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The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets

Definition

Let $λ \in P^{+}$ and let $L (λ)$ be the simple integrable highest weight module with highest weight vector $v_{λ}$ (The integrable quantum highest weight module exists and is unique); in this page we write $V (λ) : = L (λ)$ , following the notation of the crystal sources. Let $U_{q}^{-}$ be the negative half with its Kashiwara operators ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ (The $i$ -string decomposition of the negative half), and let $A_{0}$ be the crystal limiting ring (The local ring $A_{0}$ of rational functions regular at $q = 0$ ).

Define $L (λ)$ to be the $A_{0}$ -submodule of $V (λ)$ generated by all vectors ${\tilde{f}}_{i_{1}} \dots {\tilde{f}}_{i_{r}} v_{λ}$ ( $r \geq 0$ , $i_{1}, \dots, i_{r} \in I$ ), where for $r = 0$ the vector is $v_{λ}$ ; and $L (\infty)$ to be the $A_{0}$ -submodule of $U_{q}^{-}$ generated by all ${\tilde{f}}_{i_{1}} \dots {\tilde{f}}_{i_{r}} 1$ ( $r \geq 0$ ), where $1$ is the unit of $U_{q}^{-}$ . Here ${\tilde{f}}_{i}$ on $V (λ)$ are the module Kashiwara operators and ${\tilde{f}}_{i}$ on $U_{q}^{-}$ are the negative-half operators. Define $B (λ) : = {{\tilde{f}}_{i_{1}} \dots {\tilde{f}}_{i_{r}} v_{λ} + q L (λ) : r \geq 0} ∖ {0} \subseteq L (λ) / q L (λ),$ $B (\infty) : = {{\tilde{f}}_{i_{1}} \dots {\tilde{f}}_{i_{r}} 1 + q L (\infty) : r \geq 0} ∖ {0} \subseteq L (\infty) / q L (\infty) .$

Then, with $π_{λ} : U_{q}^{-} \to V (λ)$ , $π_{λ} (P) = P v_{λ}$ :

(a) $L (λ) = ⨁_{μ \in P} L (λ)_{μ}$ with $L (λ)_{μ} = L (λ) \cap V (λ)_{μ}$ , and $L (\infty) = ⨁_{ν \in Q_{+}} L (\infty)_{- ν}$ with $L (\infty)_{- ν} = L (\infty) \cap U_{q}^{-} [- ν]$ ; correspondingly $B (λ) = ⨆_{μ} B (λ)_{μ}$ and $B (\infty) = ⨆_{ν} B (\infty)_{- ν}$ .

(b) ${\tilde{f}}_{i} L (λ) \subseteq L (λ)$ and ${\tilde{f}}_{i} L (\infty) \subseteq L (\infty)$ .

(c) (Recursive spanning.) For $α \in Q_{+} ∖ {0}$ , $L (λ)_{λ - α} = \sum_{i \in I} {\tilde{f}}_{i} L (λ)_{λ - α + α_{i}}, B (λ)_{λ - α} = (⋃_{i \in I} {\tilde{f}}_{i} B (λ)_{λ - α + α_{i}}) ∖ {0},$ and the same formulas with $L (\infty), B (\infty)$ . Each $L (λ)_{λ - α}$ and each $L (\infty)_{- α}$ is generated over $A_{0}$ by finitely many monomials of total length at most $ht (α)$ , hence is a finitely generated $A_{0}$ -module.

No basis property, scalar-extension spanning or crystal-lattice property of $L (λ)$ or $L (\infty)$ is asserted here; these are conclusions of the grand loop. The script letters avoid the collision with the simple module $L (λ)$ . No choice principle is used. Each fixed component is torsionfree inside a field vector space and generated by finitely many words of its fixed height, so it is free over the DVR $A_{0}$ ; its full field-span is proved later. The field map $π_{λ}$ is surjective, but exact equality on Kashiwara monomial generators is not asserted.

Facts & Assumptions

Given: $λ \in P^{+}$ , the simple integrable highest weight module $V (λ) = L (λ)$ with highest vector $v_{λ}$ , the negative half $U_{q}^{-}$ , the $A_{0}$ -spans $L (λ), L (\infty)$ generated by the $\tilde{f}$ -words, and the residue sets $B (λ), B (\infty)$ .

[F1]

${\tilde{f}}_{i}$ on $V (λ)$ and on $U_{q}^{-}$ is $Q (q)$ -linear, preserves the lattice $L (λ)$ , respectively $L (\infty)$ , once (b) is known, and lowers the weight, respectively the root degree, by $α_{i}$ ; on $U_{q}^{-}$ it is injective with ${\tilde{e}}_{i} {\tilde{f}}_{i} = i d$ (The Kashiwara operators ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ of an integrable module, The $i$ -string decomposition of the negative half).

[F2]

$V (λ) = ⨁_{μ \in P} V (λ)_{μ}$ and $U_{q}^{-} = ⨁_{ν \in Q_{+}} U_{q}^{-} [- ν]$ are direct sums of weight, respectively root, spaces (Quantum weight modules, weights in $P$ , and integrability, The $i$ -string decomposition of the negative half).

[F3]

Every word ${\tilde{f}}_{i_{1}} \dots {\tilde{f}}_{i_{r}} v_{λ}$ is homogeneous of weight $λ - α_{i_{1}} - \dots - α_{i_{r}}$ , and every word ${\tilde{f}}_{i_{1}} \dots {\tilde{f}}_{i_{r}} 1$ is homogeneous of root degree $- α_{i_{1}} - \dots - α_{i_{r}}$ (Kac moody integral and dominant integral weights, [F1], [F2]).

[F4]

A finitely generated torsionfree module over the PID $A_{0}$ is free (Every finitely generated torsion-free module over a PID is free, The local ring $A_{0}$ of rational functions regular at $q = 0$ ).

[F5]

$V (λ)$ is universal among integrable cyclic highest weight modules of highest weight $λ$ : every such module receives a unique map of $U_{q} (g)$ -modules sending its highest vector to a prescribed highest vector, and $π_{λ} : U_{q}^{-} \to V (λ)$ , $P \mapsto P v_{λ}$ , is a surjective $Q (q)$ -linear map sending $1$ to $v_{λ}$ (The integrable quantum highest weight module exists and is unique).

Verification

technique · direct. Homogeneity gives the graded decompositions, prepending a color gives the recursive spanning and stability, and finite generation plus torsionfreeness gives freeness of the components

1.1F2F3given

Grading (a). Each generator ${\tilde{f}}_{i_{1}} \dots {\tilde{f}}_{i_{r}} v_{λ}$ is homogeneous of weight $λ - α_{i_{1}} - \dots - α_{i_{r}}$ , and each generator of $L (\infty)$ is homogeneous of root degree $- α_{i_{1}} - \dots - α_{i_{r}}$ , by [F3]. Taking $A_{0}$ -linear combinations and using [F2], every element of $L (λ)$ lies in $⨁_{μ} V (λ)_{μ}$ with only finitely many nonzero components, and its weight components lie in $L (λ)$ ; hence $L (λ) = ⨁_{μ} L (λ)_{μ}$ with $L (λ)_{μ} = L (λ) \cap V (λ)_{μ}$ , and similarly $L (\infty) = ⨁_{ν} L (\infty)_{- ν}$ . The residue sets decompose accordingly by their definition.

1.2F1given

Stability (b). A generator is carried by ${\tilde{f}}_{i}$ to the generator with the color $i$ prepended; since ${\tilde{f}}_{i}$ is $Q (q)$ -linear, it commutes with $A_{0}$ -coefficients, so ${\tilde{f}}_{i} L (λ) \subseteq L (λ)$ and ${\tilde{f}}_{i} L (\infty) \subseteq L (\infty)$ .

2.1F1step 1.1step 1.2given

Recursive spanning (c). Let $α \in Q_{+} ∖ {0}$ and let $w$ be a word of total degree $α$ ; writing $w = {\tilde{f}}_{i_{1}} w^{'}$ exhibits the class of $w$ as ${\tilde{f}}_{i_{1}}$ of the class of the shorter word $w^{'}$ , which is a generator of degree $α - α_{i_{1}}$ . This gives $\subseteq$ ; the reverse inclusion holds because ${\tilde{f}}_{i}$ preserves $L$ by step 1.2 and lowers the degree by $α_{i}$ . For the residue sets, a word class that is nonzero remains nonzero when shortened (if $w^{'} \in q L$ then $w = {\tilde{f}}_{i_{1}} w^{'} \in q L$ ), and the displayed union minus zero therefore equals $B (λ)_{λ - α}$ ; the same argument applies to $B (\infty)$ .

3.1F4step 2.1algebra

Finite generation and freeness. Iterating (c) from degree $α$ down to degree $0$ expresses every $L (λ)_{λ - α}$ as the sum of the images of $A_{0} v_{λ}$ under the words of length exactly $ht (α)$ ; there are finitely many such words because $I$ is finite, and for $α = 0$ the component is $A_{0} v_{λ}$ . Hence each component is finitely generated over $A_{0}$ by monomials of length at most $ht (α)$ . It is torsionfree because it is contained in the $Q (q)$ -vector space $V (λ)$ , respectively $U_{q}^{-}$ , so [F4] makes it a free $A_{0}$ -module.

4.1F5step 3.1

Choice and the field map. The words are indexed by finite sequences in the finite set $I$ , so no choice is needed to name the generators. The map $π_{λ}$ is surjective by [F5], because $U_{q}^{-}$ generates $V (λ)$ over $U_{q} (g)$ from $v_{λ}$ ; no equality between $π_{λ}$ and the lattice projections, and no spanning over $A_{0}$ of the field-spanning property, is asserted here.

5.1givenstep 3.1∎

No crystal-lattice claim yet. The statements proved above are the graded decompositions, stability, recursive spanning, finite generation and freeness of the components; whether $B (λ)$ and $B (\infty)$ are crystal bases, and whether $L (λ) \otimes_{A_{0}} Q (q) \to V (λ)$ is an isomorphism, is not asserted and is deferred to the grand loop.

Depends on

Used by

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Sources

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)
Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (book-length lecture notes, last updated 18 January 2024) (standard reference, not scraped)

The Kashiwara lattices L(λ)L(λ) and L(∞)L(∞) and their generating sets