Divided-power integrality of the quasi-$R$ action on based lattices

Divided coproducts. If $yx=q_i^{2ϵ}xy$ with $ϵ\in \{1,-1\}$, induction on $r$ gives $(x+y{)}^r\mathrm{/}[r{]}_i!={\sum }_{a+b=r}{q}_i^{ϵab}{x}^a{y}^b\mathrm{/}([a{]}_i![b{]}_i!)$. Multiplication on the right by $x+y$ gives the recurrence $c_{r+1,a}=q_i^{2ϵ(r-a+1)}{c}_{r,a-1}+c_{r,a}$ for the ordinary-power coefficients $c_{r,a}=q_i^{ϵa(r-a)}{\left(\genfrac{}{}{0px}{}{r}{a}\right)}_i$ when $0\le a\le r$, extended by zero outside that range; substitution reduces it to $[r+1{]}_i=q_i^{ϵ(r+1-a)}[a{]}_i+q_i^{-ϵa}[r+1-a{]}_i$, which follows by expanding quantum integers. For $F_i$, take $x=F_i\otimes 1$, $y=K_i\otimes F_i$, $ϵ=-1$ and $a=r-t$; for $E_i$, take $x=E_i\otimes K_i^{-1}$, $y=1\otimes E_i$, $ϵ=1$ and $a=t$. This proves the two formulas in [F1], including $r=0$ and both endpoint terms. Toral factors act on homogeneous vectors by Laurent units, so these coproducts preserve the relevant integral tensor lattices.

Highest-first integral action. Let $m=F_i^{(r)}{m}^{\mathrm{\prime }}$ be a homogeneous divided word of first height $h$, with $r\ge 1$, and let $n\in N_A$. The base $v_{\lambda }\otimes n$ is preserved integrally by $\mathrm{\Psi }$, since all positive components kill $v_{\lambda }$. For $t\ge 1$ put $z_t=F_i^{(r-t)}{m}^{\mathrm{\prime }}\otimes F_i^{(t)}n$ and $a_t=q_i^{t(\mu (h_i)-r+t)}\in A$, where $m^{\mathrm{\prime }}$ has weight $\mu$. Formula [F1] gives $\mathrm{\Delta }(F_i^{(r)})(m^{\mathrm{\prime }}\otimes n)=m\otimes n+{\sum }_{t=1}^r{a}_t{z}_t$. Semilinearity and bar compatibility give the exact recursion $\mathrm{\Psi }(m\otimes n)=\mathrm{\Delta }(F_i^{(r)})\mathrm{\Psi }(m^{\mathrm{\prime }}\otimes n)-{\sum }_{t=1}^r{\bar{a}}_t\mathrm{\Psi }(z_t)$. All tensors on the right have first height less than $h$, so induction makes their bars integral; the coproduct preserves the integral tensor lattice by [F1]. Divided words span $V_A(\lambda )$, hence $\mathrm{\Psi }$ is integral on the whole tensor lattice. The factor bar is an integral bijection, so $\mathrm{\Theta }$ is integral as well. The action is finite because first-factor raising is bounded by its highest cone.

Lowest-second action (b). Let $n=E_i^{(r)}{n}^{\mathrm{\prime }}$ be a homogeneous divided word in the lowest module, with $r\ge 1$, and let $m\in M_A$ be homogeneous. If $n^{\mathrm{\prime }}$ has weight $\mu$, put $z_t=E_i^{(t)}m\otimes E_i^{(r-t)}{n}^{\mathrm{\prime }}$ and $a_t=q_i^{-t(\mu (h_i)+r-t)}$ for $t\ge 1$. Indeed $K_i^{-t}$ in [F1] acts on the second factor after $E_i^{(r-t)}$, with scalar $q_i^{-t(\mu (h_i)+2(r-t))}$, which combines with $q_i^{t(r-t)}$ to give $a_t$. Thus $\mathrm{\Delta }(E_i^{(r)})(m\otimes n^{\mathrm{\prime }})=m\otimes n+{\sum }_{t=1}^r{a}_t{z}_t$ and $\mathrm{\Psi }(m\otimes n)=\mathrm{\Delta }(E_i^{(r)})\mathrm{\Psi }(m\otimes n^{\mathrm{\prime }})-{\sum }_{t=1}^r{\bar{a}}_t\mathrm{\Psi }(z_t)$. All bars on the right have smaller second divided-word height. At height zero every positive-degree quasi-$R$ term kills the second lowest vector, so its bar is the integral factor bar. Induction and step 1.1 prove preservation of $M_A\otimes \omega V_A(\lambda )$; composing with the integral factor bar gives integral $\mathrm{\Theta }$-action. The lower-cone support makes the action finite before evaluation. This proves (b).

Finite-type action finiteness without multiplicity bounds. First an integrable upper-cone module has reflection-stable label support: a nonzero admissible string position at label $\eta$ with top $\eta +n{\alpha }_i$ and top coroot $a=⟨\eta ,h_i⟩+2n$ has a nonzero mirrored position $a-n$ of label $\eta -⟨\eta ,h_i⟩{\alpha }_i$. Thus support is invariant under each $s_i$, and applying $w_0$ puts support in both ${\bigcup }_j({\lambda }_j-Q_{+})$ and ${\bigcup }_k(w_0{\lambda }_k+Q_{+})$. Each upper/lower cone intersection has only finitely many labels: writing a label as $w_0{\lambda }_k+\gamma ={\lambda }_j-\beta$ gives $\beta ,\gamma \in Q_{+}$ with fixed sum ${\lambda }_j-w_0{\lambda }_k$, whose independent simple-root coordinates bound both terms. In particular the highest module of [F5] has finite label support and finite-dimensional label spaces, hence finite total dimension. Now let $m$ be any homogeneous vector of any integrable module, with no support or multiplicity restriction. Choose integers $a_i\ge 0$ with $E_i^{a_i+1}m=0$. A sufficiently large dominant $\lambda \in Q\subseteq P$ exists even for a nonperfect datum: solve $Ax=\mathbf{1}$ over $\mathbb{Q}$, clear denominators and scale, so $\lambda ={\sum }_j{x}_j{\alpha }_j$ after this scaling has every $⟨\lambda ,h_i⟩$ a positive integer at least $a_i$. The map $U^{+}\to U^{+}m$, $u\mapsto um$, kills ${\sum }_i{U}^{+}{E}_i^{(⟨\lambda ,h_i⟩+1)}$, and by the half-exchange and highest presentation in [F5] its quotient is a copy of the finite-dimensional negative-half highest quotient as a vector space. Hence $U^{+}m$ is finite dimensional. Choose a finite homogeneous basis of this span; local nilpotence of every $F_i$ on its finitely many vectors gives common exponents. The same argument with the negative half shows that $U^{-}{U}^{+}m$ is finite dimensional. The torus preserves $U^{+}m$ because it is label graded, so triangular decomposition gives $Um=U^{-}{U}^0{U}^{+}m=U^{-}{U}^{+}m$. A general vector has finitely many homogeneous components and therefore also a finite-dimensional $U$-orbit. In particular $U^{+}m$ has only finitely many labels; if $U^{+}[\nu ]m\mathrm{\ne }0$, its label is $\mathrm{wt}(m)+\nu$, so only finitely many $\nu \in Q_{+}$ can act. This proves vectorwise finite quasi-$R$ action on arbitrary integrable finite-type modules. The divided coproducts in [F1] also prove tensor integrability: on a homogeneous pure tensor, every term of a sufficiently high simple-generator divided power has an exponent beyond the nilpotence bound of at least one factor, and toral factors only rescale homogeneous vectors. Finite maxima over finite sums and induction over the factors give integrability of finite tensor products, so the same argument proves finite successive quasi-$R$ actions there. It does not assert a finite raw-basis correction closure for arbitrary two-sided-unbounded based lattices.

Finite-type universal coefficients (c). In finite Cartan type choose a longest Weyl element $w_0$ and sufficiently large regular dominant weights as in [F4]. Building the lowest extremal vector by the reduced $w_0$-string of divided powers uses complete rank-one top-to-bottom strings, whose normalized norm factor at the end is $d_{a,a}=1$, and the extremal weight spaces are one-dimensional by the classical character comparison; hence the extremal vector is integral, bar fixed and of norm $1$, and by the norm-$1$ characterization it is the unique signed global element of its weight. The unsigned Chevalley map $\omega$ commutes with the normalized bar, so the normalized field isomorphism from the corresponding lowest module to the highest module is a form isometry and bar compatible, and its source integral lattice maps into the target because the extremal vector is integral and $U_A$-stable. Transported global vectors are integral, bar fixed, of residue norm $1$, hence are signed target global vectors; equal finite dimensions make this a signed integral lattice identification, sufficient for the coefficient test. Expanding ${\mathrm{\Theta }}_{\nu }$ in global half bases and applying it to the lowest extremal vector of a large highest module and the highest extremal vector of a large lowest module, the highest-first action of step 2.1 proves the output integral, and since distinct degrees have distinct weight pairs and the outputs are distinct actual $A$-basis vectors, every expansion coefficient lies in $A$. Hence ${\mathrm{\Theta }}_{\nu }\in U_A^{+}[\nu ]\otimes U_A^{-}[-\nu ]$, and the vectorwise finiteness proved in step 2.3 makes its action a finite sum on every arbitrary $U_A$-stable based pair. Each summand preserves the tensor lattice, so (c) follows without any weight-multiplicity restriction.

The constructed highest and lowest classes (d). Form right-associated products of highest modules. Highest-first action proves integrality of the normalized bar at every binary step, while the field cocycle of [F2] gives all parenthesizations the same bar without assuming integral action for intermediate arbitrary pairs. In the transfer order in which each positive-root component raises the first weight and lowers the second, only the degree-$0$ component fixes a label pair, so the bar is triangular with diagonal $1$. Every upward interval is finite by the upper-cone or finite-total-weight hypothesis, and for the lowest classes the lower-cone bound replaces finiteness of the whole mixed weight space. In a finite interval solve for a bar-fixed lift $g_t=e_t+{\sum }_{s>t}{a}_{ts}{e}_s$ with $a_{ts}\in q\mathbb{Z}[q]$, starting from the top: after subtracting the already constructed larger lifts, involutivity makes the remainder satisfy $f+\bar{f}=0$, whose unique positive $q$-part supplies the next correction. Finite induction gives existence, uniqueness and a unitriangular $A$-matrix with finite-interval inverse; the corrected family is an $A$-basis and an $A_0$-basis with the raw tensor crystal residues, is bar fixed, and has $A_{\mathrm{\infty }}$-span the actual tensor bar chart.

Closure under tensor constructions and based subquotients. For a highest tensor object $P=V({\lambda }_1)\otimes \cdots \otimes V({\lambda }_m)$ and any based $N$, right-associate $P\otimes N$. Repeated highest-first action from step 2.1 makes every binary bar integral. The cocycle identifies this bar with the grouped one using the constructed bar of $P$, so $\mathrm{\Theta }$ acts integrally on $P\otimes N$. The mirror argument uses step 2.2 for arbitrary first factor and a lowest tensor object. This includes pairs of constructed objects and proves the arbitrary-other-factor cases before inheritance. Based subobjects are spanned by subsets of the actual constructed bases and are therefore saturated, $(P_A\otimes Q_A)\cap (P^{\mathrm{\prime }}\otimes Q^{\mathrm{\prime }})=P_A^{\mathrm{\prime }}\otimes Q_A^{\mathrm{\prime }}$; $\mathrm{\Theta }$ preserves their field submodules by its $U$-monomial terms, so integral action follows from the equality. Based quotient maps are $U$-linear, send basis to basis or $0$, have basis-subset kernel and commute with bars because both bars fix those bases; their integral tensor maps are surjective onto free complementary quotient lattices, so integral action descends. Direct sums are handled componentwise with the stated weight bounds. This proves closure of $C_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}$ and, dually, of $C_{\mathrm{l}\mathrm{o}\mathrm{w}}$, hence (d) for the pure classes, and for an ordered highest-prefix, based-middle, lowest-suffix product each binary split has either a pure highest-class left block or a pure lowest-class right block, so one of the two action cases applies; every prefix transfer is bounded by a prefix upper cone or a complementary lowest suffix lower cone, so all correction intervals are finite.

The affine boundaries. For $A_1^{(1)}$, with $a_{01}=a_{10}=-2$, the degree-$\delta$ Kashiwara Gram matrix in the basis $F_0{F}_1,F_1{F}_0$ is $\left(\begin{array}{cc}1& q^2\\ q^2& 1\end{array}\right)$, so the correct $D$-coevaluation carries the factor $(q-q^{-1}{)}^2\mathrm{/}(1-q^4)=q^{-2}(1-q^2)\mathrm{/}(1+q^2)\mathrm{\notin }A$: the universal algebra-coefficient claim fails in affine type. For the stronger arbitrary-pair failure take $\lambda ={\mathrm{\Lambda }}_0$, $\mu =\lambda -\delta$, $p=[2]$, and the global element $g=F_1{F}_0{v}_{\lambda }$; then $E_{0}g=0$, $E_{1}g=p{F}_{0}v$, $F_0^{(r)}g=0$ for $r\ge 2$ by the length-three divided Serre relation, and $F_{1}g=p{F}_1^{(2)}{F}_{0}v$, $F_1^{(r)}g=0$ for $r\ge 2$ by its rank-one length-two string. The integral based-quotient presentation therefore gives a $U_A$-map $V_A(\mu )\mathrm{/}p\to V_A(\lambda )\mathrm{/}p$ sending the highest vector to $g<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p$. For each global $h_b$ its image has bar-fixed coefficients in $A\mathrm{/}(p)=\mathbb{Z}[i]$, whose bar-fixed subring is $\mathbb{Z}$; let $T_b$ be its unique integer-coordinate lift in the $\lambda$-global basis and set $c_b=h_b+T_b\mathrm{/}p$. The free module $M_A$ with basis ${\mathcal{B}}_{\lambda }\cup \{c_b\}$ inside $V_{\lambda }\oplus V_{\mu }$ is $U_A$-stable because $U{T}_b-T(U{h}_b)$ is divisible by $p$, all basis vectors are homogeneous and bar fixed, and because $1\mathrm{/}p=q\mathrm{/}(1+q^2)\in q{A}_0$ its exact chart and residue data equal the standard direct-sum based module, so all four based axioms hold. On the tensor $c_{\mathrm{t}\mathrm{o}\mathrm{p}}\otimes h_{\mu }$ the $\mathrm{\Theta }$-action has the term $c_{\mathrm{t}\mathrm{o}\mathrm{p}}\otimes h_{\mu }-\frac{1-q^2}{1+q^2}{v}_{\lambda }\otimes g_{\mu }$, whose coefficient is not in $A$ and lies on a distinct tensor basis vector (the simple-root terms vanish by the displayed relations and every higher first raising degree vanishes). Hence unrestricted arbitrary-based $O_{\mathrm{i}\mathrm{n}\mathrm{t}}$ action is false, and the ordered or constructed hypotheses of (a)–(d) are necessary.

Finite-type action does not imply supported globalization. In [F6] every algebraic normalized lift of the specified seed would have finite raw support and coefficients in $q\mathbb{Z}[q]$ away from its seed. Project such a bar-fixed lift to a cut $J$ greater than its maximal constituent index by two. The projection has the same normalization, and finite-cut uniqueness in [F6] identifies it with the cut lift. That lift has a nonzero coordinate at constituent index at least $J-1$, outside the putative finite support, a contradiction. Thus the unrestricted construction assertion fails even in rank one, although the evaluated quasi-$R$ action is finite and integral. For balancedness, any element $g$ of the natural integral zero/bar intersection has both $g$ and $\mathrm{\Psi }g$ with polynomial raw coefficients. The finite difference $w=\mathrm{\Psi }g-g$ is anti-invariant; at its least first weight strict triangularity gives $\bar{s}=-s$ for a polynomial coefficient $s$, forcing $s=0$ and therefore $w=0$. Consequently a balanced seed lift would be such a finite fixed lift and is impossible. The residue condition and the least-first-weight argument eliminate all polynomial corrections at or below the seed weight, so such a balanced lift has the triangular normalization just contradicted. Finite-dimensional based pairs remain fully covered: their entire raw tensor basis is a finite correction closure. More generally integral finite-column triangularity with finite raw-basis correction closure gives the same finite correction and inverse used in step 3.2. This leaves all supported constructions of step 4.1 intact.

Conclusion. Steps 2.1–2.2 prove the highest-first and lowest-second action with integral lattice preservation; step 2.3 proves arbitrary-multiplicity finite-type action finiteness; step 3.1 proves the finite-type algebra coefficients ${\mathrm{\Theta }}_{\nu }\in U_A^{+}[\nu ]\otimes U_A^{-}[-\nu ]$; steps 3.2–4.1 construct the corrected tensor global basis, prove its triangularity with diagonal $1$ and establish the bounded highest/lowest construction and closure classes in (d); step 5.1 records the exact affine failure of the unrestricted statements. step 5.2 refutes only the unrestricted supported-globalization inference from case (c). The only Choice use is [C1].