Completing the $\imath$Hall-algebra realization of $\imath$quantum loop algebras of weighted projective lines: injectivity for arbitrary Kac–Moody type, the composition algebra, PBW bases, and higher-genus extensions

The $\imath$Hall algebra $\imath\widetilde{\mathcal{H}}(\mathbb{X}_{\mathbf{k}})$ of a weighted projective line — the semi-derived Ringel–Hall algebra of the category of $1$-periodic complexes of coherent sheaves — realizes the $\imath$quantum loop algebra ${}^{\mathrm{Dr}}\widetilde{\mathbf{U}}^{\imath}$ of split type through a morphism $\Omega$ that is proved injective only for finite and affine Kac–Moody type. The problem asks to complete this realization program: prove $\Omega$ injective for arbitrary Kac–Moody $\mathfrak{g}$, determine the composition subalgebra of the $\imath$Hall algebra and its structure, construct PBW-type bases for $\imath$quantum loop algebras via coherent sheaves (including the $2$-toroidal types $D_4^{(1,1)}$–$E_8^{(1,1)}$), and extend the $\imath$Hall construction to higher-genus curves, notably elliptic curves.

Background

A Hall algebra reconstructs quantum groups from abelian categories: to a finitary hereditary abelian category one associates an algebra whose structure constants count short exact sequences, and Ringel's theorem realizes the positive part of a quantum group from the representations of its Dynkin quiver. Geigle–Lenzing weighted projective lines $\mathbb{X}_{\mathbf{k}}$ (a projective line over a finite field $\mathbf{k}$ with weights $(p_1,\dots,p_t)$ at $t$ distinguished points, defined via a graded coordinate ring) yield the hereditary category $\mathrm{coh}(\mathbb{X})$ of coherent sheaves; the simply-laced Kac–Moody algebra $\mathfrak{g}$ attached to the star-shaped graph $T_{p_1,\dots,p_t}$ (a central vertex with $t$ arms of lengths $p_i$) governs its indecomposables. Schiffmann used the twisted hall algebra of $\mathbb{X}_{\mathbf{k}}$ to realize the positive part of the quantum loop algebra (the quantum affine algebra in Drinfeld's new presentation) of $\mathfrak{g}$; Burban–Schiffmann treated affine $\mathfrak{sl}_2$ via Beilinson's derived equivalence and later built the Hall algebra of an elliptic curve. Bridgeland's Hall algebra of $2$-periodic complexes realizes the entire quantum group, and the semi-derived Ringel–Hall algebra of Lu–Peng extends this machine to periodic complexes over hereditary categories.

A quantum symmetric pair consists of a quantum group $\widetilde{\mathbf{U}}$ together with a coideal subalgebra $\widetilde{\mathbf{U}}^{\imath}$ fixed by an involution of the Dynkin diagram symmetry type; the subalgebra $\widetilde{\mathbf{U}}^{\imath}$, called an $\imath$quantum group ($\imath$quantum algebra), is a one-sided generalization of the quantum group (the diagonal case recovers $\widetilde{\mathbf{U}}$ itself). Lu–Ruan–Wang realized $\imath$quantum groups via $\imath$Hall algebras, i.e. semi-derived Ringel–Hall algebras of categories attached to $\imath$quivers (quivers with an involution); for the projective line this realizes the universal $q$-Onsager algebra (the $\imath$quantum group of split affine $A_1$) in Drinfeld-type presentation.

Lu–Ruan defined the $\imath$Hall algebra $\imath\widetilde{\mathcal{H}}(\mathbb{X}_{\mathbf{k}})$ as the semi-derived Ringel–Hall algebra of the category of $1$-periodic complexes in $\mathrm{coh}(\mathbb{X}_{\mathbf{k}})$, and defined the $\imath$quantum loop algebra ${}^{\mathrm{Dr}}\widetilde{\mathbf{U}}^{\imath} = {}^{\mathrm{Dr}}\widetilde{\mathbf{U}}^{\imath}(L\mathfrak{g})$ of split type: a Drinfeld-type presentation with generators $B_{\star,l}$, $H_{i,m}$, $\Theta_{i,m}$ (indexed by the vertices of $T_{p_1,\dots,p_t}$ and integers $l,m$), toral $\mathbf{K}$-generators and a central degree generator $C$; for $\mathfrak{g}$ of affine ADE type this is the Drinfeld presentation of the $\imath$quantum group of the split affine quantum symmetric pair, and for tubular weight types ($\mathbb{X}$ of type $(2,2,2,2)$, $(3,3,3)$, $(2,4,4)$, $(2,3,6)$) it is the $\imath$analogue of the $2$-toroidal (elliptic) quantum algebra of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$, $E_8^{(1,1)}$, respectively — in each of these cases the curve is still a weighted projective line, so these $2$-toroidal algebras are genus-zero objects. They constructed an algebra morphism $\Omega:{}^{\mathrm{Dr}}\widetilde{\mathbf{U}}^{\imath}_{|v=\sqrt{q}}\to\imath\widetilde{\mathcal{H}}(\mathbb{X}_{\mathbf{k}})$ sending the Drinfeld generators to explicit Hall elements (classes of torsion sheaves $[S_{ij}]$, of line bundles twisted by the canonical degree $[\mathcal{O}(l\vec{c})]$, and of transforms $\widehat{\Theta},\widehat{H}$), proved it is an embedding for split affine type $A$ via the Geigle–Lenzing derived equivalence to an $\imath$quiver Hall algebra (Theorem C of Lu–Ruan), and posed the completion of the realization as an explicit research program in the paper's 'Further works'. The sequel (Lu–Ruan II) proved $\Omega$ injective, with image exactly the composition subalgebra $\widetilde{\mathcal{C}}\mathcal{M}\mathcal{H}(\mathbb{X}_{\mathbf{k}})$ (the subalgebra generated by those Hall elements), whenever $\mathfrak{g}$ is of finite or affine type — which covers all ADE and tubular cases. Here the composition algebra means the subalgebra generated by the images of the quantum generators, analogous to the classical composition subalgebra of a Hall algebra; a PBW-type (Poincaré–Birkhoff–Witt-type) basis is a basis ordered by root vectors respecting a triangular decomposition, as Burban–Schiffmann built for affine $\mathfrak{sl}_2$ and Burban–Schiffmann's elliptic Hall algebra for elliptic curves.

Problem Statement

Complete the coherent-sheaf Hall realization of $\imath$quantum loop algebras of weighted projective lines, in the framework of Lu–Ruan, by resolving the following linked components, all of which the sources state as open directions or expectations:

(1) Injectivity in general type. Prove (or refute) that the morphism $\Omega:{}^{\mathrm{Dr}}\widetilde{\mathbf{U}}^{\imath}_{|v=\sqrt{q}}\to\imath\widetilde{\mathcal{H}}(\mathbb{X}_{\mathbf{k}})$ from the $\imath$quantum loop algebra of split type attached to an arbitrary simply-laced Kac–Moody algebra $\mathfrak{g}$ of star-shaped-graph type $T_{p_1,\dots,p_t}$ (with $\mathbb{X}_{\mathbf{k}}$ the corresponding weighted projective line) is injective for arbitrary such Kac–Moody $\mathfrak{g}$, beyond the finite and affine types where it is already proved.

(2) Composition algebra. Determine the composition subalgebra $\widetilde{\mathcal{C}}\mathcal{M}\mathcal{H}(\mathbb{X}_{\mathbf{k}})$ of the $\imath$Hall algebra of a weighted projective line — its generators, defining relations, triangular/double structure and dependence on $\mathbb{X}_{\mathbf{k}}$ — together with its relation to ${}^{\mathrm{Dr}}\widetilde{\mathbf{U}}^{\imath}$ through $\Omega$ in general type; for $\mathfrak{g}$ of finite or affine type Part II shows that the image of $\Omega$ is exactly this subalgebra.

(3) PBW-type bases and algebraic structure. Through the coherent-sheaf Hall realization of the $\imath$quantum loop algebras of arbitrary simply-laced Kac–Moody algebras $\mathfrak{g}$ of star-shaped-graph type: (a) construct PBW-type bases, with the associated root-space filtrations; and (b) determine the algebraic structure of these $\imath$quantum loop algebras beyond the verification of their defining relations, specifically a triangular decomposition compatible with the PBW basis of (a), the double/defining relations, and the dependence of the algebra on the weight type. The cases $\mathfrak{g}$ of affine type $D_4,E_6,E_7,E_8$, where ${}^{\mathrm{Dr}}\widetilde{\mathbf{U}}^{\imath}$ is the $\imath$analogue of the $2$-toroidal (elliptic) quantum algebras of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$, $E_8^{(1,1)}$ arising from tubular weighted projective lines, are included as concrete instances of these requirements.

(4) Higher genus. Study the $\imath$Hall algebras of curves of higher genus: extend the $\imath$Hall construction — semi-derived Ringel–Hall algebras of $1$-periodic complexes — from weighted projective lines to all smooth projective curves of positive genus, with the elliptic-curve case included as the concrete instance of these requirements, where the classical (non-$\imath$) Hall algebra of an elliptic curve is already known from Burban–Schiffmann. This component concerns curves of positive genus, and is therefore distinct from the genus-zero $2$-toroidal types named in component (3), which are attached to tubular weighted projective lines and are already defined.

The common scientific objective is the completion of this Hall-algebra realization programme for $\imath$quantum loop algebras of weighted projective lines, together with its extension to curves of higher genus; components (1)–(3) concern the same morphism $\Omega$ and algebra $\imath\widetilde{\mathcal{H}}(\mathbb{X}_{\mathbf{k}})$ and are jointly the announced sequel programme of Lu–Ruan, while (4) is the direction listed in the same sources of carrying the realization paradigm beyond genus zero. Components (1)–(4) are independently answerable and jointly constitute this objective.

References

1. Ming Lu, Shiquan Ruan, $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs, Representation Theory 28 (2024) 112–188, DOI 10.1090/ert/669, https://doi.org/10.1090/ert/669 (arXiv:2110.02575, https://arxiv.org/abs/2110.02575)
2. Ming Lu, Shiquan Ruan, $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity, Mathematische Zeitschrift 307 (2024), article 53, DOI 10.1007/s00209-024-03528-2, https://doi.org/10.1007/s00209-024-03528-2 (arXiv:2402.03322, https://arxiv.org/abs/2402.03322)
3. Ming Lu, Shiquan Ruan, $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type, arXiv:2411.13078 (2024), DOI 10.48550/arXiv.2411.13078, https://arxiv.org/abs/2411.13078
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