Quiver grassmannians, quiver varieties and the preprojective algebra

Abstract

Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassmannians which are homeomorphic to the Demazure quiver varieties introduced by the first author, and others which are homeomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians allow us to describe injective objects in the category of locally nilpotent modules of the preprojective algebra. We conclude by relating our construction to a similar one of Lusztig using projectives in place of injectives. In an appendix added after the first version of the current paper was released, we show how subsequent results of Shipman imply that the above homeomorphisms are in fact isomorphisms of algebraic varieties.

Quantum loop algebras, quiver varieties, and cluster algebras

“…We then have the following result, due to Lusztig [31] in the ungraded case, and extended to the graded case by Savage and Tingley [38]. It follows that we can rewrite Nakajima's formula for standard modules of C Z as…”

Graded quiver varieties and derived categories

“…Hence wσ−C q v 0 has non negative components: these components indicate the multiplicities of the indecomposable factors of Φ(M ) by Lemma 4.13. Consider the following dimension vector of S C : So it remains to prove the assertion for M = S. In this case, it was shown by Savage-Tingley in Theorem 5.4 of [43], who used input from Shipman's [44] to improve on a bijection constructed in the non graded case by Lusztig in Theorem 2.26 of [32]. Recall that V denotes the category of all finite direct sums of objects in the image.…”

“…By Nakajima's slice Theorem (Theorem 2.4.9 of [28] based on Theorem 3.14 of [36] based on §3.3 of [35]), the fibre of π : M(v, w) → M 0 (w) over M is isomorphic, in the complex-analytic topology, to the fibre of π : M(v − v 0 , w 0 ) → M 0 (w 0 ) over S. Moreover, it follows from Lemma 4.13, that CK(M ) is isomorphic to CK(S). So it remains to prove the assertion for M = S. In this case, it was shown by Savage-Tingley in Theorem 5.4 of [43], who used input from Shipman's [44] to improve on a bijection constructed in the non graded case by Lusztig in Theorem 2.26 of [32]. √ 4.21.…”

“…Along the way, we obtain information on graded affine quiver varieties as well as on their desingularization by Nakajima's smooth (quasi-projective) graded quiver varieties. Among other results, -we determine the quiver of the singular Nakajima category S, whose representations form the (affine) graded quiver varieties; -we determine the number of minimal relations between the vertices of the quiver of S; remarkably, there are no relations if Q is a connected non Dynkin quiver; -we construct the stratifying functor Φ from the category of finite-dimensional S-modules to the derived category of Q and use it to describe the strata and their closures in terms of the derived category; -we describe the fibers of Nakajima's desingularization map using Φ in the spirit of theorems by Lusztig [32], Savage-Tingley [43] and Shipman [44]; -we extend Happel's equivalence [14] by showing that, for a Dynkin quiver Q, the singular category S is weakly Gorenstein and that its derived category of singularities is equivalent to the derived category of Q; -we vastly generalize the preceding point by showing that for any configuration C of vertices of S satisfying a certain natural condition, the associated quotient S C of S is weakly Gorenstein with associated derived category of singularities equivalent to the derived category of Q.…”

The statistics of vacuum geometry

“…Physically, the class of such theories is by far the most studied, especially in the context of string theory and AdS/CFT because of the underlying toric Calabi-Yau geometry. Mathematically, it is interesting to point out that every projective variety (note that all our affine varieties are trivially complex cones over some projective variety) is a quiver Grassmannian [36]. Thus, from both physical and mathematical motivations, our representative class of theories is highly non-trivial.…”