Perverse-sheaf categorification and canonical-type integral bases of the 2-periodic (root-category / Bridgeland) Hall algebra beyond simply-laced Dynkin type

The Hall algebra of the category of 2-periodic complexes — equivalently, the derived Hall algebra of the root category of a finitary hereditary abelian category — realizes the whole quantized enveloping algebra of the Kac–Moody algebra of the underlying quiver, and recent work of Zhang and of Chen–Lu–Ruan (both Israel J. Math. 2025) established this realization for arbitrary finitary hereditary categories within a unified periodic derived Hall algebra framework. However, the construction of a canonical-type basis and of a Lusztig-style geometric categorification by perverse sheaves is known only for Dynkin quivers (Fang–Lan–Xiao 2023, building on Qin's cyclic quiver varieties), where it yields a positive $\mathbb{Z}[v,v^{-1}]$-basis of an integral form of the whole quantum group. The open problem is to extend this categorification and construct a bar-invariant, positive integral (canonical-type) basis of the 2-periodic/derived Hall algebra beyond simply-laced Dynkin type, the target being the general case of an arbitrary finitary hereditary category (which includes the non-simply-laced Kac–Moody types), with the non-Dynkin (affine and wild) loop-free quivers as a special case.

Background

To a finitary hereditary abelian category $\mathcal{A}$ (an abelian category, Hom-finite over the finite field $\mathbb{F}_q$, with vanishing $\mathrm{Ext}^2$, e.g. finite-dimensional representations of a loop-free quiver $Q$ (a directed graph without self-loops), or coherent sheaves on a weighted projective line (an orbifold-type quotient of a projective line, with marked weighted points, whose category of coherent sheaves is still hereditary) one associates its Hall algebra: the $\mathbb{Q}$-vector space with basis the isomorphism classes $[M]$ of objects, multiplied by counting short exact sequences. Ringel (1990) and Green (1995) proved that the twisted Hall algebra (twisted by the Euler form of $\mathcal{A}$, the alternating bilinear form on classes of objects given by the difference of Hom- and Ext-dimensions) contains, as its composition subalgebra (the subalgebra generated by the classes of the simple objects), the positive half $U_v(\mathbf{n}^+)$ of the quantized enveloping algebra $U_v(\mathfrak{g})$ — the standard one-parameter deformation of the universal enveloping algebra of the Kac–Moody Lie algebra $\mathfrak{g}$ attached to $Q$ (finite-dimensional exactly in Dynkin type, infinite-dimensional otherwise). Lusztig realized $U_v(\mathbf{n}^+)$ geometrically — the Grothendieck group (the free abelian group on objects modulo the relations from exact sequences) of mixed perverse sheaves (a perverse sheaf is a constructible complex of sheaves satisfying cohomological support conditions that make simple intersection cohomology complexes behave like an orthonormal 'basis' of the variety; 'mixed' equips it with a compatible system of Frobenius actions over finite fields) on the moduli of quiver representations carries a convolution product, and the classes of simple perverse sheaves form the canonical basis of $U_v(\mathbf{n}^+)$, with remarkable integrality and positivity properties. Realizing the whole algebra $U_v(\mathfrak{g})$ categorically proved harder: Xiao's Drinfeld-double construction (the Drinfeld double is the standard Hopf-algebra construction that combines a Hopf algebra with its dual so that the two halves generate one larger Hopf algebra) glues two Hall halves by hand, and the derived Hall algebras of Toën and of Xiao–Xu, defined for triangulated categories under homological finiteness conditions, do not apply to the root category $\mathcal{R}(\mathcal{A})=D^b(\mathcal{A})/[2]$ (the 2-periodic orbit category of the bounded derived category), which is exactly the triangulated category long expected to carry the full quantum group — as Xu–Chen stated explicitly when constructing Hall algebras only for odd-periodic triangulated categories (odd period $t>1$), noting that no available method applied to the 2-periodic case. Bridgeland (2013) broke the impasse differently: he took the Hall algebra of the exact category of $\mathbb{Z}/2$-graded projective complexes, localized at the acyclic (contractible) complexes, and reduced; the resulting algebra $DH_{\mathrm{red}}(\mathcal{A})$ admits an embedding of $U_v(\mathfrak{g})$ for every loop-free quiver (an isomorphism precisely for simply-laced Dynkin $Q$), and Yanagida (2016) identified it with the reduced Drinfeld double of the twisted Hall algebra. Gorsky and Lu–Peng recast the same algebra via semi-derived Hall algebras (Hall algebras of complexes in which the contractible complexes are formally inverted) valid for all finitary hereditary $\mathcal{A}$ (no projectives needed); Xu–Zhang connected it with derived Hall algebras of $m$-cyclic complexes; and in 2022–2025 two independent constructions finally defined the Hall algebra of the root category itself — Zhang via the Xiao–Xu derived Hall numbers of $D^b(\mathcal{A})$, and Chen–Lu–Ruan by counting triangles via the octahedral axiom — each proved isomorphic to the Drinfeld double of the Hall algebra of $\mathcal{A}$ and to Bridgeland's algebra, with Zhang's periodic derived Hall algebras (JPAA 2025) providing a unified framework for all periods, and Gorsky–Haiden (2024) an intrinsic definition via homotopy cardinality (a systematic notion of the 'number of objects' of a category over a finite field) in the Calabi–Yau case (categories whose shift functor satisfies a Serre-type duality). Bridgeland closed his paper by asking whether one can 'consider Hall algebras of perverse sheaves in the style of Lusztig', with 'the potential to lead to new results on canonical bases'. The only known answers are restricted to Dynkin type: Qin's categorification via perverse sheaves on cyclic quiver varieties (a class of holomorphic-symplectic moduli spaces of quiver representations carrying a cyclic group action), and Fang–Lan–Xiao's sheaf realization of Bridgeland's Hall algebra of Dynkin type, which produces a basis with positivity and categorifies an integral form isomorphic to the Poisson integral form of $U_v(\mathfrak{g})$; Lu–Pan's 2025–2026 dual canonical bases for quantum and $\imath$quantum groups (the coideal subalgebras appearing in quantum symmetric pairs) are likewise confined to type ADE. Outside Dynkin type the obstruction is geometric: the relevant moduli of complexes then have infinitely many orbits, so Lusztig's orbit-wise intersection-cohomology construction has no direct analogue, and no positivity mechanism is available.

Problem Statement

Let $\mathcal{A}$ be a finitary hereditary abelian category — for instance the finite-dimensional representations over $\mathbb{F}_q$ of an arbitrary loop-free quiver $Q$ (not necessarily of Dynkin type), or coherent sheaves on a weighted projective line — and let $\mathfrak{g}$ be the associated Kac–Moody algebra. The algebra-level framework is now in place: Bridgeland's Hall algebra of 2-periodic complexes $DH_{\mathrm{red}}(\mathcal{A})$, the twisted semi-derived Ringel–Hall algebra (the Hall algebra of complexes in which contractible complexes are inverted), and the derived Hall algebra of the root category $\mathcal{R}(\mathcal{A})=D^b(\mathcal{A})/[2]$ are compatible realizations of (localizations/reductions of) the Drinfeld double of the twisted Hall algebra of $\mathcal{A}$, whose composition subalgebra is $U_v(\mathfrak{g})$. The open problem, posed by Bridgeland's closing question and answered so far only for Dynkin quivers, is to extend the canonical-basis side of this picture beyond simply-laced Dynkin type. Concretely, the target is the general case of an arbitrary finitary hereditary category (which includes the non-simply-laced Kac–Moody types); the non-Dynkin (affine and wild) loop-free quivers are a special case, and treating them alone does not resolve the task. Both of the following are required: (1) construct a geometric categorification of the 2-periodic/derived Hall algebra in Lusztig's style whose Grothendieck group, with its natural integral form, is identified with the generic 2-periodic Hall algebra realizing $U_v(\mathfrak{g})$; and (2) produce within this framework a canonical-type basis of the integral form: a $\mathbb{Z}[v,v^{-1}]$-basis that is bar-invariant (fixed by the bar involution, the anti-linear map that fixes the Chevalley generators and sends $v\mapsto v^{-1}$), whose structure constants lie in $\mathbb{N}[v,v^{-1}]$ (positivity).

References

1. C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990) 583–591, DOI 10.1007/BF01231516, https://doi.org/10.1007/BF01231516
2. J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995) 361–377, DOI 10.1007/BF01241133, https://doi.org/10.1007/BF01241133
3. G. Lusztig, Introduction to Quantum Groups, Birkhäuser, Boston, 1993, DOI 10.1007/978-0-8176-4717-9, https://doi.org/10.1007/978-0-8176-4717-9
4. B. Toën, Derived Hall algebras, Duke Math. J. 135 (2006) 587–615, DOI 10.1215/s0012-7094-06-13536-6, https://doi.org/10.1215/s0012-7094-06-13536-6
5. J. Xiao, F. Xu, Hall algebras associated to triangulated categories, Duke Math. J. 143 (2008) 357–373, DOI 10.1215/00127094-2008-021, arXiv:math/0608144, https://arxiv.org/abs/math/0608144
6. F. Xu, X. Chen, Hall algebras for odd periodic triangulated categories, Algebr. Represent. Theory 16 (2013) 673–687, DOI 10.1007/s10468-011-9325-z, arXiv:1001.4353, https://arxiv.org/abs/1001.4353
7. T. Bridgeland, Quantum groups via Hall algebras of complexes, Ann. of Math. 177 (2013) 739–759, DOI 10.4007/annals.2013.177.2.9, arXiv:1111.0745, https://arxiv.org/abs/1111.0745
8. S. Yanagida, A note on Bridgeland's Hall algebra of two-periodic complexes, Math. Z. 282 (2016) 973–991, DOI 10.1007/s00209-015-1573-x, arXiv:1207.0905, https://arxiv.org/abs/1207.0905
9. F. Qin, Quantum groups via cyclic quiver varieties I, Compos. Math. 152 (2016) 299–326, DOI 10.1112/s0010437x15007551, arXiv:1312.1101, https://arxiv.org/abs/1312.1101
10. S. Scherotzke, N. Sibilla, Quiver varieties and Hall algebras, Proc. Lond. Math. Soc. (3) 112 (2016) 1002–1018, DOI 10.1112/plms/pdw016, https://doi.org/10.1112/plms/pdw016
11. M. Gorsky, Semi-derived and derived Hall algebras for stable categories, Int. Math. Res. Not. IMRN 2018 (2018) 138–159, DOI 10.1093/imrn/rnv325, arXiv:1409.6798, https://arxiv.org/abs/1409.6798
12. M. Lu, L. Peng, Semi-derived Ringel-Hall algebras and Drinfeld double, Adv. Math. 383 (2021) 107668, DOI 10.1016/j.aim.2021.107668, arXiv:1608.03106, https://arxiv.org/abs/1608.03106
13. F. Xu, H. Zhang, Drinfeld doubles via derived Hall algebras and Bridgeland Hall algebras, Czechoslovak Math. J. 71 (2021) 253–267, DOI 10.21136/cmj.2020.0313-19, arXiv:1901.00257, https://arxiv.org/abs/1901.00257
14. J. Lin, L. Peng, Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories, Sci. China Math. 67 (2024) 1735–1760, DOI 10.1007/s11425-022-2145-x, arXiv:2304.11585, https://arxiv.org/abs/2304.11585
15. J. Fang, Y. Lan, J. Xiao, Sheaf realization of Bridgeland's Hall algebra of Dynkin type, arXiv:2303.04993 (2023, v3), https://arxiv.org/abs/2303.04993
16. J. Chen, M. Lu, S. Ruan, Derived Hall algebras of root categories, Israel J. Math. 270 (2025) 185–228, DOI 10.1007/s11856-025-2770-x, arXiv:2303.01670, https://arxiv.org/abs/2303.01670
17. H. Zhang, Hall algebras associated to root categories, Israel J. Math. 270 (2025) 413–448, DOI 10.1007/s11856-025-2825-z, arXiv:2210.12351, https://arxiv.org/abs/2210.12351
18. H. Zhang, Periodic derived Hall algebras of hereditary abelian categories, J. Pure Appl. Algebra 229 (2025) 107824, DOI 10.1016/j.jpaa.2024.107824, arXiv:2303.02912, https://arxiv.org/abs/2303.02912
19. M. Gorsky, F. Haiden, Counting in Calabi–Yau categories, with applications to Hall algebras and knot polynomials, arXiv:2409.10154 (2024), https://arxiv.org/abs/2409.10154
20. M. Lu, X. Pan, Dual canonical bases of quantum groups and $\imath$quantum groups I: Hall algebras, arXiv:2504.19073 (2025), https://arxiv.org/abs/2504.19073
21. M. Lu, X. Pan, Dual canonical bases of quantum groups and $\imath$quantum groups II: geometry, arXiv:2603.01350 (2026), https://arxiv.org/abs/2603.01350
22. X. Feng, F. Xu, Enveloping algebras via motivic Hall algebras, arXiv:2603.06965 (2026), https://arxiv.org/abs/2603.06965