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Bar invariance, congruence and uniqueness of the lower global basis

Statement

For the divided-power integral form $U_{A}^{-}$ of The Lusztig divided-power integral form $U_{A}$ and for every dominant $λ$ , let $M_{A}$ be $U_{A}^{-}$ or $V_{A} (λ) = U_{A}^{-} v_{λ}$ , let $L$ be $L (\infty)$ or $L (λ)$ with bar lattice $L^{-} = ψ (L)$ , and let $B$ be $B (\infty)$ or $B (λ)$ . Then there is a unique lower global family $G (B) \subseteq M_{A} \cap L \cap L^{-}$ with the property that the residue class of $G (b)$ in $(M_{A} \cap L) / (M_{A} \cap q L)$ is $b$ for every $b \in B$ , namely the family of The lower global basis of $U_{q}^{-}$ and of a highest weight module. It is bar invariant, a genuine $A = Z [q, q^{- 1}]$ basis of $M_{A}$ , and a basis of both rational chart lattices $L$ and $L^{-}$ over $A_{0}$ and $A_{\infty}$ ; its $Z$ -span is the simultaneous balanced intersection $M_{A} \cap L \cap L^{-}$ and its field span is the whole module. It is adapted to every divided string-image ideal in the exact sense of the integral balanced theorem, and the same family is obtained with residue $b$ over either chart. No word family is assumed to be a basis, no unitriangular characterization is used, and no lower lattice self-duality is asserted.

Facts & Assumptions

Given: The balanced triple $(M_{A}, L, L^{-})$ over $A_{0}, A_{\infty}$ of Balanced triples over the two charts of $P^{1}$ and the globalization map, the string-image ideals $I_{(i, n)}^{A}, J_{(i, n)}^{A}$ of Integrality of the string decomposition and the divided-power string ideals, the lower global definition of The lower global basis of $U_{q}^{-}$ and of a highest weight module, and the bar involution $ψ$ with its stability of $U_{A}^{-}$ and the divided-power form (The bar involution and the contravariant anti-involution preserve $U_{A}$ ).

[F1]

Integral balanced theorem (The highest weight and negative half integral crystal triples are balanced): the triples $(U_{A}^{-}, L (\infty), ψ (L (\infty)))$ and $(V_{A} (λ), L (λ), ψ (L (λ)))$ are integrally balanced componentwise; the balanced $Z$ -intersection maps isomorphically onto $Z B$ , after rational extension onto the full rational crystal space; there are bar-invariant global lifts $G (b)$ forming an $A$ -basis of $M_{A}$ , spanning $L$ over $A_{0}$ and $L^{-}$ over $A_{\infty}$ , adapted to every string-image ideal $I_{(i, n)}^{A}$ and $J_{(i, n)}^{A}$ with the displayed free decompositions.

[F2]

Balancedness and its kernel (Balanced triples over the two charts of $P^{1}$ and the globalization map): the canonical map $E_{Z} = M_{A} \cap L \cap L^{-} \to (M_{A} \cap L) / (M_{A} \cap q L)$ is an isomorphism exactly when the triple is balanced, so in the balanced case an element of $E_{Z}$ lies in $M_{A} \cap q L$ only if it is $0$ .

[F3]

Bar transport (The bar involution and the contravariant anti-involution preserve $U_{A}$ , The bar lattice and the divided-power integral lattice of a based module): $ψ$ preserves $U_{A}^{-}$ , commutes with the projection $π_{λ}$ , and exchanges $L$ and $L^{-}$ , so the balanced intersection $E_{Z}$ is stable under $ψ$ and $ψ$ transports the zero-chart residue map to the infinity-chart residue map; the bar-fixed global lifts and their corresponding residue labels are supplied by [F1], not by an action of $q \mapsto q^{- 1}$ on the zero-chart quotient.

Proof

technique · direct

1.1F1F3

Existence. Apply [F1] degree by degree, and note that the residue target is the free $Z$ -span of $B$ in each finite root or label component. The lift $G (b)$ of residue $b$ exists in $E_{Z}$ , is bar invariant by [F1], and the direct sums over degrees give a family $G (B)$ with the asserted spanning properties over $A$ , $A_{0}$ and $A_{\infty}$ ; adaptedness to the string ideals is the corresponding clause of [F1].

2.1F2step 1.1

Uniqueness. Let $g, h \in M_{A} \cap L \cap L^{-}$ have the same residue class in $(M_{A} \cap L) / (M_{A} \cap q L)$ . Then $g - h \in E_{Z} \cap q L$ lies in the kernel of the canonical map of [F2], which is injective in the balanced case; hence $g = h$ . Applied to $g = G (b)$ and an arbitrary lift of $b$ , this is exactly the uniqueness used by later consumers, and it identifies the family defined over either chart.

3.1step 1.1step 2.1∎

Conclusion. Steps 1.1 and 2.1 give existence, the residue normalization, bar invariance, the two chart-basis assertions, the balanced-intersection and field-span descriptions and uniqueness; no word family, unitriangular condition or lower self-duality enters, since the construction is the inverse of the integral globalization map of [F2].

Depends on

Used by

Dependency tree · two levels

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Sources

Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Mathematical Journal 63 (1991), 465–516 (standard reference, not scraped)
Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (standard reference, not scraped)