Local Quivers and Stable Representations

In this paper we associate to a k-qurve A (formerly known as a quasi-free algebra [J. Cuntz, D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995) 251-289] or formally smooth algebra [M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, math.AG/9812158, 1998]) the one-quiver Q 1 (A) and dimension vector α 1 (A). This pair contains enough information to reconstruct for all n ∈ N the GL n -étale local structure of the representation scheme rep n A. In an appendix we indicate how one might extend this to qurves over nonalgebraically closed fields. Further, we classify all finitely generated groups G such that the group algebra kG is a k-qurve. If char(k) = 0 these are exactly the virtually free groups. We determine the one-quiver setting in this case and indicate how it can be used to study the finite-dimensional representations of virtually free groups. As this approach also applies to fundamental algebras of graphs of separable k-algebras, we state the results in this more general setting.  2005 Elsevier Inc. All rights reserved.

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“…using techniques of [1]. Given a point M ∈ M sst d (Q) of polystable type ξ as above, we construct a smooth affine open Gd-stable subvariety X such that…”

Section: The Dimension Vectord Is Coprime Forθmentioning

confidence: 99%

Smooth models of quiver moduli

Engel

Reineke

2008

Math. Z.

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For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are constructed.

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Local Quivers and Stable Representations

Cited by 31 publications

References 8 publications

Moduli of representations of quivers

Moduli of representations of quivers

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Smooth models of quiver moduli

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