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Prof William Crawley-Boevey

Professor of Pure Mathematics
Pure Mathematics

Contact details

Room: 8.08
Tel: +44 (0)113 3435185
Email: W.Crawley-Boevey @ leeds.ac.uk

Keywords

Representations of algebras
Representations of quivers
Preprojective algebras
Weighted projective lines

Prof Bill Crawley-Boevey

Research interests

My research has mainly been on the representation theory of finite-dimensional associative algebras and related questions in linear algebra, ring and module theory, and algebraic geometry.

In recent years I have concentrated on representations of quivers and preprojective algebras. A quiver is essentially the same thing as a directed graph, and a representation associates a vector space to each vertex and a linear map to each arrow. The subject was started by P. Gabriel in 1972, when he discovered that the quivers with only finitely many indecomposable representations are exactly the ADE Dynkin diagrams which occur in Lie theory. Quivers and their representations now appear in all sorts of areas of mathematics and physics, including representation theory, cluster algebras, geometry (algebraic, differential, symplectic), noncommutative geometry, quantum groups, string theory, and more.

The preprojective algebra associated to a quiver was invented by I. M. Gelfand and V. A. Ponomarev. Its modules are intimately related to representations of the quiver, but it is often the modules for the preprojective algebra which are of relevance in other parts of mathematics. There is beautiful geometry linked to the preprojective algebra, including Kleinian singularities and H. Nakajima's quiver varieties.

There are also links between the preprojective algebra and the classification of differential equations on the Riemann sphere. They are used in work on the Deligne-Simpson problem, which concerns the existence of matrices in prescribed conjugacy classes whose product is the identity matrix, or whose sum is the zero matrix.

Useful links

Personal homepage
FDLIST Information list for representation theory of finite dimensional algebras
Front for the arXiv

Publications

Crawley-Boevey W, Sauter J On quiver Grassmannians and orbit closures for representation-finite algebras Mathematische Zeitschrift, 285, 367-395, 2017
DOI:10.1007/s00209-016-1712-z
View abstract

Chazal F, Crawley-Boevey WW, de Silva V The observable structure of persistence modules Homology, Homotopy and Applications, 18, 247-265, 2016
DOI:10.4310/HHA.2016.v18.n2.a14
View abstract

Crawley-Boevey W Decomposition of pointwise finite-dimensional persistence modules Journal of Algebra and its Applications, 14, 2015
DOI:10.1142/S0219498815500668
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Crawley-Boevey WW Kac's Theorem for equipped graphs and for maximal rank representations Proceedings of the Edinburgh Mathematical Society, 56, 443-447, 2013
DOI:10.1017/S001309151200034X
View abstract

Crawley-Boevey WW Monodromy for systems of vector bundles and multiplicative preprojective algebras Bulletin of the London Mathematical Society, 45, 309-317, 2012
DOI:10.1112/blms/bds089
View abstract

Crawley-Boevey W Poisson structures on moduli spaces of representations Journal of Algebra, 325, 205-215, 2011
DOI:10.1016/j.jalgebra.2010.09.033

Crawley-Boevey W Connections for weighted projective lines Journal of Pure and Applied Algebra, 215, 35-43, 2011
DOI:10.1016/j.jpaa.2010.02.030

Crawley-Boevey W Kac’s theorem for weighted projective lines Journal of the European Mathematical Society (JEMS), 12, 1331-1345, 2010
DOI:10.4171/JEMS/232

Crawley-Boevey W General sheaves over weighted projective lines Colloquium Mathematicum, 113, 119-149, 2008
DOI:10.4064/cm113-1-8

Crawley-Boevey WW, Etingof P, Ginzburg V Noncommutative Geometry and Quiver algebras Advances in Mathematics, 209, 274-336, 2007
DOI:10.1016/j.aim.2006.05.004
View abstract

Crawley-Boevey WW Quiver algebras, weighted projective lines, and the Deligne-Simpson problemInternational Congress of Mathematicians - Madrid, August 22-30, 2006, 117-129 2007
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Crawley-Boevey WW, Jensen BT A note on sub-bundles of vector bundles Glasgow Mathematical Journal, 48, 459-462, 2006
DOI:10.1017/S0017089506003259
View abstract

Crawley-Boevey WW Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity Publications Mathématiques / Institut des Hautes Études Scientifiques, 100, 171-207, 2004
DOI:10.1007/s10240-004-0025-7
View abstract

Crawley-Boevey WW, Van den Bergh M Absolutely indecomposable representations and Kac-Moody Lie algebras Inventiones Mathematicae, 155, 537-559, 2004
DOI:10.1007/s00222-003-0329-0
View abstract

Crawley-Boevey WW Normality of Marsden-Weinstein reductions for representations of quivers Mathematische Annalen, 325, 55-79, 2003
DOI:10.1007/s00208-002-0367-8
View abstract

Crawley-Boevey WW On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero Duke Mathematical Journal, 118, 339-352, 2003
DOI:10.1215/S0012-7094-03-11825-6
View abstract

Crawley-Boevey WW, Schroer JH Irreducible components of varieties of modules Journal für die Reine und Angewandte Mathematik, 553, 201-220, 2002
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Crawley-Boevey WW; Geiss C Horn's problem and semi-stability for quiver representationsRepresentations of Algebras, 40-48 2002

Crawley-Boevey WW Decomposition of Marsden-Weinstein reductions for representations of quivers Compositio Mathematica, 130, 225-239, 2002
DOI:10.1023/A:1013793632709
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Crawley-Boevey WW Geometry of the moment map for representations of quivers Compositio Mathematica, 126, 257-293, 2001
DOI:10.1023/A:1017558904030
View abstract

Crawley-Boevey WW On the exceptional fibres of Kleinian singularities American Journal of Mathematics, 122, 1027-1037, 2000

Crawley-Boevey WW Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities Commentarii Mathematici Helvetici, 74, 548-574, 1999

Crawley-Boevey WW, Holland MP Noncommutative deformations of Kleinian singularities Duke Mathematical Journal, 92, 605-635, 1998

Crawley-Boevey WW On homomorphisms from a fixed representation to a general representation of a quiver Transactions of the American Mathematical Society, 348, 1909-1919, 1996

Crawley-Boevey WW Classification of modules for infinite-dimensional string algebras Transactions of the American Mathematical Society
DOI:10.1090/tran/7032
View abstract