The lower global basis of $U_q^{-}$ and of a highest weight module

Let $M_A$ be $U_A^{-}$ or $V_A(\lambda )=U_A^{-}{v}_{\lambda }$, let $L$ be $\mathcal{L}(\mathrm{\infty })$ or $\mathcal{L}(\lambda )$, let $L^{-}:=\psi (L)$ be the bar lattice, all as in The bar lattice and the divided-power integral lattice of a based module and The integral residue lattices $L_A(\mathrm{\infty })$ and $L_A(\lambda )$, and let $E_{\mathbb{Z}}:=M_A\cap L\cap L^{-}$ be the balanced intersection. Let $\mathcal{B}$ be $\mathcal{B}(\mathrm{\infty })$ or $\mathcal{B}(\lambda )$.

By The highest weight and negative half integral crystal triples are balanced the integral triple $(M_A,L,L^{-})$ is balanced componentwise: for every root or label degree the canonical map $E_{\mathbb{Z}}\to (M_A\cap L)\mathrm{/}(M_A\cap qL)$ is an isomorphism onto the free $\mathbb{Z}$-span ${⨁}_{b\in \mathcal{B}}\mathbb{Z}b$ of the crystal classes of that degree. For $b\in \mathcal{B}$ define the lower global lift $G(b)$ to be the unique element of $E_{\mathbb{Z}}$ whose residue class in $(M_A\cap L)\mathrm{/}(M_A\cap qL)$ is $b$. The family $G(\mathcal{B})$ is the lower global basis of $M_A$ (of $U_A^{-}$ or of $V_A(\lambda )$).

With these definitions, the theorem's extensions give the basis assertions: the $A$-span of $G(\mathcal{B})$ is $M_A$, its $A_0$-span is $L$, its $A_{\mathrm{\infty }}$-span is $L^{-}$, and its $\mathbb{Z}$-span is $E_{\mathbb{Z}}$, while after tensoring with $\mathbb{Q}$ the residue classes of the $G(b)$ are the full rational crystal spaces; in particular $G(\mathcal{B})$ is an $A$-basis of $M_A$. Each $G(b)$ is bar fixed, $G(0)$ is not defined, and no equality of $E_{\mathbb{Z}}$ with the whole Laurent module $M_A$ is implied; the identification of the integral residue subgroup with $\mathbb{Z}\mathcal{B}(\lambda )$ for highest modules is the corresponding part of the same theorem. The lifts are characterized by the simultaneous balanced intersection, not by an unproved unitriangular word-basis condition, and all definitions apply componentwise on finite root or label components and are extended by their direct sums.