Pure Research
Algebra
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Our Research Interests
Research interests of the algebraists at Manchester University include Lie theory, representation theory and invariant theory of algebraic groups and related Lie algebras, representation theory of symmetric groups, GL(n) and the Steenrod algebra, modular group algebras of finite p-groups, the unit group of the Steenrod algebra, and infinite dimensional representations of quivers and algebras.- Lie Theory
In Lie theory, research is mainly concentrated around classifying all finite dimensional simple Lie algebras of characteristic p>3. It is expected that any such Lie algebra is either classical, or of Cartan type, or belongs to the exceptional series constructed by Melikian (in the latter case p=5). Significant progress has been achieved in this area, of which the most recent result is the classification of all simple Lie algebras of toral rank 2 for p>3 (Premet-Strade, submitted).
- Invariant Theory of Algebraic Groups
In invariant theory of algebraic groups, we work on generalising well-known results of Andreev, Vinberg, Elashvili and A.M. Popov on complex linear G-actions with nontrivial generic stabilisers to the case where G is a semisimple group over a field of positive characteristic.
- Modular Representation Theory of Lie Algebras
In modular representation theory of Lie algebras, irreducible representations of the p-Lie algebra of a simple algebraic group, having a nilpotent p-character, are studied. It was a remarkable observation by Brieskorn (pushed further by Slodowy) that subregular nilpotent orbits in such Lie algebras are closely related to Klein singularities. It is established that a similar relationship exists between the algebras arising in subregular representation theory and noncommutative deformations of Klein singularities similar to those introduced recently by Hodges, Crawley-Boevey and Holland. Modular representation theory of Lie algebras is now a very active and attractive field due to deep interactions with representation theory of quantum groups at roots of unity and some recent discoveries such as Premet's proof of the Kac-Weisfeiler conjecture and Jantzen's work on subregular nilpotent representations. The latter work inspired Lusztig to conjecture that the number of simple modules belonging to a generic block of a reduced enveloping algebra equals the Euler characteristic of the corresponding complex Springer fibre.
- Modular Representation Theory of Sn and GL(n)
In modular representation theory of Sn and GL(n), the differential operator algebra and the Steenrod algebra are used to study the natural action of the above two groups on the polynomial ring in n variables. Combinatorics and Algebraic Topology are parts of this approach. The unit group of the Steenrod algebra has been studied by applying results from the theory of group rings of finite p-groups.
- Properties of the Ziegler spectrum of an associative ring
Properties of the Ziegler spectrum of an associative ring as a topological space are studied. The points of this space are the isomorphism classes of indecomposable pure-injective right modules over the ring. The theory of Ziegler spectra incorporates quiver approach to representation theory of associative algebras along with results and ideas from Logic.
Seminars
Manchester Algebra Seminar, joint with UMIST, meets weekly providing a forum for presentation of completed results and work in progress.
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