The bar lattice and the divided-power integral lattice of a based module

Fix $A=\mathbb{Z}[q,q^{-1}]\subseteq \mathbb{Q}(q)$ and the crystal limiting ring $A_0\subseteq \mathbb{Q}(q)$ of The local ring $A_0$ of rational functions regular at $q=0$, and put $A_{\mathrm{\infty }}:=\psi (A_0)$, the image of $A_0$ under the semilinear involution $\psi (q)=q^{-1}$, i.e. the ring of rational functions regular at $q=\mathrm{\infty }$; $A_0$ and $A_{\mathrm{\infty }}$ are the rational local rings at $0$ and at infinity, and the integral affine chart coordinate rings are $\mathbb{Z}[q]$ and $\mathbb{Z}[q^{-1}]$.

Let $M$ be one of $U_q^{-}$ or $V(\lambda )=L(\lambda )$ for $\lambda \in P^{+}$. Equip $M$ with the bar involution $\psi$ of The bar involution and the contravariant anti-involution preserve $U_A$: on $U_q^{-}$ the semilinear algebra involution with $\psi (q)=q^{-1}$, $\psi (E_i)=E_i$, $\psi (F_i)=F_i$, $\psi (K_h)=K_h^{-1}$; on $V(\lambda )$ the induced semilinear involution determined by $P{v}_{\lambda }\mapsto \psi (P)v_{\lambda }$, well defined by the highest-module presentation: its negative-half kernel is ${\sum }_i{U}_q^{-}{F}_i^{\lambda (h_i)+1}$, preserved by $\psi$ because every $F_i$ is fixed (The integrable quantum highest weight module exists and is unique). Equivalently the full highest-vector relations are bar stable: $E_i$ is fixed, the singular divided powers are fixed, and $\psi (K_h-q^{⟨\lambda ,h⟩})=-q^{-⟨\lambda ,h⟩}{K}_{-h}(K_h-q^{⟨\lambda ,h⟩})$. Hence the quotient involution is compatible with the algebra action and fixes $v_{\lambda }$. Let $M_A\subseteq M$ be the divided-power integral form: $M_A=U_A^{-}$ for $M=U_q^{-}$, and $M_A=V_A(\lambda ):=U_A^{-}{v}_{\lambda }$ for $M=V(\lambda )$ (The Lusztig divided-power integral form $U_A$); both are $\psi$-stable. Let $L$ be the Kashiwara crystal lattice of $M$ over $A_0$: $L=\mathcal{L}(\mathrm{\infty })$ for $U_q^{-}$ and $L=\mathcal{L}(\lambda )$ for $V(\lambda )$ (The Kashiwara lattices $\mathcal{L}(\lambda )$ and $\mathcal{L}(\mathrm{\infty })$ and their generating sets, The grand loop: existence of the highest weight crystal bases).

Define the bar lattice $L^{-}:=\psi (L)$, an $A_{\mathrm{\infty }}$-submodule of $M$, and the balanced part $E:=M_A\cap L\cap L^{-}$.

The definition keeps three structures on the same $\mathbb{Q}(q)$-vector space strictly separate: the $A$-module $M_A$, which is not assumed free and not claimed to be contained in $L$; the free $A_0$-module $L$; and the free $A_{\mathrm{\infty }}$-module $L^{-}$. No equality $L^{-}=L$, no inclusion $M_A\subseteq L$, no freeness of $M_A$ and no identification $E=M_A$ are asserted; freeness is a conclusion of the balanced-triple construction on this page, whereas $E$ is a $\mathbb{Z}$-module of global lifts and is not equal to the Laurent lattice $M_A$. The bar lattice is not the form-dual (upper) lattice, which is defined later on this page. The intersections $M_A\cap L$ and $M_A\cap L^{-}$ are respectively $\mathbb{Z}[q]$- and $\mathbb{Z}[q^{-1}]$-modules, generally not $A$-modules. The balanced part $E$ is a $\mathbb{Z}$-module; its Laurent extension $A{\otimes }_{\mathbb{Z}}E$ is the full integral module once globalization is proved.

More generally a based integrable module $(M,B)$ here means exactly: $B$ is a homogeneous field basis; $M_A={⨁}_{b}Ab$ is $U_A$-stable; the semilinear involution fixing $B$ is compatible with the algebra action; and $L_0={⨁}_b{A}_{0}b$ with residue $B$ is a crystal local basis at $0$. These are the four conditions of Lusztig 27.1.2(a)–(d) after the transport $v=q^{-1}$, together with the integrable extension used by Bao–Wang. Such data do not automatically imply an integral $\mathrm{\Theta }$-action on an arbitrary based pair. A based subobject is a $U$-submodule spanned by a subset of $B$; a based quotient is its quotient with the complementary basis images. Based morphisms send $B$ to the target basis or to $0$, and have kernels spanned by the killed basis subset. These exact integral and bar-compatible maps, not arbitrary field summands, are used in tensor inheritance.