Completing the geometrization of quantum groups and dual semicanonical bases: multiple arrows, product structure on the almost-equal quotient, and canonical q-normalizations

Bi (arXiv:2404.00663, 2024) geometrized the semicanonical basis of the positive part $U_q(\mathfrak{n})$ of a quantum group as the basis of constant sheaves of irreducible components in Grothendieck groups of equivariant derived categories of Lusztig nilpotent varieties, and described the dual basis via Serre polynomials of quiver Grassmannians — but only for quivers with at most one arrow between any two vertices, with an algebra structure obtained only after quotienting by an 'almost equal' relation on which the induction product provably does not descend, and with shuffle-algebra images fixed only up to overall q-powers. The problem is to complete this geometrization: extend the construction — including the almost-equal quotient and its word-concatenation product — to quivers with multiple arrows, fix canonical q-power normalizations. No work addressing these gaps is known in the literature to date.

Background

Let $Q$ be a quiver (a directed graph). Its double adds an opposite arrow $\bar{\alpha}$ for each arrow $\alpha$ ; the preprojective algebra $\Lambda_Q$ is the quotient of the path algebra of the double by the relations $\sum_{\alpha}(\alpha\bar{\alpha}-\bar{\alpha}\alpha)$ . For a dimension vector $\alpha$ , the Lusztig nilpotent variety $\Lambda_\alpha$ is the closed subvariety of the representation variety of $\Lambda_Q$ consisting of nilpotent representations; these varieties, introduced by Lusztig (1991), are the geometric foundation of the canonical basis (Lusztig's distinguished basis of the positive part of the quantized enveloping algebra, built from perverse sheaves) and of the semicanonical basis. The semicanonical basis of the positive part $U(\mathfrak{n})$ of the enveloping algebra, constructed by Lusztig (2000), is indexed by irreducible components of the $\Lambda_\alpha$ : each component $Z$ defines a linear form $f_Z$ on $U(\mathfrak{n})$ via Euler characteristics of the fibres of the projection of the variety of flags of representations onto $Z$ , and the $f_Z$ form a basis. Geiss–Leclerc–Schröer (2005) recast this basis and its dual semicanonical basis (a basis of the coordinate ring $\mathbb{C}[N]$ of the maximal unipotent subgroup $N$ of the Kac–Moody group) in terms of modules over preprojective algebras, connecting it to cluster algebras; in particular, Geiss–Leclerc–Schröer (2013) showed that quantum cluster monomials in certain quantum coordinate rings $A_q(\mathfrak{n}(w))$ (each a quantization of the coordinate ring of a subvariety of the unipotent group indexed by an element $w$ of the Weyl group, the reflection group of the root system) belong to a dual semicanonical-type basis, and the conjecture that they belong to the dual canonical basis was proved by Qin (2017) for symmetrizable Kac–Moody types and by Kang–Kashiwara–Kim–Oh (2018) for symmetric types via monoidal categorification — that is, by realizing the algebra as the Grothendieck ring of a monoidal category of modules over a quiver Hecke algebra (a diagrammatic algebra categorifying the negative half of the quantum group, introduced by Khovanov–Lauda and Rouquier), with basis elements corresponding to classes of simple modules. The quantum analogue of the semicanonical basis itself, however, was long missing. Bi (2024) addresses this: writing $D^b_{G_\alpha}(\Lambda_\alpha)$ for the equivariant bounded derived category of constructible sheaves on $\Lambda_\alpha$ (the triangulated category of bounded complexes of sheaves whose cohomology is locally constant on the strata of an algebraic stratification; with $G_\alpha=\prod_i GL_{\alpha_i}$ acting by change of basis) and $K(\Lambda_Q)$ for the direct sum of its Grothendieck groups, an induction functor between these categories (a version of Lusztig's convolution) defines a product $m$ on $K(\Lambda_Q)$ . Because a Serre-type relation acquires an unwanted lower-order term, Bi quotients $K(\Lambda_Q)$ by the almost equal relation: two complexes are identified when they agree, after restriction, on a dense open subset of each irreducible component; the resulting quotient is denoted $\widetilde{K}(\Lambda_Q)$ , and Bi's Remark 2.6 shows that the induction product does not descend to it. Instead, on the subalgebra $\widetilde{K}^{nil}(\Lambda_Q)$ of $\widetilde{K}(\Lambda_Q)$ generated by the images of the constant sheaves of the $\Lambda_{\alpha_i}$ , equipped with a separately defined word-concatenation product, Bi proves (Theorem 1.1) a $\mathbb{C}(q)$ -algebra isomorphism with $U_q(\mathfrak{n}_Q)$ under which constant sheaves of irreducible components give the quantum semicanonical basis. Dually (Theorems 1.2, 4.1), the images of irreducible components in Leclerc's (2004) embedding of $A_q(\mathfrak{n}_Q)$ into a quantum shuffle algebra $\mathbb{C}(q)\langle Q_0\rangle$ — the shuffle algebra, a q-deformed algebra of words in the vertex alphabet — are expressed as $\delta_Z=\sum_{\mathbf{i}} q^{-d_{\mathbf{i}}}\,\chi_q(\mathrm{Gr}_{\mathbf{i}}(z))\,\mathbf{i}$ , where $z$ is a generic point of $Z$ , $\mathrm{Gr}_{\mathbf{i}}(z)$ is the quiver Grassmannian of subrepresentations of type $\mathbf{i}$ of the representation $z$ , $\chi_q$ is its Serre polynomial (the graded Poincaré polynomial whose value at $q=1$ is the Euler characteristic), and $d_{\mathbf{i}}=\dim\Lambda_{\mathbf{i},\alpha}$ — but only up to an overall q-power, and only for quivers without multiple arrows. For context, in the classical ( $q=1$ ) setting Lusztig's original geometric construction handles symmetric Kac–Moody types with multiple arrows, and Geiss–Leclerc–Schröer (2017) generalized nilpotent varieties and semicanonical functions to symmetrizable types conjecturally, so the single-arrow hypothesis is a genuine limitation of the quantum derived-category model rather than of the general program.

Problem Statement

Complete the geometrization of quantum groups and dual semicanonical bases initiated in Bi (arXiv:2404.00663, 2024), by removing its standing limitations within one coherent objective — turning the partial, hypothesis-limited model into a canonical structural theory of the quantum semicanonical basis $\mathcal{S}_q$ and its dual. Concretely, with the common goal that each requested result feeds the same Grothendieck-group geometrization, determine: (a) whether the algebra isomorphism $\widetilde{K}^{nil}(\Lambda_Q)\cong U_q(\mathfrak{n}_Q)$ with irreducible-component semicanonical basis (Theorem 1.1) and the Serre-polynomial description of the dual basis $\delta_Z=\sum_{\mathbf{i}}q^{-d_{\mathbf{i}}}\chi_q(\mathrm{Gr}_{\mathbf{i}}(z))\mathbf{i}$ (Theorems 1.2, 4.1) extend to quivers with multiple arrows between vertices, or what modifies in their place; (b) how the product structure on the almost-equal quotient extends to that setting: the induction product on $K(\Lambda_Q)$ provably does not descend to $\widetilde{K}(\Lambda_Q)$ — Bi's Remark 2.6 (Remark 2.7 in v1) shows the naive descent fails already in the single-arrow case via $I_\beta\star I_0\not\sim 0$ — and Bi instead equips the subalgebra $\widetilde{K}^{nil}(\Lambda_Q)$ with a word-concatenation product; determine whether this quotient-and-concatenation construction, and the resulting realization of $U_q(\mathfrak{n}_Q)$ , extends to quivers with multiple arrows, or what modifies in its place; (c) canonical q-power normalizations of the shuffle-algebra images $\delta_Z$ , replacing the current 'up to some q-powers' indeterminacy, compatible with the algebra structure; and (d) whether the dual semicanonical basis elements $\delta_Z$ can be identified, up to the normalizations of (c), with q-character images of simple modules over quiver Hecke algebras inside Leclerc's shuffle-algebra embedding, which would connect the geometric construction to the monoidal categorifications of Kang–Kashiwara–Kim–Oh (2018). Facets (a)–(c) are the explicitly acknowledged limitations of the cited work, taken together as the remaining gap. The comparison in (d) is motivated in the paper's introduction (a hope in v1, dropped in v2).

References

Yingjin Bi, "Geometrizations of quantum groups and dual semicanonical bases", arXiv:2404.00663 (2024), https://arxiv.org/abs/2404.00663
George Lusztig, "Quivers, perverse sheaves, and quantized enveloping algebras", Journal of the American Mathematical Society 4 (1991), DOI 10.1090/s0894-0347-1991-1088333-2, https://doi.org/10.1090/s0894-0347-1991-1088333-2
George Lusztig, "Semicanonical bases arising from enveloping algebras", Advances in Mathematics 151 (2000), DOI 10.1006/aima.1999.1873, https://doi.org/10.1006/aima.1999.1873
Christof Geiss, Bernard Leclerc, Jan Schröer, "Semicanonical bases and preprojective algebras", Annales Scientifiques de l'École Normale Supérieure 38 (2005), DOI 10.1016/j.ansens.2004.12.001, https://doi.org/10.1016/j.ansens.2004.12.001
Bernard Leclerc, "Dual canonical bases, quantum shuffles and q-characters", Mathematische Zeitschrift 246 (2004), DOI 10.1007/s00209-003-0609-9, https://doi.org/10.1007/s00209-003-0609-9
Christof Geiß, Bernard Leclerc, Jan Schröer, "Cluster structures on quantum coordinate rings", Selecta Mathematica 19 (2013), DOI 10.1007/s00029-012-0099-x, https://doi.org/10.1007/s00029-012-0099-x
Fan Qin, "Triangular bases in quantum cluster algebras and monoidal categorification conjectures", Duke Mathematical Journal 166 (2017), DOI 10.1215/00127094-2017-0006, https://doi.org/10.1215/00127094-2017-0006
Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, Se-jin Oh, "Monoidal categorification of cluster algebras", Journal of the American Mathematical Society 31 (2018), DOI 10.1090/jams/895, https://doi.org/10.1090/jams/895
Christof Geiß, Bernard Leclerc, Jan Schröer, "Quivers with relations for symmetrizable Cartan matrices IV: Crystal graphs and semicanonical functions", arXiv:1702.07570 (2017), https://arxiv.org/abs/1702.07570
Yingjin Bi, "Monoidal categorification on open Richardson varieties", arXiv:2409.04715 (2024), https://arxiv.org/abs/2409.04715
Yingjin Bi, "On cluster structures of bosonic extensions", arXiv:2506.00882 (2025), https://arxiv.org/abs/2506.00882
Yingjin Bi, "Towards Monoidal Categorifications of Twisted Products of Flag Varieties", arXiv:2602.11559 (2026), https://arxiv.org/abs/2602.11559
Yingjin Bi, "Monoidal Categories associated with Kac-Moody Open Richardson Varieties in Symmetric Type", arXiv:2606.05184 (2026), https://arxiv.org/abs/2606.05184