An algebraic study of extension algebras

By Syu Kato. 2012

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We present simple conditions which guarantee a geometric extension algebra to behave like a variant of quasi-hereditary algebras. In particular, standard modules of affine Hecke algebras of type $\sf{BC}$, 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 $\sf{B}$ [T. Shoji, {\it Adv. Stud. Pure Math.} 40 (2004)], and the purity of the exotic Springer fibers [S. Kato, {\it Duke Math. J.} 148 (2009)]. Using this, we describe the leading terms of the $C^{\infty}$-realization of a solution of the Lieb-McGuire system in the appendix. In [S. Kato, {\it Duke Math. J.} 163 (2014)], we apply the results of this paper to the KLR algebras of type $\sf{ADE}$ to establish Kashwara's problem and Lusztig's conjecture.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8ee94c499e4912f745c4c4951b46b5c7ca7977bf

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In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Alexander S. Kleshchev

Citations in the corpus

Affine highest weight categories and affine quasihereditary algebras (2014, background): “S. Kato, ‘An algebraic study of extension algebras’, Preprint, 2012, arXiv:1207.4640v4.”