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Projection from $L (\infty)$ to $L (λ)$ and its compatibility with the Kashiwara operators

Statement

Let $λ \in P^{+}$ and let $L (λ), B (λ), L (\infty), B (\infty)$ be the lattices and bases of The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets, with $π_{λ} : U_{q}^{-} \to V (λ)$ , $P \mapsto P v_{λ}$ , and Kashiwara operators ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ . Then for every $α \in Q_{+}$ , every $i \in I$ and every $λ \in P^{+}$ :

(a) Lattice projection. $π_{λ} (L (\infty)_{- α}) = L (λ)_{λ - α}$ .

(b) Operator compatibility modulo $q$ . For every $P \in L (\infty)_{- α + α_{i}}$ , ${\tilde{f}}_{i} (P v_{λ}) \equiv ({\tilde{f}}_{i} P) v_{λ} m o d q L (λ)$ , and for every $b \in B (\infty)_{- α}$ with $π_{λ} (b) \neq 0$ , ${\tilde{e}}_{i} π_{λ} (b) = π_{λ} ({\tilde{e}}_{i} b)$ in $L (λ) / q L (λ)$ . Equivalently ${\tilde{f}}_{i} π_{λ} = π_{λ} {\tilde{f}}_{i}$ on $B (\infty)_{- α + α_{i}}$ , where the left-hand side is computed in $V (λ)$ .

(c) Basis projection. The map induced by $π_{λ}$ is a bijection ${b \in B (\infty)_{- α} : π_{λ} (b) \neq 0} \tilde{\to} B (λ)_{λ - α}$ .

(d) Large-weight form. If all $λ (h_{j})$ , $j \in I$ , are sufficiently large compared with $α$ , then $π_{λ}$ induces isomorphisms $L (\infty)_{- α} \tilde{\to} L (λ)_{λ - α}$ and $B (\infty)_{- α} \tilde{\to} B (λ)_{λ - α}$ , and the congruences of (b) for such $λ$ are the computation used inside the grand loop to deduce the general case; the general statements (a)-(c) hold for every $λ \in P^{+}$ , without a largeness hypothesis.

Choice. The axiom of choice is declared via The Axiom of Choice: clause (c) is clause N(r) of The grand loop: existence of the highest weight crystal bases, whose proof applies clause J(r), and J(r) is where the AC-dependent form comparison of The negative-half bilinear form is symmetric, nondegenerate and integral is used. Clauses (a), (b) and (d) are themselves choice-free.

Facts & Assumptions

Given: A dominant integral weight $λ$ , a simple root height $α \in Q_{+}$ , and the lattices, residue bases and operators of The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets.

[F1]

The grand loop proves, for every $r$ and hence for $r = ht (α)$ : clause L(r), $π_{λ} (L (\infty)_{- α}) = L (λ)_{λ - α}$ ; clause M(r), ${\tilde{f}}_{i} (P v_{λ}) \equiv ({\tilde{f}}_{i} P) v_{λ} m o d q L (λ)$ for $P \in L (\infty)_{- α + α_{i}}$ ; clause O(r), $π_{λ} ({\tilde{e}}_{i} b) = {\tilde{e}}_{i} π_{λ} (b)$ for surviving $b$ ; and clause N(r), the bijection on surviving residue elements; the proof of N(r) applies J(r), which uses the negative-half form comparison; the large-weight case gives a lattice isomorphism and a residue bijection in the degree (The grand loop: existence of the highest weight crystal bases, The negative-half bilinear form is symmetric, nondegenerate and integral, The Axiom of Choice).

[F2]

The projection $π_{λ}$ is $Q (q)$ -linear with $π_{λ} (1) = v_{λ}$ , and on the negative half ${\tilde{e}}_{i} {\tilde{f}}_{i} = i d$ ; on $V (λ)$ the operators are the string operators with $\ker {\tilde{e}}_{i} = \ker E_{i}$ and $\ker {\tilde{f}}_{i} = \ker F_{i}$ (The Kashiwara operators ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ of an integrable module, The $i$ -string decomposition of the negative half).

[F3]

$L (λ)$ and $L (\infty)$ are free $A_{0}$ -modules with the graded decompositions of The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets, and the residue bases $B (λ), B (\infty)$ are $Q$ -bases of the respective crystal limits (Crystal lattices, crystal bases and their crystals).

Proof

technique · direct. Each clause is the corresponding clause of the grand loop at height $r=\operatorname{ht}(\alpha)$

1.1F1

Clause (a). Since the grand loop holds for every $r$ , clause L(r) at $r = ht (α)$ is exactly $π_{λ} (L (\infty)_{- α}) = L (λ)_{λ - α}$ .

1.2F1F2

Clause (b). Clause M(r) is the first congruence, and clause O(r) is the second equality in the crystal limit; the equivalence with ${\tilde{f}}_{i} π_{λ} = π_{λ} {\tilde{f}}_{i}$ on the residue classes of $B (\infty)_{- α + α_{i}}$ follows because both sides of the congruence are computed modulo $q L (λ)$ and ${\tilde{f}}_{i}$ preserves both lattices by their defining lowering-word construction.

1.3F1F3

Clause (c). Clause N(r) states precisely that $π_{λ}$ induces a bijection from the surviving elements of $B (\infty)_{- α}$ onto $B (λ)_{λ - α}$ ; its proof applies J(r) and therefore the AC-dependent form comparison, while nothing in (a), (b) or (d) needs it.

1.4F1

Clause (d). For large weights the grand-loop large-weight computation gives that $π_{λ}$ is an isomorphism on the component and a bijection on its residue bases; those congruences are used as an input inside the loop to derive the general clauses, and the present clauses (a)-(c) do not assume the largeness.

2.1F1F3∎

Choice. All congruences in (a), (b) and (d) take place between finitely generated $A_{0}$ -modules and use only the operators and the string decomposition, so they are choice-free; clause (c) inherits the declared axiom of choice through J(r) and the negative-half form comparison.

Depends on

Used by

Dependency tree · two levels

35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516 (standard reference, not scraped)

Projection from L(∞)L(∞) to L(λ)L(λ) and its compatibility with the Kashiwara operators