Finite-support regularization of quantum label-weight modules

Choosing a free basis of $P$, let $S$ be the finite set of basis elements occurring in the finitely many elements ${\lambda }_j,{\alpha }_i$, and put $X_f:={⨁}_{s\in S}\mathbb{Z}{e}_s$. Then $X_f$ is free of finite rank, contains all ${\lambda }_j$ and ${\alpha }_i$, and $P=X_f\oplus X_f^{\mathrm{\prime }}$ for the summand $X_f^{\mathrm{\prime }}$ spanned by the remaining basis elements, by [F1]. Consequently $X_f$ contains the whole support of $M$: every weight is some ${\lambda }_j-\nu$ with $\nu \in {⨁}_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$ by [F4], and both ${\lambda }_j$ and the ${\alpha }_i$ lie in $X_f$.

By [F2] the evaluation pairing $X_{\mathrm{a}\mathrm{u}\mathrm{x}}\times Y_{\mathrm{a}\mathrm{u}\mathrm{x}}$ is perfect, and the map $j$ is an isomorphism of $X_f$ onto the subgroup $\{j(x)\}$ of $X_{\mathrm{a}\mathrm{u}\mathrm{x}}$; hence $j$ is injective, so the ${\alpha }_i^{\mathrm{a}\mathrm{u}\mathrm{x}}=j({\alpha }_i)$ are linearly independent (as the ${\alpha }_i$ are, Symmetrizable Cartan data for quantum groups). The $H_i$ are basis elements of the first summand, hence independent and primitive. For all $i,j$, $⟨{\alpha }_j^{\mathrm{a}\mathrm{u}\mathrm{x}},H_i⟩=j({\alpha }_j)(H_i)=⟨{\alpha }_j,h_i⟩=a_{ij}$ by the row convention of the datum, and $j({\alpha }_j)(f)=f({\alpha }_j)$ on the second summand; hence the generalized Cartan matrix of the auxiliary datum is $A$ and the symmetrizer $D$ still satisfies $d_i{a}_{ij}=d_j{a}_{ji}$. The elements ${\mathrm{\Lambda }}_j\in X_{\mathrm{a}\mathrm{u}\mathrm{x}}$ with ${\mathrm{\Lambda }}_j(H_i)={\delta }_{ij}$ and ${\mathrm{\Lambda }}_j{\mathrm{\mid }}_{\mathrm{Hom}(X_f,\mathbb{Z})}=0$ are well-defined fundamental weights. Thus (i) holds.

Transport of the module: grade $M^{\mathrm{a}\mathrm{u}\mathrm{x}}$ by $X_{\mathrm{a}\mathrm{u}\mathrm{x}}$ as in (ii) and let $K_{(n,f)}$ act on $M_x$ by the scalar $q^{{\sum }_i{n}_i⟨x,h_i⟩+f(x)}$, while $E_i,F_i$ act as before. This is well defined because $P=X_f\oplus X_f^{\mathrm{\prime }}$ and $M$ has no labels outside $X_f$ by step 1.1, so $M={⨁}_{x\in X_f}{M}_x$. The toral relations hold because the scalars multiply ($q^a{q}^b=q^{a+b}$ and $K_{(0,0)}$ acts by $1$). For the toral-action relations, $K_{(n,f)}{E}_j{K}_{(n,f)}^{-1}$ acts on $M_x$ by $q^{{\sum }_i{n}_i⟨x+{\alpha }_j,h_i⟩+f(x+{\alpha }_j)-{\sum }_i{n}_i⟨x,h_i⟩-f(x)}=q^{{\sum }_i{n}_i⟨{\alpha }_j,h_i⟩+f({\alpha }_j)}=q^{⟨{\alpha }_j^{\mathrm{a}\mathrm{u}\mathrm{x}},(n,f)⟩}$, which is the required scalar; the $F_j$-variant has the opposite sign. The mixed relation holds because its toral part involves only $K_{d_i{H}_i}$, which acts on $M_x$ by $q^{d_i⟨x,h_i⟩}=q_i^{⟨x,h_i⟩}$, the same scalar as the original $K_i=K_{d_i{h}_i}$; and both $E_i,F_i$ coincide with their original actions. The Serre relations hold because they involve only the unchanged $E_i,F_i$ and the unchanged Cartan matrix and $d_i$. Hence $M^{\mathrm{a}\mathrm{u}\mathrm{x}}$ is a type-one module for the auxiliary datum with support in $j(X_f)$. For uniqueness on the prescribed grading, the type-one condition forces every $K_y$ to act on $M_{j(x)}^{\mathrm{a}\mathrm{u}\mathrm{x}}=M_x$ by $q^{j(x)(y)}$; these summands span $M$, and the $E_i,F_i$ actions are fixed. Thus every generator action is determined, proving (ii).

Converse and equivalence: given an $X_{\mathrm{a}\mathrm{u}\mathrm{x}}$-graded type-one module $N$ with labels in $j(X_f)$ (i.e. $N_{\xi }=0$ for $\xi \mathrm{\notin }j(X_f)$), define $N^{\mathrm{o}\mathrm{l}\mathrm{d}}$ with the same underlying vector spaces and $N_x^{\mathrm{o}\mathrm{l}\mathrm{d}}:=N_{j(x)}$ for $x\in X_f$; the actions of $E_i,F_i$ are kept, and $K_h$ acts on $N_x^{\mathrm{o}\mathrm{l}\mathrm{d}}$ by $q^{⟨j(x),h⟩}$ where $⟨j(x),h⟩:=⟨x,h⟩$ — well defined since $j$ is injective by step 2.1. These two assignments are inverse on objects; on morphisms both are the identity map on underlying $\mathbb{Q}(q)$-linear maps, and a map preserving one grading preserves the other because $j$ is a bijection $X_f\to j(X_f)$ fixing the $E_i,F_i$ actions. This proves (iii). The construction is not an algebra embedding of tori: $\mathrm{Hom}(X_f,\mathbb{Z})$ may be nonzero, and its elements act as new labels with no preimage in $P$; the equivalence is only between the stated module categories with labels in $j(X_f)$.

Simultaneity: for a Verma module or highest weight module of highest weight $\lambda$ the support lies in $\lambda -Q_{+}$, and for a finite tensor product of such modules the labels are sums of labels, hence lie in ${\sum }_k({\lambda }^{(k)}-Q_{+})\subseteq ({\sum }_k{\lambda }^{(k)})-Q_{+}$; therefore one finite set $\{{\lambda }_j\}$ (the participating upper weights) and one $X_f$ containing them and all ${\alpha }_i$ cover all modules under consideration. Under the coproduct the $E_i$- and $F_i$-components are unchanged ($\mathrm{\Delta }(E_i)=E_i\otimes K_i^{-1}+1\otimes E_i$ and $\mathrm{\Delta }(F_i)=F_i\otimes 1+K_i\otimes F_i$ of The Drinfeld–Jimbo formulas define a Hopf algebra), and the toral factors act by the auxiliary characters on both tensor factors, so the auxiliary structure on a tensor product is the tensor product of the auxiliary structures; the collection of such constructions is therefore compatible. This proves (iv).

Rational decomposition: let $h_{\mathrm{a}\mathrm{u}\mathrm{x}}=Y_{\mathrm{a}\mathrm{u}\mathrm{x}}\otimes \mathbb{Q}$, so $\dim h_{\mathrm{a}\mathrm{u}\mathrm{x}}=n+\mathrm{rank}{X}_f$, and let $\alpha \text{\hspace{0.17em}⁣}:h_{\mathrm{a}\mathrm{u}\mathrm{x}}\to {\mathbb{Q}}^I$ be the evaluation $\alpha (y{)}_i=⟨{\alpha }_i^{\mathrm{a}\mathrm{u}\mathrm{x}},y⟩$. It is surjective because the ${\alpha }_i^{\mathrm{a}\mathrm{u}\mathrm{x}}$ are linearly independent and the pairing is perfect; its restriction to ${\mathrm{span}}_{\mathbb{Q}}\{H_i\}$ has rank $r=\mathrm{rank}A$ (the matrix $(⟨{\alpha }_j,H_i⟩)=(a_{ij})$). Choose $n-r$ rational directions in $h_{\mathrm{a}\mathrm{u}\mathrm{x}}$ whose images under $\alpha$ supplement the image of $\mathrm{span}\{H_i\}$, and let $H_{\min }$ be the span of these directions together with the $n$ independent $H_i$; then $\dim H_{\min }=n+(n-r)=2n-r$ and the coroots $H_i$ together with the chosen directions form a basis of $H_{\min }$ on which $\alpha$ has rank $n$, so after extension to $\mathbb{C}$, $H_{\min }\otimes \mathbb{C}$ is a minimal realization of $A$ in the sense of Realization of a generalized cartan matrix and Minimal realizations exist and are unique up to isomorphism. Inside $\ker \alpha$ choose a linear complement $Z$ to $H_{\min }\cap \ker \alpha$; then $h_{\mathrm{a}\mathrm{u}\mathrm{x}}=H_{\min }\oplus Z$ by construction, and $\alpha {\mathrm{\mid }}_Z=0$, so all simple roots vanish on $Z$ and the corresponding generators are central. This proves the rational decomposition.

Classical comparison. By [F5], adjoining $Z\otimes \mathbb{C}$ to the minimal Cartan introduces only central generators: all simple roots vanish there and all coroots belong to $H_{\min }$. Mapping these generators to the two summands of $\mathfrak{g}(H_{\min }\otimes \mathbb{C})\oplus (Z\otimes \mathbb{C})$, and mapping back the minimal generators and $Z$, gives mutually inverse Lie homomorphisms, since every Cartan and Serre relation is respected. In a cyclic highest weight module with highest vector $v_{\xi }$, centrality gives $z(a{v}_{\xi })=a(z{v}_{\xi })=\xi (z)a{v}_{\xi }$ for every $z\in Z$, so a subspace is a submodule for the extended algebra exactly when it is one for the minimal algebra. Moreover the restrictions ${\alpha }_i^{\mathrm{a}\mathrm{u}\mathrm{x}}{\mathrm{\mid }}_{H_{\min }}$ are independent by step 6.1. Thus different weights $\xi -\nu$ in this one highest-weight cone remain different on $H_{\min }$, while their values on $Z$ are all $\xi {\mathrm{\mid }}_Z$; their weight spaces, and hence multiplicities, are unchanged. In particular restriction identifies the Verma module and its simple highest weight quotient with their minimal-realization counterparts, preserving integrability. This proves (v); the minimal-realization proposition supplies only realization isomorphisms, and the multiplicity comparison is the argument here.