Dual canonical bases of ıquantum groups beyond type ADE: all-types construction, positivity, and geometric realization

The 2026 papers of Lu–Pan and Chen–Lu–Pan–Ruan–Wang established the dual canonical basis for quasi-split universal ıquantum groups: geometrically and via ıHall algebras in Dynkin type ADE, and algebraically (with an algorithm) in arbitrary finite type. The open problem is the completion of this program to all types: construct the dual canonical basis of the universal ıquantum group for general (non-quasi-split) Satake diagrams of finite type and for quasi-split Kac–Moody (especially affine) types, via both ıHall algebras and perverse-sheaf/quantum-Grothendieck-ring geometry, and settle the positivity properties (structure constants, embedding and coproduct transition coefficients) that are proved only in type ADE and in rank one.

Background

A quantized enveloping algebra (quantum group) $\mathbf{U}$ deforms the universal enveloping algebra of a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$ , depending on a parameter $v$ . Lusztig and Kashiwara showed that the positive half $\mathbf{U}^{+}$ admits a canonical basis with remarkable integrality and positivity properties; Lusztig constructed it geometrically from simple perverse sheaves (a class of sheaves built by the decomposition theorem, with IC sheaves as the local building blocks) on varieties of quiver representations, and algebraically from PBW bases (bases indexed by conjugacy classes of the Weyl group, produced by braid group symmetries). The dual canonical basis is the basis of $\mathbf{U}^{+}$ dual to it; it governs, for example, the transition to cluster monomials and has structure constants in $\mathbb{N}[v^{\pm 1}]$ . Extending such bases from $\mathbf{U}^{+}$ to the whole quantum group requires passing to a doubled object: the Drinfeld double $\widetilde{\mathbf{U}}$ (quantum groups being Hopf algebras, their double pairs positive and negative halves with matched Cartan parts). Qin realized $\widetilde{\mathbf{U}}$ of type ADE through the quantum Grothendieck ring of perverse sheaves on cyclic (Nakajima-type, graded) quiver varieties (the quantum Grothendieck ring is a $v$ -deformation of the Grothendieck group of a monoidal category of sheaf complexes, with multiplication given by a convolution product), obtaining an integral positive basis there; Berenstein and Greenstein constructed algebraically a double canonical basis of $\widetilde{\mathbf{U}}$ of every finite type via Heisenberg doubles, posed several conjectures on it (braid invariance, positivity, coincidence of variants), and proved them only in special cases. An ıquantum group (the coideal subalgebra $\mathbf{U}^{\imath}$ of a quantum symmetric pair, introduced by Letzter and Kolb) is the quantum analogue of the fixed-point subalgebra of $\mathfrak{g}$ under a Lie algebra involution; its type is encoded by a Satake diagram, a Dynkin diagram with a diagram involution and a chosen set of “black” fixed vertices. The Satake diagram is quasi-split when there are no black vertices; the universal ıquantum group $\widetilde{\mathbf{U}}^{\imath}$ (a coideal of the Drinfeld double, introduced by Lu and Wang) specializes to $\mathbf{U}^{\imath}$ for all parameters by central reduction, and includes $\widetilde{\mathbf{U}}$ itself as the diagonal case. Two realizations of $\widetilde{\mathbf{U}}^{\imath}$ are known for Dynkin ıquivers (a quiver with involution): the ıHall algebra, a twisted semi-derived Ringel–Hall algebra of modules over the ıquiver algebra (an algebraic construction from the category of representations, following Ringel's Hall-algebra realization of $\mathbf{U}^{+}$ and Bridgeland's of $\widetilde{\mathbf{U}}$ ), and the quantum Grothendieck ring of perverse sheaves on Nakajima–Keller–Scherotzke (NKS) quiver varieties, whose dual IC basis plays the geometric role. From these, Lu–Pan (arXiv:2504.19073 and arXiv:2603.01350) constructed the dual canonical basis of $\widetilde{\mathbf{U}}^{\imath}$ of type ADE in both ways — via Lusztig's lemma (a triangularity fixed-point argument producing a unique bar-invariant basis, where bar-involution is the Q-linear anti-involution with $\bar v=v^{-1}$ ), proving the two constructions coincide, that the basis is invariant under the relative braid group action (the ıanalogue of Lusztig's braid symmetries), and that its structure constants are positive, i.e. lie in $\mathbb{N}[v^{\pm 1/2}]$ . Chen–Lu–Pan–Ruan–Wang (arXiv:2601.00524, building on arXiv:2511.11291) then gave an algebraic construction via iHopf algebras (a new associative structure on a Hopf algebra with Hopf pairing, realizing $\widetilde{\mathbf{U}}^{\imath}$ on a Borel) for quasi-split $\widetilde{\mathbf{U}}^{\imath}$ of arbitrary finite type, with an explicit algorithm and ibraid invariance, and identified the resulting Drinfeld-double basis with the double canonical basis of Berenstein–Greenstein, settling their main finite-type conjectures; they also recovered the geometry for ADE but could not re-establish positivity beyond it. Non-simply-laced (valued) Cartan data, Satake diagrams with black vertices, and infinite (affine/Kac–Moody) root systems all fall outside every existing construction of this basis.

Problem Statement

Construct the dual canonical basis of the universal ıquantum group $\widetilde{\mathbf{U}}^{\imath}$ in the generality beyond the already-covered quasi-split, simply-laced finite (ADE) case, with the characteristic properties established there — bar-invariance, integrality, independence of choices, invariance under the relative braid group action, and positivity (structure constants in $\mathbb{N}[v^{\pm 1/2}]$ ) in the types where positivity is required, as fixed by the component-specific requirements below. Concretely, following the problem list of Lu–Pan (arXiv:2605.13578, §7) and the Perspectives of arXiv:2603.01350: (1) for Satake diagrams of finite type beyond the quasi-split ones, provide an ıHall-algebra realization and a geometric (perverse-sheaf/quantum-Grothendieck-ring) realization of $\widetilde{\mathbf{U}}^{\imath}$ and construct its dual canonical basis; (2) for non-simply-laced types, formulate the valued ıquiver algebras and their ıHall algebras, and extend the Nakajima–Keller–Scherotzke quiver-variety construction so that the perverse sheaves correspond to the dual canonical basis, proving positivity of the ıHall-to-basis transition coefficients in the spirit of Berenstein–Greenstein's Conjecture 1.21; (3) settle positivity for the remaining quasi-split finite types, the outstanding instance being type $\mathrm{AIII}_{2r}$ , and the embedding and coproduct positivity conjectures for the basis, so far verified only in rank one ( $\mathfrak{sl}_2$ ); and (4) for quasi-split ıquantum groups of Kac–Moody type, especially affine type, construct the dual canonical basis from the ıHall algebras of Lu–Wang and extend the perverse-sheaf geometric realization to that setting, where positivity of the structure constants is expected, namely for the symmetric (simply-laced) types. These four components pursue one objective — completing the dual canonical basis theory across all types of quantum symmetric pairs — but they are four independent open sub-problems, each posed separately in the cited sources: each may be resolved or refuted on its own, and resolving one of them establishes only its own component and leaves the others open, so that a complete resolution of the task is the conjunction of all four and settling only type $\mathrm{AIII}_{2r}$ , or only the finite-type components, does not resolve the task. Each component is measured against the ADE template: a basis that is integral, bar-invariant, and braid-invariant, together with the particular positivity property required for its type as stated in that component.

References

Ming Lu and Xiaolong Pan, "Dual canonical bases of quantum groups and $\imath$ quantum groups II: geometry", arXiv:2603.01350 (2026), DOI 10.48550/arXiv.2603.01350, https://arxiv.org/abs/2603.01350
Ming Lu and Xiaolong Pan, "Dual canonical bases of quantum groups and $\imath$ quantum groups I: Hall algebras", arXiv:2504.19073 (2025), https://arxiv.org/abs/2504.19073
Jiayi Chen, Ming Lu, Xiaolong Pan, Shiquan Ruan and Weiqiang Wang, "iQuantum groups and iHopf algebras II: dual canonical bases", arXiv:2601.00524 (2026), https://arxiv.org/abs/2601.00524
Jiayi Chen, Ming Lu, Xiaolong Pan, Shiquan Ruan and Weiqiang Wang, "iQuantum groups and iHopf algebras I: foundation", arXiv:2511.11291 (2025), https://arxiv.org/abs/2511.11291
Ming Lu and Xiaolong Pan, "Quiver varieties and dual canonical bases", arXiv:2605.13578 (2026), https://arxiv.org/abs/2605.13578
Ming Lu and Weiqiang Wang, "Hall algebras and quantum symmetric pairs of Kac-Moody type", Advances in Mathematics 430 (2023), 109215, DOI 10.1016/j.aim.2023.109215
Ming Lu and Weiqiang Wang, "Hall algebras and quantum symmetric pairs III: Quiver varieties", Advances in Mathematics 393 (2021), 108071, DOI 10.1016/j.aim.2021.108071
Ming Lu and Shiquan Ruan, " $\imath$ Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type", arXiv:2411.13078 (2024), https://arxiv.org/abs/2411.13078
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