Professor William Crawley-Boevey


The representation theory of quivers brings together all sorts of algebra and geometry, and has applications to many areas, including differential equations and physics.

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FDLIST Information list for representation theory of finite dimensional algebras

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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.

Research groups and institutes

  • Pure Mathematics