Explicit Deformation Algebras for Operated Structures

Definitions and mathematical statement

An operated algebraic structure here means an algebraic structure equipped with prescribed distinguished operators and relations. The supplied record does not specify the operators, their arities, their relations, the ground ring, or the ambient category. Consequently, it specifies a research programme rather than a single fully determined deformation problem.

In the operadic setting described by the excerpt, let $\mathcal P$ denote the operad encoding the chosen structure and let $A$ be a $\mathcal P$-algebra. These are parameters whose admissible class must still be specified. Write $\operatorname{End}_A$ for the endomorphism operad; in a specified linear setting its arity-$n$ component is $\operatorname{Hom}(A^{\otimes n},A)$. The algebra structure is encoded by an operad morphism $\alpha\colon\mathcal P\to\operatorname{End}_A$.

A differential graded Lie algebra is a graded module $\mathfrak g$ with a degree-$1$ differential $d$, satisfying $d^2=0$, and a degree-$0$ graded Lie bracket for which $d$ is a graded derivation. In cohomological grading, an $L_\infty$-algebra has graded skew-symmetric operations \[ \ell_n\colon\mathfrak g^{\otimes n}\longrightarrow\mathfrak g, \qquad |\ell_n|=2-n,\qquad n\geq 1, \] satisfying the higher Jacobi identities. A differential graded Lie algebra is the special case with $\ell_n=0$ for $n\geq3$. These descriptions do not supply the missing hypotheses on the ground ring needed for a particular deformation-theoretic application.

Requested construction. For the intended operated structures $\mathcal P$ and their algebras $A$, construct explicitly a differential graded Lie algebra or an $L_\infty$-algebra $\mathfrak g_A$ governing deformations of the structure on $A$. An explicit construction should specify the underlying graded object, its differential and bracket or higher brackets, and explain how its deformation data and equivalences correspond to the intended deformations and equivalences of $A$. The record does not determine whether the underlying object is fixed, which operations are allowed to deform, or which notion of equivalence is intended; thus a precise controlling correspondence cannot yet be prescribed.

Also develop homotopy versions of these operated structures, analogous to $A_\infty$-algebras for associative algebras. In an ambient category of operads with a specified model structure, a cofibrant resolution is a weak equivalence \[ q\colon\mathcal Q\longrightarrow\mathcal P \] with $\mathcal Q$ cofibrant. Algebras over such a resolution provide the operadic interpretation of homotopy $\mathcal P$-algebras. The supplied excerpt states that, given a cofibrant resolution, in particular a minimal model when available, one can form the deformation complex and construct its $L_\infty$-structure. The requested work concerns explicit constructions and homotopy versions; neither a particular resolution nor a definition of minimality is supplied. Existence of a minimal model for every possible operated structure is not asserted here.

The supplied dated assessment additionally identifies a general explicit small model as a remaining research target, but does not define the intended class of structures or what counts as small. These qualifications remain unresolved.

Short English statement

Construct explicit differential graded Lie or $L_\infty$-algebras controlling deformations of the intended operated algebraic structures, and describe their homotopy versions using suitable operadic resolutions. The record leaves the operators, relations, base, deformation equivalences, and meaning of a small model unspecified, so this is a qualified research programme requiring further specification. Its partially solved label is supplied dated curation.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied record for permanent UnsolvedMath ID $30005910$, title "Explicit Deformation Algebras for Operated Structures," exact source problem code catalogue entry.
• Supplied report citation: Hochschild (Co)Homology and Applications (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/20, pp. 1097--1162. https://doi.org/10.4171/owr/2024/20. The excerpt includes the title "The homotopy theory of operated algebras" and the name Guodong Zhou. The record supplies no separate proposer attribution.
• Supplied literature source: Dinh Van, Lowen and Hermans, "box operads." https://arxiv.org/2305.20036. No fuller bibliographic details are supplied.
• Supplied literature source: Oberwolfach Report 20/2024. https://doi.org/10.4171/owr/2024/20.
• The original excerpt attributes the construction of deformation complexes and their $L_\infty$-structures from cofibrant resolutions to Kontsevich and Soibelman [14] and van der Laan [24, 25]. These are source-local reference labels; their bibliographic entries are absent from the supplied record.
• Literature assessment dated exactly 2026-08-22: the supplied status is "partially solved." The supplied assessment says that general deformation theory provides controlling $L_\infty$-structures, explicit constructions exist for broad classes of operated algebras, and a general explicit small model remains a research problem. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, with the qualification that only formula delimiters, notation, grammar, and typography were repaired while preserving mathematical content and scope. This supplied label does not constitute mathematical review of this draft.