On representation schemes and Grassmanians of finite dimensional algebras and a construction of Lusztig

Abstract

Let I be a finite set and C{I} be the algebra of functions on I. For a finite dimensional C algebra A with C{I} ⊂ A we show that certain moduli spaces of finite dimsional modules are isomorphic to certain Grassmannian (quot-type) varieties. There is a special case of interest in representation theory. Lusztig defined two varieties related to a quiver and gave a bijection between their C-points, [4, Theorem 2.20]. Savage and Tingley raised the question [7, Remark 4.5] of whether these varieties are isomorphic as algebraic varieties. This question has been open since Lusztig's original work. It follows from the result of this note that the two varieties are indeed isomorphic.Date: November 2, 2018.

Quiver grassmannians, quiver varieties and the preprojective algebra

“…For u ∈ NQ 0 , we definẽ Proof. This is proven in [41,Corollary 3.2]. Note that, in [41], a different stability condition is used in the definition of L(v, w).…”

“…After an earlier version [39] of the current paper was released, it was proven in [41] that the grassmannian type varietiesGr P (v, p w ) defined by Lusztig are indeed isomorphic as algebraic varieties to the lagrangian Nakajima quiver varieties L(v, w). A simple "duality" map gives an isomorphism of varieties between the quiver grassmannian Gr P (v,q w ) andGr P (v, p w ).…”

The Mirković–Vilonen basis and Duistermaat–Heckman measures

“…This quiver variety has a "core" L(w), and there is a homotopy retraction of M(w) onto L(w). We have the following result of Shipman [Sh,Corollary 3.2].…”

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.…”

“…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.…”