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Lieven Le Bruyn

Publications and source records attributed to Lieven Le Bruyn.

44 records · Page 3Linked to original sources
arXiv2002

Smooth order singularities

In this paper we give methods to classify the central singularities of Cayley-Hamilton smooth orders up to smooth equivalence in arbitrary central dimension. We prove that there is just one type in dimension 3 (the conifold singularity), three types in dimension 4, ten types in dimension 5 and 53 types in dimension 6. As the classification in dimension 5 and 6 is xy-pic intensive, we refer for the details to the full version of this paper, available at ftp://wins.uia.ac.be/pub/preprints/02/SOSfull.pdf

math.RA↗
arXiv2001

Simple roots of deformed preprojective algebras

W. Crawley-Boevey has given a description of the set of dimension vectors of simple representations of the deformed preprojective algebras. In this note we give alternative descriptions. Note however that our descriptions depend on irreducibility of the quotient varieties, so they do NOT give a shorter proof of Crawley-Boevey's result.

math.RA↗
arXiv2001

Noncommutative smoothness and coadjoint orbits

Raf Bocklandt and the author have proved in math.AG/0010030 that certain quotient varieties of representations of deformed preprojective algebras are coadjoint orbits for the necklace Lie algebra of the corresponding quiver. A conjectural ringtheoretical explanation of these results was given in terms of noncommutative smoothness in the sense of C. Procesi. In this paper we prove these conjectures. The main tool in the proof is the etale local description due to W. Crawley-Boevey in math.AG/0105247. Along the way we determine the smooth locus of the so called Marsden-Weinstein reductions for quiver representations.

math.AG↗
arXiv2001

Formal structures and representation spaces

M. Kapranov introduced and studied in math.AG/9802041 the noncommutative formal structure of a smooth affine variety. In this note we show that his construction is a special case of microlocalization and extend it in a functorial way to representation schemes of affine algebras. We describe the formal completion in the case of path algebras of quivers and initiate the study of their finite dimensional representations.

math.RA↗
arXiv2000

Local quivers and stable representations

In this paper we introduce and study the local quiver as a tool to investigate the etale local structure of moduli spaces of theta-stable representations of quivers. As an application we determine the dimension vectors associated to irreducible representations of torus knot groups.

math.RT↗
arXiv2000

Necklace Lie algebras and noncommutative symplectic geometry

Recently V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry. In this note we generalize this argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg.

math.AG↗
arXiv1999

Noncommutative geometry at n

Kontsevich and Rosenberg propose to study smooth noncommutative spaces by approximation at level n by representation spaces. In this note we make some comments about their proposal.

math.AG↗