Cohomological Hall algebras, their categorification, and their representations via torsion pairs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| author | Diaconescu, Duiliu-Emanuel Porta, Mauro Sala, Francesco |
| author_facet | Diaconescu, Duiliu-Emanuel Porta, Mauro Sala, Francesco |
| contents | In this paper we provide a systematic way of producing representations of cohomological, K-theoretical and categorified Hall algebras, and study the output of our construction in several cases. We thus recover and categorify in a unified framework the action of the COHA of a quiver on the cohomology of Nakajima quiver variety, the action of the COHA of zero-dimensional sheaves on the the cohomology of Hilbert schemes of points and moduli spaces of Gieseker-stable sheaves on smooth surfaces, recovering the constructions of Neguţ and DeHority. We also obtain new examples, associated to Pandharipande-Thomas stable pairs.
Along the way, we explain carefully under which conditions one can associate to a pair $(\mathscr{C},τ)$ consisting of a stable $\infty$-category with a t-structure a COHA. This requires a careful analysis and extension of Khan's theory of motivic Borel-Moore homology to the non quasi-compact setting, and it allows to produce new examples of COHAs arising from Bridgeland's stability conditions.
The representations that we construct take an extra categorical input: that of a torsion pair $(\mathscr{T},\mathscr{F})$ on the heart $\mathscr{C}^\heartsuit$ of $τ$. Under favorable conditions, the homology of the moduli stack associated to $\mathscr{T}$ acquires a Hall multiplication, that acts both on the left and on the right on the homology of the moduli stack associated to $\mathscr{F}$. The left action generalizes and categorifies Nakajima's positive operators, while the right action corresponds to negative operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08926 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cohomological Hall algebras, their categorification, and their representations via torsion pairs Diaconescu, Duiliu-Emanuel Porta, Mauro Sala, Francesco Algebraic Geometry Quantum Algebra Representation Theory 14A20 (Primary), 14A30, 14F08, 17B37 (Secondary) In this paper we provide a systematic way of producing representations of cohomological, K-theoretical and categorified Hall algebras, and study the output of our construction in several cases. We thus recover and categorify in a unified framework the action of the COHA of a quiver on the cohomology of Nakajima quiver variety, the action of the COHA of zero-dimensional sheaves on the the cohomology of Hilbert schemes of points and moduli spaces of Gieseker-stable sheaves on smooth surfaces, recovering the constructions of Neguţ and DeHority. We also obtain new examples, associated to Pandharipande-Thomas stable pairs. Along the way, we explain carefully under which conditions one can associate to a pair $(\mathscr{C},τ)$ consisting of a stable $\infty$-category with a t-structure a COHA. This requires a careful analysis and extension of Khan's theory of motivic Borel-Moore homology to the non quasi-compact setting, and it allows to produce new examples of COHAs arising from Bridgeland's stability conditions. The representations that we construct take an extra categorical input: that of a torsion pair $(\mathscr{T},\mathscr{F})$ on the heart $\mathscr{C}^\heartsuit$ of $τ$. Under favorable conditions, the homology of the moduli stack associated to $\mathscr{T}$ acquires a Hall multiplication, that acts both on the left and on the right on the homology of the moduli stack associated to $\mathscr{F}$. The left action generalizes and categorifies Nakajima's positive operators, while the right action corresponds to negative operators. |
| title | Cohomological Hall algebras, their categorification, and their representations via torsion pairs |
| topic | Algebraic Geometry Quantum Algebra Representation Theory 14A20 (Primary), 14A30, 14F08, 17B37 (Secondary) |
| url | https://arxiv.org/abs/2207.08926 |