# School of Mathematics

## 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

## 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

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**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

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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

DOI:10.1112/blms/bds089

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**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

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**Crawley-Boevey WW ** *Quiver algebras, weighted projective lines, and the Deligne-Simpson problem*International 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

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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

DOI:10.1007/s10240-004-0025-7

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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

DOI:10.1007/s00222-003-0329-0

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**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

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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

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 representations*Representations 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

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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*

DOI:10.1090/tran/7032

View abstract

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