Two-dimensional categorified Hall algebras

Porta, Mauro; Sala, Francesco

doi:10.4171/jems/1303

Cited by 20 publications

(10 citation statements)

References 34 publications

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“…Equality ( 39) is a straightforward computation involving pushforward and pullback morphisms; we will explain the main idea of the proof, and leave the details as an exercise to the interested reader (see [32,Proposition 2.7] for a closely related computation). Consider the following diagram (40) where 𝔸 𝑛 2 −𝑛+1 and 𝔸 2𝑛 2 are the affine spaces of coordinates of the matrices 𝑋 and 𝑌 in (37) and (27), respectively, and the maps 𝑗 ′ and 𝑗 denote the respective closed embeddings of 𝑈 and 𝑉 into these affine spaces. Using (30), we have…”

Section: Constructing Subalgebrasmentioning

confidence: 99%

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The trace of the affine Hecke category

Gorsky

Neguţ

2023

Proceedings of London Math Soc

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We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an “affine” version of the construction of Gorsky et al. (Int. Math. Res. Not. IMRN 2022 (2022) 11304–11400). Explicitly, we show that the aforementioned trace is generated by the objects Ed=Tr(Y1d1⋯YndnT1⋯Tn−1)$E_{\mathbf {d}} = {\rm Tr}(Y_1^{d_1} \dots Y_n^{d_n} T_1 \dots T_{n-1})$ as boldd=(d1,⋯,dn)∈Zn$\mathbf {d}= (d_1,\dots ,d_n) \in \mathbb {Z}^n$, where Yi$Y_i$ denote the Wakimoto objects of Elias and Ti$T_i$ denote Rouquier complexes. We compute certain categorical commutators between the Eboldd$E_{\mathbf {d}}$'s and show that they match the categorical commutators between the sheaves scriptEboldd$\mathcal {E}_{\mathbf {d}}$ on the flag commuting stack that were considered in Neguț (Publ. Math. Inst. Hautes Études Sci. 135 (2022) 337–418). At the level of K$K$‐theory, these commutators yield a certain integral form scriptA∼$\widetilde{\mathcal {A}}$ of the elliptic Hall algebra, which we can thus map to the K$K$‐theory of the trace of the affine Hecke category.

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Section: Constructing Subalgebrasmentioning

confidence: 99%

“…where the induction is necessary because we consider 𝐵 𝑛 (instead of 𝐺𝐿 𝑛 ) equivariant 𝐾-theory on the bottom line of (40). By [11,Lemma 5.4.9], we have for any class 𝛼:…”

Section: Constructing Subalgebrasmentioning

confidence: 99%

The trace of the affine Hecke category

Gorsky

Neguţ

2023

Proceedings of London Math Soc

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“…which is a special case of the product of categorical Hall algebras for surfaces defined by Porta-Sala [PS23].…”

Section: Complexes In Quasi-bps Categoriesmentioning

confidence: 99%

“…There exist evaluation morphisms where p is proper and q is quasi-smooth. The above diagram for defines the categorical Hall product which is a special case of the product of categorical Hall algebras for surfaces defined by Porta–Sala [PS23].…”

Section: Preliminariesmentioning

confidence: 99%

Categorical and K-theoretic Donaldson–Thomas theory of (part II)

Pădurariu,

Toda

2023

Forum of Mathematics, Sigma

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Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory.We first prove a categorical analogue of Davison’s support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three-dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight.We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one-dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three-dimensional affine space.

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“…Let X be one of the derived stacks above. As shown in [PS22], the dg-category 3 Coh b (X) has the structure of an E 1 -monoidal dg-category (a categorified Hall algebra of X) which induces, after passing to K-theory, a structure of an associative algebra on G 0 (X) (a K-theoretical Hall algebra of X). By using the construction of Borel-Moore homology for higher stacks developed in [KV19], one can show that H BM * (X) has the structure of an associative algebra (a cohomological Hall algebra of X).…”

Section: Introductionmentioning

confidence: 99%

Untitled

2023

Represent. Theory

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Let π : Y → X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of ωsemistable properly supported sheaves on Y with fixed slope μ and ζ-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on ζ depending on the polarization ω and μ. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, Ktheoretical and categorified Hall algebra of ω-semistable properly supported sheaves on Y with fixed slope μ: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.

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Two-dimensional categorified Hall algebras

Cited by 20 publications

References 34 publications

The trace of the affine Hecke category

The trace of the affine Hecke category

Categorical and K-theoretic Donaldson–Thomas theory of (part II)

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