Homological Description of Crystal Structures on Lusztig’s Quiver Varieties

Abstract

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions are closely related by studying sufficiently generic representations of the preprojective algebra.This research was supported by the Graduiertenkolleg "Global Structures in Geometry und Analysis" and the DFG Priority program Darstellungstheorie 1388.The author is deeply indebted to Volker Genz and Markus Reineke for several valuable and clarifying discussion and comments. Further a big thanks goes to Michael Ehrig for careful proofreading and helpful advise and Yoshiyuki Kimura for his comments on an earlier version of this paper.

Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers

“…Proof of Theorem 4 From Lemma 4.3 and Proposition 4.3 we obtain that the restriction of the crystal isomorphism B g (∞) → B h (∞), X M → M (see [16,Theorem 3.26]) induces an isomorphism of crystals B g (λ) → B h (λ).…”

“…where the second equality follows from the crystal isomorphism in [16,Theorem 3.26]. Conversely, assume that ϕ i (X M ) = 0, i.e.…”

“…we denote this realization by B h (λ). We first give in Theorem 3 a homological description of the stable irreducible components and then make use of the homological description of B g (∞) developed by the second author in [16] and show in Theorem 4 that the embedding of B g (λ) into B g (∞) is compatible with this description when g is of finite type A or D.…”

“…A quiver Q is called cospecial if dim Hom CQ (S(i), M ) ≤ 1 for all i ∈ I and all indecomposable CQ-modules M . Equivalently, this means that each sink i ∈ I corrspond to a miniscule weight of the Lie algebra of Q (see [16,Corollary 2.22]). We call Q special if Q * is cospecial.…”

“…(1) The explicit description of ẽi can be found in [16,Proposition 2.19]. (2) The isomorphism between B h (∞) and B(∞) implies that the module V M is unique.…”