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The based quotient: projection of the lower global basis to a highest weight module

Statement

Let $λ \in P^{+}$ , let $π_{λ} : U_{q}^{-} \to V (λ)$ , $π_{λ} (P) = P v_{λ}$ , and let $G_{\infty} (b)$ , $G_{λ} (β)$ be the lower global bases of $U_{A}^{-}$ and $V_{A} (λ) = U_{A}^{-} v_{λ}$ of The lower global basis of $U_{q}^{-}$ and of a highest weight module. Then:

(i) $π_{λ} : U_{A}^{-} \to V_{A} (λ)$ is a based quotient over $A = Z [q, q^{- 1}]$ : $π_{λ} (G_{\infty} (b)) = 0$ if its crystal image $π_{λ} (b) \in B (λ) \cup {0}$ is $0$ , and $π_{λ} (G_{\infty} (b)) = G_{λ} (π_{λ} (b))$ otherwise; consequently the kernel of $π_{λ}$ on $U_{A}^{-}$ is the $A$ -span of exactly the killed lower global elements, the surviving elements map bijectively onto the integral highest-module basis $G_{λ} (B (λ))$ , the module weight components are finite free over $A$ , and the compatible zero/infinity chart and residue projections have the crystal compatibility of the completed grand loop, all for every symmetrizable datum with the explicit label and auxiliary-realization convention;

(ii) more precisely, the integral kernel is $\ker (π_{λ} ∣_{U_{A}^{-}}) = \sum_{i \in I} \sum_{s > λ (h_{i})} U_{A}^{-} F_{i}^{(s)} = \sum_{i \in I} (U_{A}^{-} \cap U_{q}^{-} F_{i}^{λ (h_{i}) + 1}),$ so the integral highest-module presentation uses all divided right powers beyond each highest limit, not just an ordinary power with a nonunit factorial.

Facts & Assumptions

Given: The balanced triples of The highest weight and negative half integral crystal triples are balanced with lower global bases and the projection rule $G_{\infty} (b) \mapsto G_{λ} (π_{λ} (b))$ or $0$ ; the string ideals $I_{(i, n)}^{A} = U_{A}^{-} \cap F_{i}^{n} U_{q}^{-}$ of Integrality of the string decomposition and the divided-power string ideals with their adapted global bases (The string-ideal triples are balanced: the induction step (Kashiwara 7.4.1 / JKK 10.4)); the reversal anti-involution $⋆$ (Reversal preserves the negative-half integral and crystal lattices); the projection compatibility and lattice statements of Projection from $L (\infty)$ to $L (λ)$ and its compatibility with the Kashiwara operators and The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets; and the field Serre kernel of The integrable quantum highest weight module exists and is unique.

[F1]

Based projection (The highest weight and negative half integral crystal triples are balanced): $π_{λ} (G_{\infty} (b)) = 0$ exactly when $π_{λ} (b) = 0$ , and $= G_{λ} (π_{λ} (b))$ otherwise; $G_{\infty} (B (\infty))$ is an $A$ -basis of $U_{A}^{-}$ and $G_{λ} (B (λ))$ is an $A$ -basis of $V_{A} (λ)$ ; the surviving residues $π_{λ} (b)$ biject onto $B (λ)$ .

[F2]

Saturated string ideals and reversal (Integrality of the string decomposition and the divided-power string ideals, The string-ideal triples are balanced: the induction step (Kashiwara 7.4.1 / JKK 10.4), Reversal preserves the negative-half integral and crystal lattices): the ideal $I_{(i, n)}^{A} = U_{A}^{-} \cap F_{i}^{n} U_{q}^{-}$ has the free adapted decomposition over the global basis; reversal preserves $U_{A}^{-}$ , commutes with $ψ$ , preserves the crystal lattice and maps $I_{(i, n)}^{A}$ onto the saturated right ideal $R_{(i, n)}^{A} : = U_{A}^{-} \cap U_{q}^{-} F_{i}^{n} = \sum_{s \geq n} U_{A}^{-} F_{i}^{(s)},$ adapted to a signed subset $\pm G_{\infty} (B_{(i, n)}^{R})$ of one global basis: each $⋆ G_{\infty} (b)$ is bar fixed with integral residue of norm $1$ , hence residue $\pm b^{'}$ , and balanced uniqueness gives $⋆ G_{\infty} (b) = \pm G_{\infty} (b^{'})$ .

[F3]

Projection compatibility (Projection from $L (\infty)$ to $L (λ)$ and its compatibility with the Kashiwara operators, The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets): $π_{λ} (L (\infty)_{- α}) = L (λ)_{λ - α}$ , the map on residues sends each $b \in B (\infty)$ to an element of $B (λ) \cup {0}$ and restricts to a bijection ${b \in B (\infty) : π_{λ} (b) \neq 0} \to B (λ)$ , and the bar involution commutes with $π_{λ}$ ; the grand loop supplies these statements for every $λ \in P^{+}$ .

[F4]

Field Serre kernel (The integrable quantum highest weight module exists and is unique, The dominant quantum Serre quotient is nonzero and integrable): on $U_{q}^{-}$ the kernel of $π_{λ}$ is $\sum_{i} U_{q}^{-} F_{i}^{λ (h_{i}) + 1}$ , the sum of the singular-vector ideals at each simple root.

[C1]

Choice: the Axiom of Choice is used only through the completed grand loop and the negative-half form suppliers of [F1] and [F3] (The Axiom of Choice); the ideal, reversal and intersection computations below are choice-free.

Proof

technique · direct

1.1F1F3

The based quotient. Since $G_{\infty} (B (\infty))$ is an $A$ -basis of $U_{A}^{-}$ and $π_{λ}$ is $A$ -linear, the kernel of $π_{λ}$ on $U_{A}^{-}$ is the $A$ -span of the basis elements mapped to $0$ , and the surviving basis elements map onto $V_{A} (λ)$ ; by [F1] the killed elements are exactly those with $π_{λ} (b) = 0$ , the surviving images are exactly the $A$ -basis $G_{λ} (B (λ))$ , and the correspondence is bijective on residues. The weight components of $V_{A} (λ)$ are finite free over $A$ because they have the finite basis $G_{λ} (B (λ)_{μ})$ , and the chart and residue projections are the compatible maps of [F3]. This proves (i).

1.2F2F4

The field kernel and its saturated ideals. By [F4] the field kernel of $π_{λ}$ on $U_{q}^{-}$ is $\sum_{i} U_{q}^{-} F_{i}^{s_{i}}$ with $s_{i} : = λ (h_{i}) + 1$ . By [F2] each $U_{q}^{-} F_{i}^{s_{i}}$ is the field span of the signed global family of the saturated right ideal $R_{(i, s_{i})}^{A} = U_{A}^{-} \cap U_{q}^{-} F_{i}^{s_{i}}$ , and the field span of a sum of subspaces is the span of the union of their spanning subsets of the single global basis. Intersecting with the free $A$ -lattice $U_{A}^{-}$ gives exactly the $A$ -span of that union, i.e. $\ker (π_{λ} ∣_{U_{A}^{-}}) = \sum_{i} R_{(i, s_{i})}^{A}$ .

2.1F2step 1.2

The divided-power form of the kernel. By [F2] again, $R_{(i, s)}^{A} = \sum_{t \geq s} U_{A}^{-} F_{i}^{(t)}$ for every $s$ , since every element of $U_{A}^{-} \cap U_{q}^{-} F_{i}^{s}$ is an integral combination of products $u F_{i}^{(t)}$ with $t \geq s$ and conversely each $U_{A}^{-} F_{i}^{(t)} \subseteq U_{A}^{-} \cap U_{q}^{-} F_{i}^{s}$ for $t \geq s$ . Substituting $s = s_{i} = λ (h_{i}) + 1$ into step 1.2 gives $\ker (π_{λ} ∣_{U_{A}^{-}}) = \sum_{i} \sum_{s > λ (h_{i})} U_{A}^{-} F_{i}^{(s)}$ , the asserted integral presentation by all divided right powers beyond each highest limit.

3.1step 1.1step 1.2step 2.1C1∎

Conclusion. Step 1.1 proves the based quotient, kernel-description, basis bijection, finiteness and chart compatibility of (i); steps 1.2 and 2.1 prove the exact integral kernel of (ii), using the reversal symmetry between left string ideals and saturated right ideals and avoiding any unproved commutation of integral intersections with sums. The Choice use is confined to [C1].

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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)