Hopf Algebra Extensions and Their Conformal Field Theories

Definitions and mathematical statement

Let $k$ denote the common ground field, which is not specified in the supplied record. A Hopf algebra $u$ over $k$ is pointed if every simple subcoalgebra of $u$ is one-dimensional, equivalently if its coradical is spanned by its group-like elements. Here a group-like element is an element $x$ satisfying $\Delta(x)=x\otimes x$ and $\varepsilon(x)=1$. The finite-dimensionality requirement is $\dim_k u<\infty$.

For a Lie algebra $\mathfrak g''$ over $k$, its universal enveloping algebra is \[ U(\mathfrak g'')=T_k(\mathfrak g'')/\langle x\otimes y-y\otimes x-[x,y]:x,y\in\mathfrak g''\rangle, \] with its usual Hopf structure, in which the elements of $\mathfrak g''$ are primitive.

Classification problem. Classify all extensions of the form \[ u\longrightarrow H\longrightarrow U(\mathfrak g'') \] where $u$ is a finite-dimensional pointed Hopf algebra, and determine the corresponding conformal field theories.

Unresolved specifications. This preserves the source's classification target, but the record does not specify the ground field or its characteristic, the permitted class of Lie algebras $\mathfrak g''$, the exactness conditions meant by an extension, or the equivalence relation for classification. In particular, the displayed arrows alone do not specify kernel or coinvariant conditions. The record also supplies no definition of the conformal field theories under consideration or of their correspondence with these extensions. These choices must be resolved before the target can be made into a fully specified classification problem; no particular choices are imposed here.

Short English statement

Classify every extension $u\longrightarrow H\longrightarrow U(\mathfrak g'')$ with $u$ a finite-dimensional pointed Hopf algebra, and determine its corresponding conformal field theories. The source leaves the ground field, extension and equivalence conventions, permitted Lie algebras, and conformal-field-theory correspondence unspecified.

Sources

• Local UnsolvedMath registry, supplied source record with permanent ID $30002800$, exact source code catalogue entry, title Hopf Algebra Extensions and Their Conformal Field Theories. The record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2015, and names no individual proposer.
• Supplied source citation: Mini-Workshop: Coideal Subalgebras of Quantum Groups (2015), Oberwolfach Reports, pp. 533--569, DOI 10.4171/owr/2015/10. https://doi.org/10.4171/owr/2015/10. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Andruskiewitsch and Schneider. https://annals.math.princeton.edu/2010/171-1/p07.
• Supplied literature source: Wang, connected Hopf algebras. https://doi.org/10.1016/j.aim.2015.05.009.
• Supplied literature assessment, checked 2026-08-21: the status is recorded as open; classification is said to exist for important restricted families of pointed Hopf algebras, while no general classification of all requested extensions and corresponding conformal field theories is said to be available. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report and preserves the report's hypotheses, notation, and open target. This is a supplied verification claim; the present draft does not independently verify it.