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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 @


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


Crawley-Boevey W, Sauter J On quiver Grassmannians and orbit closures for representation-finite algebras Mathematische Zeitschrift, 285, 367-395, 2017
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Chazal F, Crawley-Boevey WW, de Silva V The observable structure of persistence modules Homology, Homotopy and Applications, 18, 247-265, 2016
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Crawley-Boevey W Decomposition of pointwise finite-dimensional persistence modules Journal of Algebra and its Applications, 14, 2015
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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
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Crawley-Boevey WW Monodromy for systems of vector bundles and multiplicative preprojective algebras Bulletin of the London Mathematical Society, 45, 309-317, 2012
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Crawley-Boevey W Poisson structures on moduli spaces of representations Journal of Algebra, 325, 205-215, 2011

Crawley-Boevey W Connections for weighted projective lines Journal of Pure and Applied Algebra, 215, 35-43, 2011

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

Crawley-Boevey W General sheaves over weighted projective lines Colloquium Mathematicum, 113, 119-149, 2008

Crawley-Boevey WW, Etingof P, Ginzburg V Noncommutative Geometry and Quiver algebras Advances in Mathematics, 209, 274-336, 2007
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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
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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
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Crawley-Boevey WW, Van den Bergh M Absolutely indecomposable representations and Kac-Moody Lie algebras Inventiones Mathematicae, 155, 537-559, 2004
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Crawley-Boevey WW Normality of Marsden-Weinstein reductions for representations of quivers Mathematische Annalen, 325, 55-79, 2003
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Crawley-Boevey WW On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero Duke Mathematical Journal, 118, 339-352, 2003
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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
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Crawley-Boevey WW Geometry of the moment map for representations of quivers Compositio Mathematica, 126, 257-293, 2001
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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
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