MathDB

Kontsevich–Soibelman conjecture on the structure of the cohomological Hall algebra of the m-loop quiver

About 15 years old · traced to

Let mm be the number of loops in the quiver, and let

H=⨁n≥0HGCn∗(ECn)\mathcal{H}=\bigoplus_{n\geq 0}H^{*}_{G_{\mathbf{C}^n}}(E_{\mathbf{C}^n})

be its bigraded cohomological Hall algebra, with HGCnk(ECn)H^{k}_{G_{\mathbf{C}^n}}(E_{\mathbf{C}^n}) in bidegree (n,(m−1)(n2)−k/2)(n,(m-1)\binom{n}{2}-k/2). A bigraded supervector space C=⨁n,kCn,kC=\bigoplus_{n,k}C_{n,k} has only finitely many nonzero homogeneous components Cn,kC_{n,k} for each fixed n≥1n\geq 1, and these satisfy k≥0k\geq 0. Let zz be homogeneous of bidegree (0,−1)(0,-1).

Kontsevich–Soibelman conjecture. The bigraded algebra H\mathcal{H} is isomorphic to

Sym⁡(C⊗Q[z]),\operatorname{Sym}(C\otimes\mathbf{Q}[z]),

the graded symmetric algebra over the bigraded supervector space CC.

This conjectural free-generator description controls the structure and Poincaré–Hilbert series of the cohomological Hall algebra of the mm-loop quiver. It is attributed to Kontsevich and Soibelman; the supplied text does not state whether it has been proved or disproved.

References

Primary source

Markus Reineke, “Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers”, arXiv:1102.3978 (2011).

cohomological-hall-algebrasdonaldson-thomas-invariantsquiver-representations
Posted

Progress summary

Refreshed
Claimed solved

A paper claims a complete proof for all such quivers, while independent verification is not recorded here.

Attributed to Kontsevich and Soibelman, the conjecture predicts a free-generator description of the cohomological Hall algebra for the mm-loop quiver. The original formulation appears as Conjecture 4.4 in the 2011-era source.

Known results

  • The original paper gives the shuffle-algebra construction and formulates the conjecture as Sym⁡(C⊗Q[z])\operatorname{Sym}(C\otimes\mathbf{Q}[z]), with zz of bidegree (0,−1)(0,-1) (authors and date not stated in the retrieved text).

Efimov proof claim (date not stated)

Efimov's paper Cohomological Hall algebra of a symmetric quiver states that it proves the conjecture for every finite symmetric quiver, covering the mm-loop case. A related source attributes the proof to Efimov and gives the equivalent generator form Vprim⊗Q[x]V^{\mathrm{prim}}\otimes\mathbf{Q}[x]. The claim is reported here but not independently verified.

Current status (as of September 2026): The conjecture is claimed proved by Efimov for finite symmetric quivers, including the mm-loop quiver, but this automated scan records no independent verification.

Sources

Solutions 0

No solutions have been posted yet.