Problem 12.13 — (J.

Definitions and mathematical statement

J. Roberts asks the foundational question: "What are quantum groups?" The excerpt gives no specified classification target, construction, universal property or theorem to establish.

Standard examples include Drinfeld--Jimbo Hopf-algebra deformations of universal enveloping algebras. In a formal deformation convention, one has a Hopf algebra $U_\hbar(\mathfrak g)$ with classical limit $U_\hbar(\mathfrak g)/\hbar U_\hbar(\mathfrak g)\cong U(\mathfrak g)$. Other usages concern deformations of function algebras or related structures. These examples explain the supplied summary's context but do not select a unique definition answering Roberts's question.

The source does not state whether it seeks an intrinsic characterization, geometric or topological meaning, classification, or a unifying framework for those usages. It must not be silently changed into the known construction of one quantum enveloping algebra, the assertion that every Hopf algebra is a quantum group, or a demand to classify all Hopf algebras. The qualitative question is retained, with its intended conceptual success criterion unresolved.

Short English statement

Roberts asks for conceptual understanding of quantum groups. The source provides no precise theorem or criterion, and standard deformation examples do not determine the full intended question.

Sources

• Local UnsolvedMath registry, permanent ID 10400218, source code catalogue entry; statement attribution J. Roberts; proposer field and proposal year absent; update 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.13, PDF page 165: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access date 2026-07-29; PDF font-structure extraction. Provenance distinguishes 231 numbered bold headings. Present primary-literature status and redistribution terms require review.
• The supplied summary credits Drinfeld and Jimbo for enveloping-algebra deformations and describes that as partial conceptual progress, without exact references or a formal resolution criterion. The registry's open-triage label is not independently assessed here.