Primitive Generators of Preprojective Cohomological Hall Algebras

Definitions and mathematical statement

Let $Q$ be the quiver in the cited source, let $Q_0$ denote its vertex set, and let $\Pi_Q$ denote its preprojective algebra. A dimension vector is a tuple $d=(d_i)_{i\in Q_0}\in\mathbb N^{Q_0}$, recording the dimensions at the vertices. The base field, hypotheses on $Q$, and convention for $\mathbb N$ are not specified in the supplied excerpt and must be recovered from the source.

Write $A_{\Pi_Q}$ for the preprojective cohomological Hall algebra mentioned in the record, and $\mathfrak g_{\Pi_Q,d}$ for the degree-$d$ generator object occurring in its quoted Theorem 3. Let $I_d$ denote the object written $\operatorname{IC}(X_d)$ in the clean statement. Here $\operatorname{IC}$ denotes an intersection complex; the space $X_d$, its coefficients, normalization, and the passage from this complex to a space of generators are not defined in the supplied record.

Conjectural completeness target. For a dimension vector $d$ such that a simple $\Pi_Q$-module of dimension vector $d$ exists, the supplied account of Theorem 3 describes a canonical inclusion of the intersection-complex contribution into the degree-$d$ primitive-generator object. Subject to the source's precise identification of these objects, write this inclusion as \[ \iota_d:I_d\hookrightarrow\mathfrak g_{\Pi_Q,d}. \] The intended completeness assertion is that this canonical inclusion exhausts the primitive degree-$d$ generators: \[ \operatorname{im}(\iota_d)=\mathfrak g_{\Pi_Q,d}, \qquad\text{equivalently, }\iota_d\text{ is an isomorphism}. \] This is a conditional reconstruction of the target, not a fully specified conjecture: the supplied text does not define the generator object or the meaning of primitive, and does not establish that $\mathfrak g_{\Pi_Q,d}$ is literally a primitive subspace of a specified coalgebra. The original excerpt also contains a corrupted canonical direct-sum decomposition with an intersection-complex term involving $\mathbb Q^{\mathrm{vir}}$; its exact coefficients, shifts, and complementary summand cannot be recovered. The simple-module existence hypothesis is explicitly attached to Theorem 3; whether the conjecture is intended to include other dimension vectors remains unresolved.

Short English statement

The conjecture asks whether the canonical contribution denoted $\operatorname{IC}(X_d)$ accounts for every primitive generator of degree $d$ in the preprojective cohomological Hall algebra. The supplied theorem discussion assumes that a simple module of dimension vector $d$ exists. Definitions of $X_d$, the generator object, and the relevant conventions are missing, so the displayed formulation requires source consultation and human mathematical review.

Sources

• Supplied UnsolvedMath local registry record, permanent ID $30004883$, title Primitive Generators of Preprojective Cohomological Hall Algebras, exact source problem code catalogue entry. The record gives proposed year 2021 and names no proposer.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/52, pp. 2827--2891. Source URL and literature-source link labeled "OWR report": https://doi.org/10.4171/owr/2021/52.
• Supplied literature-source link labeled "Davison": https://arxiv.org/abs/2007.03289. The record supplies no further bibliographic details or precise location for the relevant claim.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem "partially solved" and reports that Theorem 3 constructs a canonical intersection-complex contribution, while completeness remains conjectural. This is supplied dated curation, not an independently verified assessment of the literature or present status.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly. That verification claim is supplied provenance; it has not been independently verified for this draft.