BPS Lie algebras and the less perverse filtration on the preprojective CoHA
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866917647329787904 |
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| author | Davison, Ben |
| author_facet | Davison, Ben |
| contents | The affinization morphism for the stack $\mathfrak{M}(Π_Q)$ of representations of a preprojective algebra $Π_Q$ is a local model for the morphism from the stack of objects in a general 2-Calabi-Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem.
We introduce a new perverse filtration on the Borel-Moore homology of $\mathfrak{M}(Π_Q)$, using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel-Moore homology of $\mathfrak{M}(Π_Q)$ is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra $\mathfrak{g}_{Π_Q}$. This Lie algebra is defined via the Kontsevich-Soibelman theory of critical cohomological Hall algebras for 3-Calabi-Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of $Π_Q$-modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable $Π_Q$-modules provide "cuspidal cohomology" - a conjecturally complete subspace of canonical generators for $\mathfrak{g}_{Π_Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_03289 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | BPS Lie algebras and the less perverse filtration on the preprojective CoHA Davison, Ben Representation Theory High Energy Physics - Theory Algebraic Geometry Quantum Algebra 81R10 The affinization morphism for the stack $\mathfrak{M}(Π_Q)$ of representations of a preprojective algebra $Π_Q$ is a local model for the morphism from the stack of objects in a general 2-Calabi-Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem. We introduce a new perverse filtration on the Borel-Moore homology of $\mathfrak{M}(Π_Q)$, using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel-Moore homology of $\mathfrak{M}(Π_Q)$ is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra $\mathfrak{g}_{Π_Q}$. This Lie algebra is defined via the Kontsevich-Soibelman theory of critical cohomological Hall algebras for 3-Calabi-Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of $Π_Q$-modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable $Π_Q$-modules provide "cuspidal cohomology" - a conjecturally complete subspace of canonical generators for $\mathfrak{g}_{Π_Q}$. |
| title | BPS Lie algebras and the less perverse filtration on the preprojective CoHA |
| topic | Representation Theory High Energy Physics - Theory Algebraic Geometry Quantum Algebra 81R10 |
| url | https://arxiv.org/abs/2007.03289 |