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Finitely generated free integral weight lattices and deformation of the classical weight character

Statement

For every dominant label weight $λ$ , let $V_{A} = U_{A} v_{λ} = U_{A}^{-} v_{λ}$ be the integral highest lattice inside the simple integrable module $L (λ)$ , over $A = Z [q, q^{- 1}]$ . Then each label-weight component of $V_{A}$ is finite free of rank $\dim_{Q (q)} L (λ)_{μ}$ , not merely torsionfree, and $V_{A}$ spans the field module. For every characteristic-zero field $k$ specialized at $q = 1$ , the fibre has the same weight dimensions and is the classical simple integrable highest weight module of the explicitly regularized realization, transported back to the original label lattice; hence its character is the Weyl–Kac character, the finite Weyl character in finite type. The integral quotient $U_{A}^{-} / \sum_{i} \sum_{s > λ (h_{i})} U_{A}^{-} F_{i}^{(s)}$ is $V_{A} (λ)$ ; its field extension is $L (λ)$ in every symmetrizable type.

Facts & Assumptions

Given: The auxiliary primitive-coroot regular datum and finite-support transport of Finite-support regularization of quantum label-weight modules; the exact highest-module existence and Serre presentation of The integrable quantum highest weight module exists and is unique and Serre presentation of a kac moody algebra; the local $q = 1$ comparison theorems; and the integral global bases of this page.

[F1]

Based quotient and freeness (The based quotient: projection of the lower global basis to a highest weight module, Bar invariance, congruence and uniqueness of the lower global basis, The lower global basis of $U_{q}^{-}$ and of a highest weight module): the projection $π_{λ} : U_{A}^{-} \to V_{A}$ is a based quotient, its surviving lower global basis elements form the $A$ -basis $G_{λ} (B (λ))$ of $V_{A}$ , and each finite weight component of $V_{A}$ is free over $A$ of rank its field dimension.

[F2]

Local $q = 1$ comparison and classical fibre (Local q=1 deformation proves integrable highest weight universality, Specialization at q=1 recovers the classical enveloping relations, The q=1 toral specialization identifies the classical Cartan generators, Quantum weight modules, weights in $P$ , and integrability): the local deformation proves the simple quantum highest weight module and its field character with discrete-valuation-ring-free lattices, and at characteristic-zero $q = 1$ the specialization identifies the fibre with the classical simple integrable highest weight module of the regularized realization.

[F3]

Integral and field quotients (The based quotient: projection of the lower global basis to a highest weight module, The integrable quantum highest weight module exists and is unique): the integral divided-right-power quotient is $V_{A} (λ)$ , and after field extension the singular-vector quotient is the simple integrable module $L (λ)$ .

[F4]

Auxiliary transport (Finite-support regularization of quantum label-weight modules): every fixed finite degree is transported into an explicit primitive-coroot regular datum with integral dominant labels and back, preserving ranks and characters, so the perfect-pairing, regularity and fundamental-weight hypotheses used in the source comparisons are available.

[F5]

Weyl–Kac character (Weyl Kac character formula): the classical simple integrable highest weight module in the regularized realization has the Weyl–Kac character, which in finite type is the finite Weyl character.

[C1]

Choice: declared through The Axiom of Choice and used only as inherited by the grand loop and the local deformation suppliers; the rank and base-change arguments are choice-free.

Proof

technique · direct

1.1F1

Free weight lattices. By [F1] the surviving lower global elements $G_{λ} (b)$ form an $A$ -basis of $V_{A}$ , so every label-weight component, being the span of the finitely many basis elements of that weight, is a finite free $A$ -module of rank equal to the number of surviving crystal classes of that weight; by the bijection of [F1] this number is $\dim_{Q (q)} L (λ)_{μ}$ . This proves freeness and spanning, not merely torsionfreeness.

1.2F3algebra

The integral quotient. By [F3] the kernel of $U_{A}^{-} \to V_{A} (λ)$ is exactly $\sum_{i} \sum_{s > λ (h_{i})} U_{A}^{-} F_{i}^{(s)}$ , so its quotient is the integral lattice $V_{A} (λ)$ . Extending to $Q (q)$ identifies it with $L (λ)$ by the field Serre presentation. The lattice and field module are not identified before that extension.

2.1F2F4step 1.1algebra

The $q = 1$ fibre. With $R = Q [q]_{(q - 1)}$ , the divided-word lattice satisfies $R \otimes_{A} V_{A} = V_{R}$ , since each divided-power factorial is an $R$ -unit. The local comparison [F2] identifies $V_{R} / (q - 1) V_{R}$ with the classical simple integrable highest module. A fibre over a characteristic-zero field $k$ is its scalar extension, and [F4] transfers its labels back to the original datum. Thus the fibre identification applies to this actual integral lattice, and step 1.1 preserves every finite weight rank.

3.1F5step 2.1

Character. By [F5] the classical module has the Weyl–Kac character, which equals the finite Weyl character in finite type; since the weight dimensions of the fibre agree with those of the classical module by step 2.1, the specialized character is exactly this Weyl–Kac character.

4.1step 1.1step 1.2step 2.1step 3.1F4C1∎

Conclusion. The global basis gives finite free integral weight pieces; the exact based kernel gives their integral quotient presentation, and the ordinary-word local lattice identifies their characteristic-zero fibre and its Weyl–Kac character. The auxiliary regularity and inherited Choice hypotheses are those already stated in [F4] and [C1].

Depends on

Used by

Integral quantum highest modules are the normalized q-Schur Weyl cells Lemma

Dependency tree · two levels

55 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

Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Mathematical Journal 63 (1991), 465–516 (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)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (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)