PBW bases and KLR algebras

Abstract

We generalize Lusztig's geometric construction of the PBW bases of finite quantum groups of type $\mathsf{ADE}$ under the framework of [Varagnolo-Vasserot, J. reine angew. Math. 659 (2011)]. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the KLR-algebras. This enables us to prove Lusztig's conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases in the $\mathsf{ADE}$ case. In addition, we verify Kashiwara's problem on the finiteness of the global dimensions of the KLR-algebras of type $\mathsf{ADE}$.

35pp, v3: separate out arXiv:1207.4640 in order to stress the Shoji conjecture and "affine quasi-hereditary categories". v4: properties of the Saito reflection functors gathered (Theorem 3.6), v5: Modified arguments to correct an error in the proof of the desired property of the PBW basis in section 4. The last page contains more detailed description

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