KLR-Schur algebra of coherent sheaves on the projective line: Tilting and PBW bases
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| Format: | Preprint |
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2026
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| _version_ | 1866915825930207232 |
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| author | Schiffmann, Olivier Yang, Fang |
| author_facet | Schiffmann, Olivier Yang, Fang |
| contents | We begin the study of Khovanov-Lauda-Rouquier type algebras associated to moduli stacks of coherent sheaves on smooth projective curves. We consider the case of $\mathbb{P}^1$ and define, for any pair $(r,d)$ of a rank and a degree, the KLR and Schur algebras $A_{r,d}, \mathcal{R}_{r,d}$ as suitable convolution algebras in the Borel-Moore homology of an analog of the Steinberg stack built from the stacks $Coh_{r,d}(\mathbb{P}^1)$. We use the tilting equivalence and Bridgeland stability conditions to construct an interpolation between the KLR or Schur algebras of the categories of coherent sheaves on $\mathbb{P}^1$ and the KLR or Schur algebras of the categories of representations of the Kronecker quiver. We also introduce a stratification of the Steinberg stacks into cohomologically pure pieces and use this to construct a PBW basis of the corresponding algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_01265 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | KLR-Schur algebra of coherent sheaves on the projective line: Tilting and PBW bases Schiffmann, Olivier Yang, Fang Representation Theory Algebraic Geometry We begin the study of Khovanov-Lauda-Rouquier type algebras associated to moduli stacks of coherent sheaves on smooth projective curves. We consider the case of $\mathbb{P}^1$ and define, for any pair $(r,d)$ of a rank and a degree, the KLR and Schur algebras $A_{r,d}, \mathcal{R}_{r,d}$ as suitable convolution algebras in the Borel-Moore homology of an analog of the Steinberg stack built from the stacks $Coh_{r,d}(\mathbb{P}^1)$. We use the tilting equivalence and Bridgeland stability conditions to construct an interpolation between the KLR or Schur algebras of the categories of coherent sheaves on $\mathbb{P}^1$ and the KLR or Schur algebras of the categories of representations of the Kronecker quiver. We also introduce a stratification of the Steinberg stacks into cohomologically pure pieces and use this to construct a PBW basis of the corresponding algebra. |
| title | KLR-Schur algebra of coherent sheaves on the projective line: Tilting and PBW bases |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2603.01265 |