PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Specialization of the global bases and the root-of-unity boundary

Statement

For every dominant $λ$ and every symmetrizable datum, the integral highest lattice $V_{A} (λ) = U_{A}^{-} v_{λ}$ has its genuine lower global $A$ -basis $G_{λ} (B (λ))$ of The lower global basis of $U_{q}^{-}$ and of a highest weight module. Hence for any unital base change $A \to S$ of $A = Z [q, q^{- 1}]$ , including nonflat specializations at a root of unity with $ζ$ invertible, the tensor fibre $V_{A} (λ) \otimes_{A} S$ has the corresponding $S$ -basis $G_{λ} (b) \otimes 1$ , indexed by the same crystal elements, and for $S \neq 0$ has identical finite weight ranks.

At a characteristic-zero field specialization $q = 1$ , the positive toral quotient and the local $q = 1$ comparison identify the fibre with the classical simple integrable highest weight module and its Weyl–Kac character. The zero and infinity chart lattices are obtained separately from the $Z [q]$ - and $Z [q^{- 1}]$ -spans of the global vectors followed by rational discrete-valuation-ring extension; there is no map $A \to A_{0}$ sending the unit $q$ to $0$ . At roots of unity the free integral fibre basis still survives, but no simplicity, semisimplicity or equality with a naive ordinary-generator specialization is asserted. For a specialization to a field, if $ζ_{i}^{2}$ has order $l > 1$ , then the Gaussian factorial $[r]_{q_{i}^{2}}!$ vanishes once $r \geq l$ ; if $ζ_{i}^{2} = 1$ in characteristic zero every factor equals $j$ and that vanishing argument does not apply. Thus the algebra or module root-of-unity boundary is distinguished from the unconditional free-basis base-change fact.

Facts & Assumptions

Given: The lower global $A$ -basis $G_{λ} (B (λ))$ of $V_{A} (λ)$ with its finite free weight components, the balanced chart factorization for $V_{A} (λ)$ , the local $q = 1$ comparison theorems, and the geometric Gaussian factorials of The Lusztig–Kashiwara form, parameter and coproduct dictionary.

[F1]

Integral global basis (Bar invariance, congruence and uniqueness of the lower global basis, The based quotient: projection of the lower global basis to a highest weight module, The lower global basis of $U_{q}^{-}$ and of a highest weight module): $V_{A} (λ)$ is a free $A$ -module with basis $G_{λ} (B (λ))$ , each label-weight component $V_{A} (λ)_{μ}$ is a finite free $A$ -module, and $V_{A} (λ)$ spans the field module $V (λ)$ .

[F2]

Base change for free modules: if a module over a commutative ring $A$ is free with basis $(e_{b})_{b \in B}$ and $A \to S$ is a unital ring map, then $M \otimes_{A} S$ is free with basis $(e_{b} \otimes 1)$ and the same index set; no flatness or injectivity of the map is used. For finitely generated free $A$ -modules the rank is preserved under every unital base change to a nonzero ring.

[F3]

The $q = 1$ comparison (Specialization at q=1 recovers the classical enveloping relations, The q=1 toral specialization identifies the classical Cartan generators, Integrable bounded-above quantum modules with finite-dimensional weight spaces are direct sums of simple highest weight modules, Local q=1 deformation proves integrable highest weight universality, Quantum weight modules, weights in $P$ , and integrability): at a characteristic-zero field specialization $q = 1$ the positive toral quotient and the local deformation comparison identify the specialized module with the classical simple integrable highest weight module of the chosen realization, weight by weight, with the classical Weyl–Kac character.

[F4]

Chart factorization (Balanced triples over the two charts of $P^{1}$ and the globalization map, Bar invariance, congruence and uniqueness of the lower global basis): the balanced triple of $V_{A} (λ)$ gives $V_{A} (λ) \cap L (λ) ≅ Z [q] \otimes_{Z} E_{Z}$ and $V_{A} (λ) \cap ψ (L (λ)) ≅ Z [q^{- 1}] \otimes_{Z} E_{Z}$ with the same free $Z$ -basis $E_{Z} = ⨁_{b} Z G_{λ} (b)$ ; the genuine chart lattices over $A_{0}, A_{\infty}$ are the localizations of these $Z [q]$ - and $Z [q^{- 1}]$ -forms, and $q$ is a unit of $A$ , so no specialization $A \to A_{0}$ with $q \mapsto 0$ exists.

[F5]

Root-of-unity scalars: at a specialization $q = ζ$ to a field with $ζ$ invertible the Gaussian factorial is $[r]_{q_{i}^{2}}! = \prod_{j = 1}^{r} (1 + q_{i}^{2} + \dots + q_{i}^{2 (j - 1)})$ . If $ζ_{i}^{2}$ has finite order $l > 1$ , the factor with $j = l$ is $0$ , so $[r]_{q_{i}^{2}}! = 0$ for all $r \geq l$ ; if $ζ_{i}^{2} = 1$ and the ground field has characteristic zero, each factor equals $j$ , so $[r]_{q_{i}^{2}}! = r! \neq 0$ . These are exact evaluations of the defining product.

Proof

technique · direct

1.1F1F2

Unconditional base change. By [F1] and [F2] the fibre $V_{A} (λ) \otimes_{A} S$ is free over $S$ with basis $G_{λ} (b) \otimes 1$ , for every unital $S$ ; the rank of each finite weight component over $S$ equals $\dim_{Q (q)} V (λ)_{μ}$ when $S$ is nontrivial, and the fibre still spans the extended module. No flatness is needed because the basis itself tensors.

2.1F1F2F3step 1.1algebra

The $q = 1$ specialization. Put $R = Q [q]_{(q - 1)}$ . The $R$ -span of $V_{A} (λ)$ is exactly the ordinary-word lattice $V_{R}$ of the local comparison in [F3]: every divided word becomes an ordinary word times a product of the $R$ -unit inverses $[r]_{i}!^{- 1}$ , and ordinary words already belong to $V_{A} (λ)$ . Hence a characteristic-zero field fibre is the scalar extension of $V_{R} / (q - 1) V_{R}$ , which [F3] identifies with the classical simple integrable highest module. Weyl Kac character formula gives its Weyl–Kac character; in finite type the formal sum and product are finite, giving the Weyl character.

2.2F4

Chart lattices. The two chart lattices are read off from the $Z [q]$ - and $Z [q^{- 1}]$ -forms of the same free $Z$ -basis in [F4] and then localized at $0$ and $\infty$ ; since $q$ is a unit of $A$ , there is no ring map $A \to A_{0}$ sending $q$ to $0$ , so the charts are genuinely distinct from the base changes of step 1.1 and no specialization to $0$ is asserted.

2.3F5step 1.1

The root-of-unity boundary. At a field specialization $q = ζ$ with $ζ$ invertible, [F5] vanishes the geometric factorial as soon as the index reaches the order $l > 1$ of $ζ_{i}^{2}$ ; since the divided powers in the integral basis are normalized by these factorials, the naive ordinary-generator description of the specialization may then fail, while the free basis $G_{λ} (b) \otimes 1$ of step 1.1 is unaffected. If instead $ζ_{i}^{2} = 1$ in characteristic zero, [F5] gives $r!$ and no vanishing occurs, so that particular root-of-unity obstruction is absent.

3.1step 1.1step 2.1step 2.2step 2.3∎

Conclusion. Step 1.1 gives the unconditional free-basis base change with preserved finite weight ranks; step 2.1 gives the classical $q = 1$ module and character; step 2.2 separates the chart lattices from specializations; step 2.3 identifies the exact scalar root-of-unity boundary, asserting no simplicity or semisimplicity claim.

Depends on

Used by

The highest weight Fock submodule carries the quantum-group global bases Theorem

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

George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (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)
Math 263C Quantum Groups (Stanford, 2022), lecture slides 'Lusztig's quantum group. Tilting modules. The Fusion Category in Action' (standard reference, not scraped)