Quantum weight modules, weights in $P$, and integrability

The structural conditions are mutually consistent. If $v\in M_{\mu }$, then by [F1] $K_h(E_{i}v)=q^{⟨{\alpha }_i,h⟩}{E}_i{K}_{h}v=q^{⟨{\alpha }_i,h⟩}{q}^{⟨\mu ,h⟩}{E}_{i}v=q^{⟨\mu +{\alpha }_i,h⟩}{E}_{i}v$, so the prescribed containment $E_i{M}_{\mu }\subseteq M_{\mu +{\alpha }_i}$ agrees with the scalar prescription on $M_{\mu +{\alpha }_i}$; similarly $K_h(F_{i}v)=q^{-⟨{\alpha }_i,h⟩}{q}^{⟨\mu ,h⟩}{F}_{i}v=q^{⟨\mu -{\alpha }_i,h⟩}{F}_{i}v$. Moreover $K_0$ acts on $M_{\mu }$ by $q^{⟨\mu ,0⟩}=1$ and $K_h{K}_{h^{\mathrm{\prime }}}$ by $q^{⟨\mu ,h+h^{\mathrm{\prime }}⟩}=q^{⟨\mu ,h⟩}{q}^{⟨\mu ,h^{\mathrm{\prime }}⟩}$ by [F2], as the toral relations require.

The grading-preserving module maps of the definition form a category: identities preserve each $M_{\mu }$, and a composite of grading-preserving maps is grading-preserving.

For $\mu \in P$ define its toral character ${\chi }_{\mu }\text{\hspace{0.17em}⁣}:P^{\vee }\to \mathbb{Q}(q{)}^{\times }$ by ${\chi }_{\mu }(h):=q^{⟨\mu ,h⟩}$; the defining condition says that $K_h$ acts on $M_{\mu }$ by the scalar ${\chi }_{\mu }(h)$. Since $M$ is the internal direct sum of the $M_{\mu }$, the simultaneous eigenspace $M^{\chi }=\{v:K_{h}v=\chi (h)v\text{ for all }h\}$ is $M^{\chi }={⨁}_{\mu :{\chi }_{\mu }=\chi }{M}_{\mu }$. Two labels have the same character, ${\chi }_{\mu }={\chi }_{{\mu }^{\mathrm{\prime }}}$, if and only if $⟨\mu -{\mu }^{\mathrm{\prime }},h⟩=0$ for all $h\in P^{\vee }$; by [F3] this is exactly equality of the integer functions $h\mapsto ⟨\cdot ,h⟩$. If the pairing separates $P$ (that is, $\mu \mathrm{\ne }{\mu }^{\mathrm{\prime }}$ implies $⟨\mu -{\mu }^{\mathrm{\prime }},h⟩\mathrm{\ne }0$ for some $h$), each $M^{\chi }$ equals a single label space $M_{\mu }$; if it does not separate, $M^{\chi }$ is the sum of all label spaces sharing the character. For an explicit nonseparating witness, take $P={\mathbb{Z}}^2$, $P^{\vee }=\mathbb{Z}h$, ${\alpha }_1=(2,0)$, $h_1=h$, $d_1=1$, with $⟨(a,b),h⟩=a$: the distinct labels $(0,0)$ and $(0,1)$ have the same character, so a spectral eigenspace can merge two label spaces.

Kernels, images and quotients inherit gradings. For a grading-preserving $f\text{\hspace{0.17em}⁣}:M\to N$, the restriction $f_{\mu }:=f{\mathrm{\mid }}_{M_{\mu }}$ maps $M_{\mu }$ into $N_{\mu }$, and $\ker f={⨁}_{\mu }\ker f_{\mu }$, $\mathrm{im}f={⨁}_{\mu }\mathrm{im}{f}_{\mu }$ by [F4]; for a graded submodule $L\subseteq M$ (one with $L={⨁}_{\mu }(L\cap M_{\mu })$) the quotient has $M\mathrm{/}L={⨁}_{\mu }(M_{\mu }+L)\mathrm{/}L$. Hence the weight submodules and quotients used on this page carry induced label gradings, and the category is closed under the usual linear algebra of grading-preserving maps. The zero module has the empty grading $M_{\mu }=0$ for all $\mu$.

Integrability is exactly local nilpotence of the two families of generators on each vector: it is inherited by grading-preserving submodules and quotients, since $E_i^{N}v=0$ in $M$ implies $E_i^{N}w=0$ for $w$ in a submodule and $E_i^N(w+L)=0$ in a quotient; no uniform exponent and no finite dimension is required. The zero module is integrable: its only vector is $0$, and $E_i^{1}0=F_i^{1}0=0$, so the exponents $N_e(0,i)=N_f(0,i)=1$ witness the condition. Finite-dimensionality of the $M_{\mu }$ and boundedness of the support are not part of the definition and are not implied by it: for any datum, the direct sum of countably many trivial modules concentrated at label $0$ is integrable and has an infinite-dimensional label space. Unbounded support is possible for the rank-one datum $P=\mathbb{Z}\omega$, $\alpha =2\omega$, $P^{\vee }=\mathbb{Z}h$, $⟨\omega ,h⟩=1$: the direct sum of the finite strings $L(n)$, $n\ge 0$, is integrable (each vector occupies finitely many summands) and has arbitrarily high labels, by Divided-power commutation and the simple $U_{q_i}({\mathfrak{s}\mathfrak{l}}_2)$ string modules(iii).

Suppose finally that the pairing separates $P$. Then a $U_q(\mathfrak{g})$-module structure with a decomposition $M={⨁}_{\mu \in P}{M}_{\mu }$ satisfying the three displayed conditions is the same data as a module structure with $M_{\mu }=\{v:K_{h}v=q^{⟨\mu ,h⟩}v\text{ for all }h\in P^{\vee }\}$ for each $\mu$, together with $E_i{M}_{\mu }\subseteq M_{\mu +{\alpha }_i}$ and $F_i{M}_{\mu }\subseteq M_{\mu -{\alpha }_i}$. Indeed, given the conditions one has, by step 2.1, $M_{\mu }\subseteq \{v:K_{h}v=q^{⟨\mu ,h⟩}v\mathrm{\forall }h\}\subseteq M_{\mu }$ (the second inclusion because any vector of the direct sum with $K_{h}v=q^{⟨\mu ,h⟩}v$ for all $h$ has all its nontrivial label components in characters equal to ${\chi }_{\mu }$, hence in $M_{\mu }$ by separation); conversely the eigenspace equations for a decomposition into the $M_{\mu }$ directly give the scalar condition, and the containments are assumed. This is the usual type-one definition for a separating datum, and it completes the discussion of the definition.