Perverse-sheaf realization of Bridgeland's Hall algebra beyond Dynkin quivers and comparison of its IC basis with canonical bases

Bridgeland (2013) realized the whole quantized enveloping algebra $\mathbf{U}_v$ of a Dynkin Lie algebra as a localized, reduced Hall algebra of two-periodic projective complexes of quiver representations, and asked for a Lusztig-style categorification of this algebra by perverse sheaves. Fang, Lan and Xiao (arXiv:2303.04993) recently gave such a categorification for Dynkin quivers: via induction and restriction functors on equivariant mixed constructible sheaves on varieties parametrizing two-periodic complexes, they obtained a Grothendieck-group algebra realizing Bridgeland's Hall algebra, a semisimple-complex coalgebra, and a bar-invariant positive integral basis of IC (intersection cohomology) type, which they compare with Lusztig's canonical basis only for rigid modules. The open problem is to extend this sheaf-theoretic realization and its positive integral basis beyond the Dynkin case, and to advance the comparison with Lusztig's canonical basis beyond rigid modules — a precise, untouched core.

Background

For a finite quiver $Q$ over a finite field $\mathbb{F}_q$ , the Ringel–Hall algebra has as a basis the isomorphism classes of finite-dimensional representations of $Q$ , with multiplication defined by counting extensions; by Ringel's theorem it realizes the positive half $\mathbf{U}_v^+$ of the quantized enveloping algebra of the Kac–Moody Lie algebra attached to the graph of $Q$ (a Dynkin diagram $A_n$ , $D_n$ , $E_6$ , $E_7$ , $E_8$ gives a simple Lie algebra; affine and wild graphs give infinite-dimensional Kac–Moody algebras). Lusztig categorified this picture: using induction and restriction functors on equivariant perverse sheaves on the representation varieties of $Q$ (perverse sheaves being a category of constructible complexes of sheaves adapted to stratified varieties, in which the intersection cohomology complexes naturally live), he showed that a suitable Grothendieck group is isomorphic to the integral form of $\mathbf{U}_v^+$ , and the classes of simple perverse sheaves of the form $\mathrm{IC}(\mathcal{O})$ — the intersection cohomology complex attached to an orbit $\mathcal{O}$ of the natural group action — form the canonical basis, a bar-invariant positive basis central to Lie theory. Bridgeland (2013) then realized the whole quantized enveloping algebra: he considered the category $C_2(\mathrm{proj})$ of two-periodic projective complexes, meaning complexes of projective representations whose differential repeats with period 2, formed its Hall algebra twisted by the Euler form (the bilinear form on the Grothendieck group given by the alternating Euler characteristics of the extension groups), localized at the classes of the contractible complexes $K_P$ (the total acyclic two-periodic complexes attached to each projective representation $P$ ) and their shifts, and reduced by a relation identifying the two; for Dynkin $Q$ he proved the resulting algebra $DH_{\mathrm{red}}$ is isomorphic to the full quantized enveloping algebra $\mathbf{U}_v$ , and for general $Q$ his construction yields a Drinfeld-double-type algebra of the positive half, the Drinfeld double being the canonical Hopf algebra obtained by pairing a Hopf algebra with its dual, which here combines the positive and negative halves of the quantum group. Bridgeland explicitly asked whether his Hall algebra admits a Lusztig-style categorification by perverse sheaves; Qin (2016) gave one for Dynkin $Q$ using perverse sheaves on cyclic quiver varieties (varieties of representations of a framed cyclic quiver), obtaining a basis that contains the dual canonical basis up to scalars, and Scherotzke–Sibilla (2016) gave a related geometric construction using Nakajima quiver varieties. Fang–Lan–Xiao (arXiv:2303.04993) answered Bridgeland's question directly for Dynkin $Q$ : they introduced affine varieties $C_{\underline{e}}$ with group actions parametrizing two-periodic projective complexes, defined induction and restriction functors on equivariant derived categories of mixed Weil $\ell$ -adic sheaves, and proved through the sheaf–function correspondence (which matches a Weil sheaf with the trace function on $\mathbb{F}_q$ -points) that the resulting Grothendieck-group algebra realizes the twisted Hall algebra of $C_2(\mathrm{proj})$ and, after localization and reduction, Bridgeland's algebra. Dually, the Grothendieck group of the semisimple complexes carries a coalgebra structure, and the IC classes over the (finitely many) orbits give a bar-invariant positive integral basis — bar-invariant meaning fixed by the bar involution, the anti-linear involution sending the quantum parameter $v$ to $v^{-1}$ , and positive meaning that its structure constants are Laurent polynomials in $v$ with non-negative integer coefficients — where these properties are established for the IC bases of the Grothendieck group of semisimple complexes and of its graded dual, before localization; after localization and reduction the resulting algebra is identified with a Poisson integral form of $\mathbf{U}_v$ , but bar-invariance is not established at that reduced level even in Dynkin type, since the reduction ideal is not closed under the bar involution. Two features of this work delimit the present problem. First, both the finiteness of orbits on $C_{\underline{e}}$ and the identification with the full quantized enveloping algebra use the Dynkin hypothesis essentially: for affine or wild quivers the representation varieties of two-periodic complexes carry infinitely many orbits, and the Dynkin machinery breaks down. Second, the constructed IC basis behaves markedly differently from Lusztig's canonical basis — for example, the constant sheaf on the orbit of $C_{S_i}$ , the minimal projective resolution of a simple module $S_i$ , which plays in this construction the role of the Chevalley generator $E_i$ , is not an IC sheaf and does not give a basis element — and the comparison with the canonical basis is established only for rigid modules, that is, modules $M$ with $\mathrm{Ext}^1(M,M)=0$ , where the relevant orbits are maximal.

Problem Statement

Extend the Lusztig-style sheaf-theoretic categorification of Bridgeland's Hall algebra beyond Dynkin type, and determine the relationship of its IC-type basis with Lusztig's canonical and dual canonical bases. Concretely, for every quiver $Q$ that is not of Dynkin type (affine, such as the Kronecker quiver, or wild), construct moduli varieties or stacks parametrizing the two-periodic complexes of projective representations of $Q$ with compatible algebraic group actions, together with induction and restriction functors on the equivariant derived categories of mixed constructible sheaves, such that the resulting Grothendieck-group algebra realizes the twisted Hall algebra of two-periodic complexes and, after localization at the classes of the contractible complexes and the reduced quotient, realizes Bridgeland's Hall algebra $DH_{\mathrm{red}}$ (recovering, in general type, its Drinfeld-double-type description). The construction should include the dual semisimple-complex coalgebra and produce a bar-invariant positive integral basis of IC type generalizing the Dynkin basis of Fang–Lan–Xiao, where bar-invariance and positivity are required at the level at which they are established in the Dynkin case — for the IC basis of the Grothendieck group of semisimple complexes and its graded dual, before localization — since at the localized reduced level these properties remain open even for Dynkin quivers, the reduction ideal not being closed under the bar involution. As the second, linked component of the same objective, determine the precise relationship between these IC-type bases — already in Dynkin type — and Lusztig's canonical and dual canonical bases, beyond the currently known agreement on rigid modules and maximal orbits. These two linked components support one scientific objective: a geometric categorification of the derived-category realization of quantum groups valid for general quivers, with a well-understood global basis.

References

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