Research in Algebra

### Non-commutative noetherian rings and representation theory

#### Introductions

Happily, noetherian rings and their modules occur in many different areas of mathematics. A hundred years ago Hilbert, in the commutative setting, used properties of noetherian rings to settle a long-standing problem of invariant theory. Later, it was realised that commutative noetherian rings are one of the building blocks of modern algebraic geometry, leading to their study both abstractly and in examples. It was not until the late 1950's, with the appearance of Goldie's theorem, that it became clear that non-commutative noetherian rings constitute an interesting class of rings in their own right. As in the commutative case, non-commutative noetherian rings are studied in abstraction and in examples.

Classic examples of noetherian rings include the co-ordinate rings of affine varieties, rings of differential operators on smooth algebraic varieties, universal enveloping algebras of finite dimensional Lie algebras and group algebras of polycyclic-by-finite groups. More recently, quantum groups have provided a new class of noetherian rings and there has also been much interest in non-commutative geometry and the non-commutative noetherian rings this throws up, such as Sklyanin algebras.

At around the same time as Hilbert, various mathematicians began to consider what is nowadays called representation theory. The pioneers, notably Frobenius, Schur and Burnside, were interested in representations of groups over the real or complex numbers. Much progress was made, particularly for symmetric groups and general linear groups. Later, Brauer made great strides in the study of representations of groups over fields of positive characteristic, a subject which remains very active today.

It turns out that the representation theory of groups such as the general linear group and symmetric group is closely connected with Lie theory, through topics like the representation theory of algebraic groups and Lie algebras. In Lie theory, it became clear that representation theory is intimately related with geometry, through such spaces as flag varieties. Further relations between representation theory and geometry continue to be found to this day and there are many outstanding questions or incomplete theories in this area.

#### Interests

At the moment we are interested in the following subjects in Glasgow (in no particular order).

• Noetherian Hopf algebras. Examples are provided by the universal enveloping algebras of finite dimensional Lie algebras and by quantum groups. It is apparent that the axioms of a Hopf algebra place strong restrictions on the behaviour of noetherian algebras, but it is still unclear exactly what these restrictions are. As a basic example, a commutative noetherian domain is necessarily smooth. Whatever these restrictions are in general, they have interesting ramifications for the representation theory of the algebra.
• P.I. algebras. These are algebras which satisfy a polynomial identity. For example commutative algebras always satisfy the identity XY-YX = 0. Examples are provided by the universal enveloping algebras of finite dimensional Lie algebras in positive characteristic and by quantum groups at roots of unity. Typically, the representation theory of such algebras is closely related to the geometry of the prime spectrum of centre of the algebra. The combination of the Hopf and P.I. algebra hypotheses can have strong consequences for representation theory.
• Reduced enveloping algebras. These are finite dimensional quotients of universal enveloping algebras of reductive Lie algebras in positive characteristic. The representation theory of these algebras is quite poorly understood at the moment. For instance, it is unknown in general how many simple modules such an algebra has. It is clear, however, that these algebras are intimately related to the geometry of the so-called Springer resolution. The geometry of this resolution is, amongst other things, related to Hecke algebras and there are very recent speculations of Lusztig, trying to pin down a relationship between the representation theory of the reduced enveloping algebras and Hecke algebras.
• Quantum groups at roots of unity. These are quite beautiful algebras with relations to almost every subject in mathematics: knot theory, geometry, analysis, mathematical physics, integrable systems, combinatorics, representation theory, Hopf algebras.....
• Quivers. Quivers are efficient ways to store linear algebra information, and as such appear in many different areas of mathematics. They are related to quantum groups through work of Lusztig on canonical bases and Nakajima on quiver varieties, to singularity theory through work of Kronheimer (and then to Lie algebras), and the representation theory of any finite dimensional algebra can be understood through representations of quivers.
• Symplectic reflection algebras: These are a very interesting family of noetherian algebras which are related to differential operators, quivers, invariant theory and integrable systems. These algebras are very new (they were introduced by Etingof and Ginzburg in 2000), so the foundations for their study are still being laid. You can find a recent survey article by Ken Brown below.

Here are interesting books or survey articles on some of the above.

1.      V. Chari and A. Pressley, A Guide to Quantum Groups, C.U.P. (1995).

2.      N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhauser (1997).

3.      K.R. Goodearl and R.B. Warfield Jr., An Introduction to Noncommutative Noetherian Rings, C.U.P. (1989).

4.      J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer (1972).

5.      J.E. Humphreys, Modular representations of simple Lie algebras, Bull. Amer. Math. Soc. (N.S.) 35 (1998), no. 2, 105--122.

6.      G. James and M. Liebeck, Representations and Characters of Groups, C.U.P. (1993).

7.      J.C. Jantzen, Lectures on quantum groups, A.M.S. (1996).

8.      J.C. Jantzen, Representations of Lie algebras in prime characteristic, in Representation theories and algebraic geometry, Kluwer (1997).

9.      I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices Amer. Math. Soc. 44 (1997), no. 5, 546--556.

Here are the most recent preprints of Ken Brown and Iain Gordon, touching on some of the above. To find out about other publications, go to our homepages by clicking on our names.

1.      I.Gordon and D.Rumynin, Subregular representations of sl_n and simple singularities of A_{n-1} of type, Compositione Mathematicae, 138 (2003), 337-360.

2.      K.A.Brown and I.Gordon, Poisson orders, symplectic reflection algebras
and representation theory, Journal fuer die Reine und
Angewandte Mathematik, 559 (2003), 193-216.

3.      K.A.Brown, Symplectic reflection algebras, Bulletin of the Irish Mathematical Society, 50 (2002) 27-49.

4.      I.Gordon and J.T.Stafford, Rational Cherednik algebras and Hilbert schemes, to appear Advances in Mathematics.

5.      I.Gordon, On the quotient ring by diagonal invariants, Inventiones Mathematicae 153 (2003), 503-518.