Affine Crystal Structures on Quantum Demazure Modules

Definitions and mathematical statement

Let $D_q(m,\lambda)$ be the source's quantum Demazure module, and let $B$ be its given crystal graph. Let $I_0$ index the finite-type simple roots and let $0$ be the extra affine node. The question is whether suitable additional $0$-labelled arrows on the same vertices extend the given graph to a crystal of an irreducible representation of $U_q(\widehat{\mathfrak g})$.

More explicitly, one seeks partial operators $\widetilde e_0,\widetilde f_0$ on $B$, with an undefined value when no arrow exists, so that $b\xrightarrow{0}b'$ precisely when $\widetilde f_0b=b'$ and $\widetilde e_0b'=b$. Existing finite-colour operators must remain unchanged, and any already specified $0$-arrows must be respected. The full operators, weights and string functions must satisfy the relevant affine crystal axioms and arise from an actual irreducible module with a crystal basis. Merely connecting the finite-colour graph or producing an arbitrary labelled graph is not enough. The original Demazure crystal may already contain partial affine data; whether the source first forgets these must be checked.

The Lie type, twisted/untwisted affine algebra, admissible $m,\lambda$, generic-$q$ convention and class of representations are not defined in the excerpt. Usual constructions use a positive integral level and appropriate dominant weight/coweight data, but those domains are not silently imposed on the notation.

There is also an important quantum-affine convention: the derived/loop algebra $U_q'(\widehat{\mathfrak g})$ omits the degree operator and allows the usual finite-dimensional level-zero crystals, with weights interpreted appropriately modulo the null-root direction. A full affine algebra including the degree operator has different representation/weight requirements. Since adding arrows preserves the size of a finite Demazure vertex set, these alternatives cannot be ignored. The source notation does not resolve them; no claim of equivalence or an unrestricted finite-dimensional representation is inserted.

The literature triage checked 21 August 2026 reports no authoritative complete resolution of the extracted claim in its targeted search. It supplies no explicit construction, case restrictions or later theorem, so the general target is retained with its missing conventions rather than certified as a precise all-types statement.

Short English statement

Can the given Demazure crystal be completed, by adding only affine-colour arrows, to the crystal of an irreducible quantum-affine representation? The original arrows must remain, and genuine crystal/module axioms matter. Affine-algebra, parameter and weight conventions are missing, particularly the derived-versus-degree-operator distinction.

Sources

• Local UnsolvedMath registry, permanent ID 30000622, exact source code catalogue entry; Affine Crystal Structures on Quantum Demazure Modules. Proposed year 2006, status open; corrected_verified extraction restores the affine algebra and module notation.
• Supplied report citation: Infinite-Dimensional Lie Theory (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/55, pp. 3259--3320, https://doi.org/10.4171/owr/2006/55. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2006/55, and reports no authoritative complete resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no separate construction/case theorem is supplied.