English

An algebraic study of extension algebras

Representation Theory 2017-07-11 v5 Algebraic Geometry Quantum Algebra

Abstract

We present simple conditions which guarantee a geometric convolution algebra to behave like a variant of the quasi-hereditary algebra. In particular, standard modules of the affine Hecke algebras of type BC\mathsf{BC}, and the quiver Schur algebras are shown to satisfy the Brauer-Humphreys type reciprocity and the semi-orthogonality property. In addition, we present a new criterion of purity of weights in the geometric side. This yields a proof of Shoji's conjecture on limit symbols of type B\mathsf{B} [Shoji, Adv. Stud. Pure Math. 40 (2004)], and the purity of the exotic Springer fibers [K, Duke Math. 148 (2009)]. Using this, we describe the leading terms of the CC^{\infty}-realization of a solution of the Lieb-McGuire system in the appendix. In [K, arXiv:1203.5254], we apply the results of this paper to the KLR algebras of type ADE\mathsf{ADE} to establish Kashwara's problem and Lusztig's conjecture.

Keywords

Cite

@article{arxiv.1207.4640,
  title  = {An algebraic study of extension algebras},
  author = {Syu Kato},
  journal= {arXiv preprint arXiv:1207.4640},
  year   = {2017}
}

Comments

40pp, v1: separated out from arXiv:1203.5254, v2: major revision. title changed. v3: major revision. modified conditions, removed dg-algebra arguments, and Appendix B separated out. v4: minor revision. v5: assumption optimized, and explanation amplified