Structure of Cohomological Hall Algebras of Preprojective Algebras

Definitions and mathematical statement

The supplied question concerns a quiver $Q$, its preprojective algebra $\Pi_Q$, and its cohomological Hall algebra $A_{\Pi_Q}$. The record does not specify the allowed quivers, coefficient field, or precise construction of this cohomological Hall algebra.

For clarity of notation, under the usual finite-quiver convention, write $Q=(Q_0,Q_1,s,t)$, where $Q_0$ and $Q_1$ are the vertex and arrow sets and $s,t$ give the source and target of each arrow. Its doubled quiver $\overline Q$ is obtained by adjoining an arrow $a^*:t(a)\to s(a)$ for every $a\in Q_1$. Over a field $k$, the ordinary preprojective algebra is \[ \Pi_Q=k\overline Q\Big/\left\langle\sum_{a\in Q_1}(aa^*-a^*a)\right\rangle, \] where $k\overline Q$ is the path algebra. This conditional definition explains standard notation; it does not resolve whether this finite-quiver convention, this coefficient field, or this ordinary version is intended by the source.

Problem. Determine the structure of the entire cohomological Hall algebra $A_{\Pi_Q}$ associated with $\Pi_Q$, using the conventions and hypotheses of the cited Oberwolfach contribution. The target is the full algebra. The supplied record does not specify a proposed presentation, comparison algebra, or other criterion for a complete structural description. It also does not supply enough information to fix the quantification over $Q$, the homology theory and coefficients, equivariance, grading shifts, or multiplication conventions. Consequently this is a structural research question with unresolved formulation parameters, rather than a fully specified conjectural isomorphism.

Supplied context and qualifications. The extracted passage states an isomorphism \[ \mathcal L_0 A_{\Pi_Q}\cong U\!\left(\mathfrak g_{\widetilde Q,\widetilde W}\right), \] where $\mathcal L_0 A_{\Pi_Q}$ is described as the subalgebra of "perverse-exact raising operators," $U$ denotes a universal enveloping algebra, and $\mathfrak g_{\widetilde Q,\widetilde W}$ is described as the BPS Lie algebra for the tripled quiver. Here $\widetilde Q$ is obtained by adding a loop $\omega_i$ at each vertex of $\overline Q$. The definitions of the subalgebra and BPS Lie algebra are absent, and the extracted formula for the potential $\widetilde W$ is corrupted. The supplied literature assessment instead describes an identification of this subalgebra with a BPS Lie algebra; this wording does not distinguish that Lie algebra from its universal enveloping algebra and requires review.

The surrounding extract also mentions a lifted positive Heisenberg action, a lift of a BPS Lie algebra action to a Lie algebra object $\mathfrak g_{\Pi_Q}$ in a category of perverse sheaves, an isomorphism $H(\mathfrak g_{\Pi_Q})\cong\mathfrak g_{\widetilde Q,\widetilde W}$, and semisimplicity with a lift to a pure weight-zero mixed Hodge module. The spaces, morphisms, stability parameter, and action formulas are not adequately defined in the record. A subsequent condition concerning summands with zero-dimensional support is truncated. These are supplied contextual claims, not independently verified statements or additional hypotheses reconstructed here.

Short English statement

Describe the full cohomological Hall algebra $A_{\Pi_Q}$ of a preprojective algebra. The supplied context describes a particular subalgebra through a universal enveloping algebra, while the question asks about the entire algebra. The intended quivers and algebra conventions remain unspecified. The supplied status is partially solved; that dated assessment has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004882$, exact source problem code catalogue entry, title: Structure of Cohomological Hall Algebras of Preprojective Algebras. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2021.
• Enveloping Algebras and Geometric Representation Theory (2022), Oberwolfach Reports, pp. 2827--2891, DOI 10.4171/owr/2021/52. Supplied report reference and literature-source link: https://doi.org/10.4171/owr/2021/52. No contribution-level page or individual author is identified in the supplied record.
• Davison, supplied literature-source link: https://arxiv.org/abs/2007.03289. No paper title or precise theorem locator is supplied. The extracted passage contains a reference labelled [3], but its bibliography entry is absent; its identification with this link is not established by the supplied material.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem partially solved and says that the perverse-exact subalgebra is identified with a BPS Lie algebra while the structure of the entire algebra remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The registry also labels the statement corrected_verified and reports a direct check of the official report; those are supplied verification claims, not checks performed for this draft.