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Necklace Lie algebras and noncommutative symplectic geometry. (English) Zbl 1113.16019

Summary: Recently, V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry [Non-commutative symplectic geometry and Calogero-Moser space, preprint (1999)]. In this note we generalize his argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg [Math. Res. Lett. 8, No. 3, 377-400 (2001; Zbl 1113.17306)]. Using results of W. Crawley-Boevey and M. P. Holland [Duke Math. J. 92, No. 3, 605-635 (1998; Zbl 0974.16007), Compos. Math. 126, No. 3, 257-293 (2001; Zbl 1037.16007) and Am. J. Math. 122, No. 5, 1027-1037 (2000; Zbl 1001.14001)] we give a combinatorial description of all the relevant couples \((\alpha,\lambda)\) which are coadjoint orbits. We give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness.

MSC:

16G20 Representations of quivers and partially ordered sets
17B65 Infinite-dimensional Lie (super)algebras
53D05 Symplectic manifolds (general theory)
14A22 Noncommutative algebraic geometry
16S38 Rings arising from noncommutative algebraic geometry