Hopf structures and Drinfeld doubles for $0$-affine quantum groups, and their extension beyond type $A$

The positive part of Arkhipov–Mazin's $0$ -affine quantum group $\mathfrak{U}_n$ (a $q=0$ degeneration of the quantum affine algebra of $\mathfrak{gl}_n$ ) was recently realized by Hsu as the K-theoretic Hall algebra $\mathrm{KHA}_{A_{n-1}}$ of the type $A$ Dynkin quiver, with induced categorical actions and semiorthogonal decompositions on flag varieties. The source paper explicitly records two unaddressed directions, which remain open: (1) no coproduct or Hopf algebra structure is known on $\mathfrak{U}_n$ or on the shifted algebra $\dot{\mathcal{U}}$ , and the Drinfeld double construction has not been applied to recover the full algebra from its positive half; (2) analogues of the $0$ -affine algebra exist only in type $A$ , so neither the KHA realization nor the categorical actions has a counterpart for other Dynkin quivers. The problem asks for these missing structures and this generality; known generic- $q$ blueprints (Pădurariu's Hopf structures on K-theoretic Hall algebras; Zhu's identification of the Drinfeld double of preprojective KHAs with Maulik–Okounkov quantum loop groups) make the $q=0$ case a well-posed expected degeneration rather than a vague wish.

Background

Hall algebras realize quantum groups through extension data of quiver representations: a classical theorem of Ringel identifies the Hall algebra of representations of a Dynkin quiver $Q$ (a quiver whose underlying graph is of type $A$ , $D$ , or $E$ ) over a finite field with the positive part of the quantum group of the Lie algebra attached to $Q$ . The geometric upgrade replaces counting over finite fields by cohomology or K-theory of moduli stacks. For a quiver $Q$ , the K-theoretic Hall algebra $\mathrm{KHA}_Q$ is the direct sum of the Grothendieck groups of the moduli stacks of representations of $Q$ of each dimension vector, equipped with an associative convolution product that is computed by an explicit shuffle-type formula on Laurent polynomials; Pădurariu introduced it, more generally for quivers with potential (a quiver equipped with a formal linear combination of oriented cycles), as the Grothendieck group of a monoidal category of singularities and established Hopf algebra structures and Poincaré–Birkhoff–Witt (PBW)-type theorems in the preprojective (cotangent-stack) setting.

In this landscape, Arkhipov and Mazin introduced the $0$ -affine quantum group $\mathfrak{U}_n=\boldsymbol{U}_0(\mathfrak{gl}_n[t,t^{-1}])$ : the algebra on Drinfeld-style generators $e_{i,r},f_{i,r}$ ( $1\le i\le n-1$ , $r\in\mathbb{Z}$ ) together with commuting elements $h_{n,r}$ , subject to quadratic sign-twisted relations; it is the $q=0$ degeneration of the quantum affine algebra $\boldsymbol{U}_q(\widehat{\mathfrak{gl}_n})$ of the loop algebra of $\mathfrak{gl}_n$ . They proved that $\mathfrak{U}_n$ surjects onto each affine $0$ -Schur algebra, i.e. the convolution algebra on the $\mathrm{GL}_N(\mathbb{C})$ -equivariant K-theory of pairs of $n$ -step partial flag varieties, so $\mathfrak{U}_n$ acts on the equivariant K-theory of partial flag varieties. Hsu then defined the shifted $0$ -affine algebra $\dot{\mathcal{U}}$ , an idempotent modification of $\mathfrak{U}_n$ with only finitely many generators and relations, and constructed categorical $\dot{\mathcal{U}}$ -actions (functorial actions on graded triangulated categories lifting the K-theory-level action) on bounded derived categories of coherent sheaves on partial flag varieties.

Hsu (2025) proved the Hall-type realization at $q=0$ : an algebra isomorphism $\mathfrak{U}_n^+\cong\mathrm{KHA}_{A_{n-1}}$ from the positive part (the subalgebra generated by the $e_{i,r}$ ) to the K-theoretic Hall algebra of the type $A_{n-1}$ Dynkin quiver, sending $e_{i,r}\mapsto x^{-r}_{i,1}$ in the shuffle presentation; it defined categorical $\mathfrak{U}_n^+$ -actions and showed they induce semiorthogonal decompositions (a decomposition of a triangulated category into simpler pieces linked by orthogonality conditions) of the weight categories, with the derived category of $n$ -step partial flag varieties as the main example.

Two structural pieces are conspicuously absent, and the paper records both in its Further remarks. First, in neither the Arkhipov–Mazin paper nor Hsu's shifted-algebra paper were coproducts (algebra homomorphisms $A\to A\otimes A$ describing how modules tensor) or Hopf algebra structures on $\mathfrak{U}_n$ and $\dot{\mathcal{U}}$ investigated, and the Drinfeld double — the standard construction that, from a Hopf half with a nondegenerate pairing, produces a Hopf algebra containing the half and its dual, classically recovering the full quantum group from its positive and negative parts — has not been applied, so the realization currently reaches only the positive part and not the full algebra. Second, everything is type $A$ : $0$ -affine algebras have been defined only for $\mathfrak{gl}_n$ , while the author expects the argument to extend to any Dynkin quiver once appropriate analogues of the $0$ -affine algebra are defined. For comparison, in the generic- $q$ world the parallel statements exist: Pădurariu proved that K-theoretic Hall algebras of symmetric quivers with potential carry Hopf structures, and Zhu (2025) identified the Drinfeld double of the localized preprojective KHA with the localized Maulik–Okounkov quantum loop group for arbitrary quiver type. The $q=0$ counterpart of that blueprint is exactly what is missing.

Problem Statement

Let $\mathfrak{U}_n=\boldsymbol{U}_0(\mathfrak{gl}_n[t,t^{-1}])$ be Arkhipov–Mazin's $0$ -affine quantum group with generators $e_{i,r},f_{i,r},h_{n,r}$ and relations as published (reproduced as Definition 3.1 of the source paper), $\mathfrak{U}_n^+$ its positive part, and $\dot{\mathcal{U}}$ the shifted $0$ -affine algebra. What is known is the type- $A$ positive-part statement: $\mathfrak{U}_n^+\cong\mathrm{KHA}_{A_{n-1}}$ , with categorical $\mathfrak{U}_n^+$ -actions inducing semiorthogonal decompositions of weight categories. The open problem, exactly as recorded in the source's Further remarks, comprises two components serving one objective — turning the isolated type- $A$ positive-part statement into a full theory of $0$ -affine quantum groups:

(1) Hopf structure and Drinfeld double. Introduce natural coproducts (bialgebra or Hopf algebra structures) on $\mathfrak{U}_n$ and on $\dot{\mathcal{U}}$ , and apply the Drinfeld double construction to the positive half so as to recover the full algebra: an explicit isomorphism between the resulting double and $\mathfrak{U}_n$ (respectively the corresponding idempotent form on the shifted side).

(2) Beyond type $A$ . Define appropriate analogues of the $0$ -affine algebra for Dynkin quivers of type other than $A$ — the source expects the argument to extend to any Dynkin quiver once such analogues are defined — and extend to them the K-theoretic Hall algebra realization and the construction of categorical $U^+$ -actions; as the source anticipates, such categorical actions could then lead to semiorthogonal decompositions of derived categories of moduli spaces of (framed) quiver representations.

The two components are related but independent: component (1) supplies the Hopf-theoretic and tensor-product structure missing at $q=0$ in type $A$ ; component (2) supplies the Dynkin generality. Each component is posed in the generality stated: no restriction to particular ranks, degrees, or a single additional Dynkin type is part of the problem.

References

You-Hung Hsu, $0$ -affine quantum groups as K-theoretic Hall algebras, arXiv:2512.08272 (2025). https://arxiv.org/abs/2512.08272
Sergey Arkhipov, Mikhail Mazin, Equivariant K-theory of the variety of partial flags, Transactions of the American Mathematical Society 379 (2025), 1453-1504, DOI 10.1090/tran/9550; arXiv:2205.05184. https://doi.org/10.1090/tran/9550
You-Hung Hsu, A categorical action of the shifted $0$ -affine algebra, Forum of Mathematics, Sigma 13 (2025), DOI 10.1017/fms.2025.17; arXiv:2009.03579. https://doi.org/10.1017/fms.2025.17
You-Hung Hsu, Categorical idempotents via shifted 0-affine algebras, Algebras and Representation Theory 27 (2024), 1735-1772, DOI 10.1007/s10468-024-10279-5; arXiv:2112.13062. https://doi.org/10.1007/s10468-024-10279-5
You-Hung Hsu, Semiorthogonal decomposition via categorical action, Journal of Algebra 638 (2024), 396-440, DOI 10.1016/j.jalgebra.2023.08.041; arXiv:2108.13008. https://doi.org/10.1016/j.jalgebra.2023.08.041
Tudor Pădurariu, K-Theoretic Hall Algebras of Quivers with Potential as Hopf Algebras, International Mathematics Research Notices 2023 (7), 5781-5815, DOI 10.1093/imrn/rnac013. https://doi.org/10.1093/imrn/rnac013
Tianqing Zhu, Maulik-Okounkov quantum loop groups and Drinfeld double of preprojective $K$ -theoretic Hall algebras, arXiv:2511.02161 (2025; v3 revised January 2026). https://arxiv.org/abs/2511.02161
Woo-Seok Jung, Young-Tak Oh, Representation Theory of $0$ -Schur Algebras and Related Categories, arXiv:2509.26194 (2025). https://arxiv.org/abs/2509.26194
Claus Michael Ringel, Hall algebras and quantum groups, Inventiones Mathematicae 101 (1990), 583-591, DOI 10.1007/BF01231516. https://doi.org/10.1007/BF01231516

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Hopf structures and Drinfeld doubles for $0$ -affine quantum groups, and their extension beyond type $A$

Background

Problem Statement

References