An algebraic study of extension algebras

Abstract

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 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 [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 C ∞ -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 to establish Kashwara's problem and Lusztig's conjecture. * This is a revised and expanded version of the first part of the paper "PBW bases and KLR algebras" arXiv:1203.5254. We divided it into two pieces since we learned that this part were invisible in the previous version of the above mentioned paper.

Affine highest weight categories and affine quasihereditary algebras

“…As explained in [24], geometric extension algebras arising from affine Hecke algebras of types A and BC, the Khovanov-Lauda-Rouquier algebras (over C) of finite ADE types, the quiver Schur algebras, and the algebra which governs the BGG category all fit into this class. It is proved in [24] that the category A-mod is a P-highest weight category, see [24, Theorem C, Lemma 1.3, proof of Corollary 3.3]. So A is P-quaihereditary.…”

“…It is proved in [21] that the category A-mod is a P-highest weight category, see [21, Theorem C, Lemma 1.3, proof of Corollary 3.3]. So A is P-quasihereditary.…”

Geometric realizations of Lusztig's symmetries

“…In [4,5], Kato gave the categorification of the PBW-type bases of quantum groups of finite type. He constructed some modules (which are called standard modules) of the KLR algebras R ν and proved that there standard modules can categorify the PBW-type basis of f by using the geometric realizations of R ν given by Varagnolo, Vasserot and Rouquier.…”

“…Consider two I-graded vector spaces V and V ′ such that dimV ′ = s i (dimV). In the case of finite type, Kato introduced an equivalenceω i : i Q V,Q → i Q V ′ ,Q ′ and studied the properties of this equivalence in [4,5]. In this paper, we generalize his construction to all cases and prove that the map induced byω i realizes the Lusztig's symmetry T i : i f → i f. For the proof of the result, we shall study the relations between the map induced bỹ ω i and the Hall algebra approach to T i in [15].…”

“…In the case of finite type, Kato proved that the projective dimension of any standard module is finite ( [4,5]). Theorem 5.4 show that the projective dimensions of a kind of standard modules are also finite in the general case.…”