⟩=2$. Thus a summand $F_i^{(n)}W[a{]}_a$ lies in label weight $\lambda$ exactly when $a-2n=⟨\lambda ,h_i⟩$, and then $W[a{]}_a\subseteq M_{\lambda +n{\alpha }_i}$ with $E_i$ killing it, so it is contained in $H_{\lambda +n{\alpha }_i}$. Collecting, for each $n\ge 0$, the top vectors of all blocks with $a=⟨\lambda ,h_i⟩+2n$ into $u_n\in H_{\lambda +n{\alpha }_i}$ gives $u={\sum }_{n\ge 0}{F}_i^{(n)}{u}_n$; here $u_n\mathrm{\ne }0$ only when $0\le n\le a=⟨\lambda ,h_i⟩+2n$, which is exactly the admissibility condition $n\le ⟨\lambda +n{\alpha }_i,h_i⟩$. For a general $u\in M_{\lambda }$, apply this to the finitely many nonzero homogeneous components of $u$ and add.

Uniqueness. Suppose ${\sum }_{n=0}^N{F}_i^{(n)}{u}_n=0$ with $u_n\in H_{\lambda +n{\alpha }_i}$ admissible and $u_N\mathrm{\ne }0$. By [F4], $E_i^N{F}_i^{(n)}$ is a finite sum of terms $F_i^{(n-t)}{K}_i^m{E}_i^{N-t}$ with $0\le t\le n$; for $n<N$ every $t\le n$ satisfies $N-t\ge N-n\ge 1$, so each term kills $u_n$ (as $E_i{u}_n=0$), while for $n=N$ only $t=N$ survives and, by iterating step 1.2, $E_i^N{F}_i^{(N)}{u}_N=({\prod }_{t=1}^N[a_N-t+1{]}_i)u_N$ with $a_N=⟨\lambda +N{\alpha }_i,h_i⟩$. Admissibility gives $a_N\ge N$, so $a_N-t+1\ge 1$ for $1\le t\le N$ and the product is nonzero by [F3]. Hence $0=E_i^{N}0=({\prod }_{t=1}^N[a_N-t+1{]}_i)u_N$ forces $u_N=0$, a contradiction.

Direct sum. By step 3.1(ii) applied to $H_{\lambda +n{\alpha }_i}$, the map $F_i^{(n)}$ on $H_{\lambda +n{\alpha }_i}$ is injective for $n\le ⟨\lambda +n{\alpha }_i,h_i⟩$ and zero otherwise. Steps 8.1 and 9.1 say that every $u\in M_{\lambda }$ is a unique sum of vectors from the nonzero spaces $F_i^{(n)}{H}_{\lambda +n{\alpha }_i}$, which is exactly the statement that $M_{\lambda }$ is their internal direct sum. All selections in the argument were among finitely many vectors or finitely many scalars, so no choice principle is used.