Mathematics – Quantum Algebra
Scientific paper
2012-03-23
Mathematics
Quantum Algebra
46pp, v2: corrected the proof of Theorem 3.10, and other minor changes/corrections
Scientific paper
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 base of such quantum groups are proven to yield a orthogonal collection in the module category of 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, and Kashiwara's problem on the finiteness of the global dimensions of KLR-algebras in the $\mathsf{ADE}$ case. To achieve our goal, we develop a general formulation which guarantees nice properties of extension algebras, including a new criteria of purity of weights. (This part also applies to quiver Schur algebras.) In the appendix, we provide a proof of Shoji's conjecture on limit symbols of type $\mathsf{B}$ [Shoji, Adv. Stud. Pure Math. 40 (2004)] based on the general formulation developed in this paper.
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