- Address: School of Mathematics, University of Leeds, Leeds LS2 9JT, UK.
- E-mail: W.Crawley-Boevey@leeds.ac.uk
- Telephone: (0113) 343 5185.
- Department: Pure Mathematics
- Research Group: Algebra, Geometry and Integrable Systems

- Semester 1. MATH 2080 Further Linear Algebra
- Semester 2. MATH 3044 Number Theory

- London Mathematical Society Editorial Board 1995-2002
- Journal of Algebra Editorial Board 2004-2006
- Algebra Montpellier Announcements Editorial Board 2003-2009

- Roberto Vila Freyer completed an Oxford D. Phil. on the topic 'Biserial algebras' in 1994.
- Nicola Richmond completed a Leeds Ph. D. on the topic 'The geometry of modules over finite dimensional algebras' in 1999.
- Andrew Hubery completed a Leeds Ph. D. on the topic 'Representations of quivers respecting a quiver automorphism and a theorem of Kac' in 2002.
- Peter Shaw completed a Leeds Ph. D. on the topic 'Generalisations of Preprojective algebras' in 2005. pdf.
- Marcel Wiedemann completed a Leeds Ph. D. on the topic 'On real root representations of quivers' in 2008. pdf.
- Daniel Kirk completed a Leeds Ph. D. on the topic 'Representations of Quivers with Applications to Collections of Matrices with Fixed Similarity Types and Sum Zero' in 2013. pdf.
- Raphael Bennett-Tennenhaus started a Leeds Ph. D. in September 2013

My research has mainly been on the representation theory of finite-dimensional associative algebras (see fdlist), 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 (for example a quiver of
type
E_{6} is illustrated on the left).
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. (The
illustration on the right shows the real-valued points of
varieties associated to a quiver of extended Dynkin, that is,
affine, type D_{4}.)

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.
(The picture on the left shows loops on the punctured Riemann sphere which
generate its fundamental group. Consideration of the monodromy around
such loops links
the classification of differential equations on the Riemann sphere
to the Deligne-Simpson problem.)

In earlier times I was interested in tame algebras, matrix problems, and
infinite-dimensional modules. Finite dimensional associative algebras naturally
divide into three classes: algebras *finite representation
type* with only finitely many indecomposable modules,
*wild algebras* for which the indecomposable modules are unclassifiable
(in a suitable sense), and those on the boundary between these classes, the
*tame* algebras. There are many interesting classes of tame algebras,
and it is often a major problem to actually give the classification of the
indecomposable modules.

One way to study tame algebras is to convert the problem of classifying their
modules into a *matrix problem*: putting a partitioned matrix into
canonical form using not all elementary operations, but a subset defined
by the partition.
(The illustration on the right shows what an arbitrary matrix can be
reduced to if you allow all row and column operations; it also shows an
example of a partition of a matrix.)
Using advanced methods based on this idea, Yu. A. Drozd
proved
his wonderful *Tame and Wild Theorem* showing that there is a wide gulf between
the behaviour of tame and wild algebras.
The same methods can be used to show that tame algebras are characterized by
the behaviour of their infinite-dimensional modules. In fact, the behaviour of
infinite-dimensional modules for tame algebras is extremely interesting, and
not at all understood.

- Polycyclic-by-finite affine group schemes, Proc. London Math. Soc., 52 (1986), 495-516.
- Locally finite representations of groups of finite p-rank, J. London Math. Soc., 34 (1986), 17-25.
- (With P. H. Kropholler and P. A. Linnell) Torsion-free soluble groups and the zero-divisor conjecture, J. Pure and Appl. Algebra, 54 (1988), 181-196.
- On tame algebras and bocses, Proc. London Math. Soc., 56 (1988), 451-483.
- Functorial filtrations and the problem of an idempotent and a square zero matrix, J. London Math. Soc., 38 (1988), 385-402.
- Functorial filtrations II: clans and the Gelfand problem, J. London Math. Soc., 40 (1989), 9-30.
- Functorial filtrations III: semidihedral algebras, J. London Math. Soc., 40 (1989), 31-39.
- Maps between representations of zero-relation algebras, J. Algebra, 126 (1989), 259-263.
- (With L. Unger) Dimensions of Auslander-Reiten translates for representation-finite algebras, Comm. Algebra, 17 (1989), 837-842.
- Matrix problems and Drozd's theorem, in 'Topics in Algebra', eds S. Balcerzyk et al., Banach Center publications, vol. 26 part 1 (PWN-Polish Scientific Publishers, Warsaw, 1990), 199-222.
- Regular modules for tame hereditary algebras, Proc. London Math. Soc., 62 (1991), 490-508.
- Tame algebras and generic modules, Proc. London Math. Soc., 63 (1991), 241-265.
- Lectures on representation theory and invariant theory, Ergänzungsreihe Sonderforschungsbereich 343 'Diskrete Strukturen in der Mathematik', 90-004, Bielefeld University, 1990, 74pp.
- Matrix reductions for artinian rings, and an application to rings of finite representation type, J. Algebra, 157 (1993), 1-25.
- (With C. M. Ringel) Algebras whose Auslander-Reiten quiver has a large regular component, J. Algebra, 153 (1992), 494-516.
- (With D. Happel and C. M. Ringel) A bypass of an arrow is sectional, Arch. Math. (Basel), 58 (1992), 525-528.
- Modules of finite length over their endomorphism rings, in 'Representations of algebras and related topics', eds H. Tachikawa and S. Brenner, London Math. Soc. Lec. Note Series 168, (Cambridge University Press, 1992), 127-184.
- Additive functions on locally finitely presented Grothendieck categories, Comm. Algebra, 22 (1994), 1629-1639
- Locally finitely presented additive categories, Comm. Algebra, 22 (1994), 1641-1674.
- Exceptional sequences of representations of quivers, in 'Representations of algebras', Proc. Ottawa 1992, eds V. Dlab and H. Lenzing, Canadian Math. Soc. Conf. Proc. 14 (Amer. Math. Soc., 1993), 117-124.
- (With O. Kerner) A functor between categories of regular modules for wild hereditary algebras, Math. Ann., 298 (1994), 481-487.
- (With D. J. Benson) A ramification formula for Poincaré series, and a hyperplane formula for modular invariants, Bull. London Math. Soc., 27 (1995), 435-440.
- Subrepresentations of general representations of quivers, Bull. London Math. Soc., 28 (1996), 363-366.
- (With R. Vila-Freyer) The structure of biserial algebras, J. London Math. Soc., 57 (1998), 41-54.
- Rigid integral representations of quivers, in 'Representations of algebras', Proc. Cocoyoc 1994, eds R. Bautista et al., Canad. Math. Soc. Conf. Proc., 18 (Amer. Math. Soc., 1996), 155-163.
- Tameness of biserial algebras, Arch. Math. (Basel), 65 (1995), 399-407.
- On homomorphisms from a fixed representation to a general representation of a quiver, Trans. Amer. Math. Soc., 348 (1996), 1909-1919.
- (With M. P. Holland) Noncommutative deformations of Kleinian singularities, Duke Math. J., 92 (1998), 605-635.
- Infinite-dimensional modules in the representation theory of finite-dimensional algebras, Canadian Math. Soc. Conf. Proc., 23 (1998), 29-54 (pdf).
- Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities, Comment. Math. Helv., 74 (1999), 548-574 (pdf).
- (With R. Bautista, T. Lei and Y. Zhang) On Homogeneous Exact Categories, J. Algebra, 230 (2000), 665-675.
- On the exceptional fibres of Kleinian singularities, Amer. J. Math., 122 (2000), 1027-1037 (pdf).
- Geometry of the moment map for representations of quivers, Compositio Math., 126 (2001), 257-293 (pdf).
- Decomposition of Marsden-Weinstein reductions for representations of quivers, Compositio Math., 130 (2002), 225-239 (math.AG/0007191).
- (with Christof Geiß) Horn's problem and semi-stability for quiver representations, in 'Representations of Algebras, Vol 1', Proceedings of the Ninth International Conference, Beijing, August 21-September 1, 2000, eds. D. Happel and Y. B. Zhang (Beijing Normal University Press, 2002), 40-48 (pdf).
- (with Jan Schröer) Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002), 201-220 (math.AG/0103100).
- Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann. 325 (2003), 55-79 (math.AG/0105247).
- On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J. 118 (2003), 339-352 (math.RA/0103101).
- (with Michel Van den Bergh) Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima), Invent. Math. 155 (2004), 537-559. (pdf).
- Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity, Publ. Math. Inst. Hautes Etudes Sci. 100 (2004), 171-207. (math.AG/0307246).
- (with Peter Shaw) Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem, Adv. Math. 201 (2006), 180-208. (math.RA/0404186).
- (With Bernt Tore Jensen) A note on sub-bundles of vector bundles, Glasgow Math. J. 48 (2006), 459-462. (math.AG/0505149).
- Quiver algebras, weighted projective lines, and the Deligne-Simpson problem, in: 'Proceedings of the International Congress of Mathematicians', vol. 2, Madrid 2006, eds M. Sanz-Solé et al. (European Mathematical Society, January 2007), 117-129. (pdf | math.RA/0604273).
- (With Pavel Etingof and Victor Ginzburg) Noncommutative Geometry and Quiver algebras, Adv. Math. 209 (2007), 274-336. (math.AG/0502301 ).
- General sheaves over weighted projective lines, Colloq. Math. 113 (2008), 119-149. (pdf).
- Kac's Theorem for weighted projective lines, Journal of the European Mathematical Society, 12 (2010), 1331-1345. (math.AG/0512078).
- Connections for weighted projective lines, J. Pure Appl. Algebra, 215 (2011), 35-43. (arXiv:0904.3430v1 [math.AG]).
- Poisson structures on moduli spaces of representations, J. Algebra 325 (2011), 205-215. (pdf).
- Monodromy for systems of vector bundles and multiplicative preprojective algebras, Bulletin of the London Mathematical Society 45 (2013), 309-317 (free access: abstract, pdf).
- Kac's Theorem for equipped graphs and for maximal rank representations, Proceedings of the Edinburgh Mathematical Society 56 (2013), 443-447. (pdf).

- Decomposition of pointwise finite-dimensional persistence modules, arXiv:1210.0819 [math.RT].
- Classification of finitely generated modules for k[x,y]/(xy) and generalizations, arXiv:1308.6410 [math.RA]

- Lectures on representation theory and invariant theory
(pdf).
A graduate course given in 1989/90 at Bielefeld University.
This is an introduction to the representation theory of the symmetric
and general linear groups (in characteristic zero), and to classical
invariant theory.
List of corrections.
- Lectures on representations of quivers
(pdf |
scanned pdf - includes one extra diagram).
A graduate course given in 1992 at Oxford University.
This is an introduction to the representation
theory of quivers, and in particular the representation theory of extended
Dynkin quivers.
List of corrections.
- More lectures on representations of quivers
(scanned pdf).
Another graduate course from 1992 at Oxford University.
More about representations of quivers, including Auslander-Reiten
theory, results of Kerner and of Schofield.
- Geometry of representations of algebras
(pdf).
A graduate course given in 1993 at Oxford University.
This is a survey of how algebraic
geometry has been used to study representations of algebras (and quivers
in particular).
- Cohomology and central simple algebras
(pdf).
An MSc course given in 1996 at Leeds University.
An introduction to homological algebra
and applications to central simple algebras.
- Representations of quivers, preprojective algebras and
deformations of quotient singularities
(pdf).
Lectures from a DMV Seminar in May 1999 on "Quantizations
of Kleinian singularities" organized by R. Buchweitz, P. Slodowy
and myself at Oberwolfach.
Here is the group photo from the meeting.
(There are mistakes in Lemma 4.5 and the proof of Theorem 5.9.
The first of these has been sorted out independently by P. Etingof
and V. Ginzburg in math.AG/0011114.
For the second, the proof elsewhere in the literature is correct.)
- The website
from a Summer School on "Geometry of Quiver-representations and Preprojective
Algebras" (Isle of Thorns/UK, September 10 - 17, 2000)
- An unfinished and abandoned paper
on generic deformed preprojective algebras
(pdf),
dating from the late 1990s,
and mentioned by P. Etingof and E. Rains in
math.RT/0503393.
- The website for a conference in honour of John McConnell and Chris Robson (Leeds, May 5-6, 2006)

- International AsiaLink conference on Algebras and Representations (Beijing Normal University, May 23--28, 2005)
- Journées Solstice d'été 2005 : Groupes (Paris, June 23-25, 2005)
- Interactions between noncommutative algebra and algebraic geometry (Banff, September 10-15, 2005)
- Workshop in Non-Commutative Geometry (Copenhagen, November 7-10, 2005)
- Sklyanin Algebras and Beyond (Leeds, December 16-17, 2005)
- Ring Theory: recent progress and applications (Leeds, May 5-6, 2006)
- Interactions between Algebraic Geometry and Noncommutative Algebra (Oberwolfach, May 7-13, 2006)
- Workshop on algebraic vector bundles (Münster, June 26-30, 2006)
- Workshop on Triangulated Categories (Leeds, August 13-19, 2006)
- International Congress of Mathematicians (Madrid, August 22-30, 2006)
- Representations of Quivers, Singularities and Lie Theory (Beijing, September 13-17, 2006)
- Representations of Algebras and their Geometry (Paderborn, November 10-11, 2006)
- Recent developments in the theory of Hall algebras (CIRM, Luminy, France, November 20-24, 2006)
- Trends in Noncommutative Geometry (Newton Institute, Cambridge, December 18-22, 2006)
- Perspectives in Auslander-Reiten Theory. On the occasion of the 65th birthday of Idun Reiten (Trondheim, May 10-12, 2007)
- Arithmetic harmonic analysis on character and quiver varieties (American Institute of Mathematics, Palo Alto, June 4-8, 2007)
- XII International Conference on Representations of Algebras and Workshop (Torun, August 15-24, 2007)
- Representation Theory of Finite Dimensional Algebras (Oberwolfach, February 17-23, 2008)
- Maurice Auslander Distinguished Lectures and International Conference (Woods Hole, Cape Cod, April 25-27, 2008)
- Symmetries in Mathematics and Physics, in honor of Victor Kac (Cortona, June 22-28, 2008)
- XIII International Conference on Representations of Algebras and Workshop (Sao Paulo, July 30-August 8, 2008)
- Minisymposium on Algebras (Uppsala, February 20, 2009)
- Combinatorial Geometric Structures in Representation Theory (Durham, July 6-16, 2009)
- Summer school on Geometry of representations (Cologne, July 26-31, 2009)
- Workshop on Noncommutative Algebraic Geometry and Related Topics (Manchester, August 3-7, 2009)
- Quiver varieties, Donaldson-Thomas invariants and instantons (CIRM, Luminy, September 14-18, 2009)
- Interplay between representation theory and geometry (Beijing, May 3-7, 2010)
- Interactions between Algebraic Geometry and Noncommutative Algebra (Oberwolfach, May 9-15, 2010)
- XIV International Conference on Representations of Algebras and Workshop (Tokyo, August 6-15, 2010)
- Representation Theory of Quivers and Finite Dimensional Algebras (Oberwolfach, February 20-26, 2011)
- New developments in noncommutative algebra and its applications (Isle of Skye, June 26-July 2, 2011)
- Cluster categories and cluster tilting. A conference honoring Idun Reiten on the occasion of her 70th birthday (Trondheim, March 28-30, 2012)
- Representation Theory and Geometry (Zurich, April 10-14, 2012)
- Workshop and International Conference on Representations of Algebras (ICRA 2012) (Bielefeld, August 8-17, 2012)
- Introductory Workshop: Noncommutative Algebraic Geometry and Representation Theory (MSRI, Berkeley, January 28-February 1, 2013)
- Representation Theory, Homological Algebra, and Free Resolutions (MSRI, Berkeley, February 11-17, 2013)
- Interactions between Noncommutative Algebra, Representation Theory, and Algebraic Geometry (MSRI, Berkeley, April 8-12, 2013)
- Representation Theory of Quivers and Finite Dimensional Algebras (Oberwolfach, February 16-22, 2014)
- Interactions between Algebraic Geometry and Noncommutative Algebra (Oberwolfach, May 18-24, 2014)
- XVI International Conference on Representations of Algebras (ICRA 2014) (Sanya, Hainan Province, China, August 20-29, 2014)