Geometry of Multiplicative Preprojective Algebra

By Daisuke Yamakawa. 2007

69 citations recorded by the source corpus.

Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there exists a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parametrizes Simpson's (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight and associated graded local system around each puncture.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ec18b82a45d73a1fe1563be0eb945c20c8dad481

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. P. Boalch
2. Daisuke Yamakawa
3. T. Mochizuki
4. Dan Xie

In Mathematical Physics

1. P. Boalch
2. Daisuke Yamakawa
3. T. Mochizuki
4. Carlos Simpson
5. Dan Xie

Citations in the corpus

• Riemann–Hilbert for tame complex parahoric connections (2010, background): “Note that in the case of G = GLn(C) the quasi-Hamiltonian spaces Ĉ may be constructed differently, in terms of quivers (see [27]), although even for GLn the spaces M do not seem to arise from quivers.”
• Geometry and braiding of Stokes data; Fission and wild character varieties (2011, methodology): “B] (weighted conjugacy classes) to obtain spaces of Betti data (filtered Stokes representations) corresponding to meromorphic connections with unramified formal types on parahoric bundles, as in [59], [69], [19] in the tame case.”
• Twisted wild character varieties (2015, background): “They also yield a twisted version of the (coloured) multiplicative quiver varieties [23, 43, 19].”