KLR-Schur algebra of coherent sheaves on the projective line: Tilting and PBW bases

Abstract

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 $Co{h}_{r,d}({\mathbb{P}}^1)$. 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.

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