PBW bases and KLR algebras
Abstract
We generalize Lusztig's geometric construction of the PBW bases of finite quantum groups of type 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 case. In addition, we verify Kashiwara's problem on the finiteness of the global dimensions of the KLR-algebras of type .
Cite
@article{arxiv.1203.5254,
title = {PBW bases and KLR algebras},
author = {Syu Kato},
journal= {arXiv preprint arXiv:1203.5254},
year = {2017}
}
Comments
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