CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

An integrable based module can have no finite based envelope of a raw basis vector

Statement refuted

Every based integrable module for a quantum group of finite Cartan type contains, for each finite subset of its raw basis, a finite-dimensional $U$ -submodule spanned by a subset of that same raw basis and containing the given subset.

The assertion fails already for $U_{q} ({s l}_{2})$ and a singleton seed. Here “based” means the four axioms of The bar lattice and the divided-power integral lattice of a based module, with integrability in place of finite-dimensionality.

Facts & Assumptions

Given: $U = U_{q} ({s l}_{2})$ over $Q (q)$ , $A = Z [q, q^{- 1}]$ , and the explicit based chain of Distinct cyclotomic shears give integrable based quantum sl-two chains.

[F1]

The chain supplier gives $M = ⨁_{j \geq 0} V (n_{j})$ , $n_{j} = j (j + 3)$ , with divided string vectors $u_{j, k} = F^{(k)} u_{j, 0}$ , $0 \leq k \leq n_{j}$ . Put $ℓ_{j} = j + 3$ , $r_{j} = j + 2$ , and $p_{j} = q^{- φ (2 ℓ_{j}) / 2} Φ_{2 ℓ_{j}} (q) \neq 0$ . The supplier's shear maps satisfy $T_{j} u_{j, 0} = u_{j + 1, r_{j}}$ and vanish on positions not divisible by $ℓ_{j}$ . With $S = \sum_{j \geq 0} T_{j} / p_{j}$ , one has $S^{2} = 0$ and $C = 1 + S$ , $C^{- 1} = 1 - S$ . The vectors $b_{j, k} = C u_{j, k}$ form the raw basis $B$ of a genuine four-axiom based integrable module; all sums act with finite support (Distinct cyclotomic shears give integrable based quantum sl-two chains).

[F2]

On the divided string of $V (N)$ , $E^{(h)} u_{k} = {[\begin{array}{c} N - k + h \\ h \end{array}]}_{q} u_{k - h}$ for $0 \leq h \leq k$ , and the result is zero for $h > k$ . In particular $E^{(h)} u_{0} = 0$ for $h > 0$ . These formulas follow by iteration of Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules; each displayed Gaussian coefficient with $0 \leq h \leq k \leq N$ is nonzero in $Q (q)$ .

[F3]

A based subobject is a $U$ -submodule spanned by a subset of the raw basis (The bar lattice and the divided-power integral lattice of a based module). Consequently, if a vector belongs to a based subobject, every raw basis vector having nonzero coefficient in its basis expansion belongs to that subobject, by uniqueness of basis coordinates.

Counterexample

1.1givenF1

Take the actual based chain of [F1] and the singleton seed ${b_{0, 0}}$ . Each vector of its field direct sum belongs to finitely many finite strings, so its $U$ -orbit is finite dimensional. In particular $b_{0, 0} = u_{0, 0} + u_{1, 2} / p_{0}$ belongs to $V (0) \oplus V (4)$ . The seed therefore has a finite field orbit, while the question asks for a submodule spanned by raw basis vectors.

1.2F1F2algebra

For every $j \geq 0$ , write $G_{j} = {[\begin{array}{c} n_{j + 1} \\ r_{j} \end{array}]}_{q} \neq 0$ . Since $b_{j, 0} = u_{j, 0} + u_{j + 1, r_{j}} / p_{j}$ , [F2] gives $E^{(r_{j})} b_{j, 0} = (G_{j} / p_{j}) u_{j + 1, 0}$ . Furthermore $u_{j + 1, 0} = b_{j + 1, 0} - u_{j + 2, r_{j + 1}} / p_{j + 1}$ . The index $r_{j + 1} = j + 3$ is strictly between $0$ and $ℓ_{j + 2} = j + 5$ , so $T_{j + 2} u_{j + 2, r_{j + 1}} = 0$ and $u_{j + 2, r_{j + 1}} = b_{j + 2, r_{j + 1}}$ . Thus the exact raw expansion is $E^{(r_{j})} b_{j, 0} = (G_{j} / p_{j}) b_{j + 1, 0} - (G_{j} / (p_{j} p_{j + 1})) b_{j + 2, r_{j + 1}}$ . The two displayed raw vectors are distinct, and the first coefficient is nonzero.

2.1F3step 1.2algebra

Let $W$ be any $U$ -submodule spanned by a subset of $B$ and containing $b_{0, 0}$ . If $b_{j, 0} \in W$ , its image under $E^{(r_{j})}$ belongs to $W$ by $U$ -stability, and the nonzero $b_{j + 1, 0}$ coordinate in step 1.2 forces $b_{j + 1, 0} \in W$ by [F3]. Starting from $j = 0$ , induction therefore gives $b_{j, 0} \in W$ for every $j \geq 0$ . These are infinitely many distinct members of the field basis $B$ , so $W$ is infinite dimensional.

3.1step 1.1step 2.1F1∎

Hence no finite-dimensional based subobject of $M$ contains the singleton seed $b_{0, 0}$ . The module satisfies all four actual based axioms and is integrable in finite Cartan type by [F1], so it refutes the stated universal finite-envelope assertion.

Remarks

This counterexample concerns raw basis compatible submodules. It does not itself refute tensor globalization: a construction using finite correction supports could conceivably work without finite based submodules. The finite-dimensional based filtration in Lusztig §27.1 and the bounded extension in Bao–Wang Remark2.10 retain their own hypotheses.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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 (standard reference, not scraped)
Huancheng Bao and Weiqiang Wang, Canonical bases in tensor products revisited (standard reference, not scraped)