Prof Alexander Premet - research
My reserach interests are in Lie theory, representation theory and invariant theory of linear algebraic groups.
In 2007 Helmut Strade and I have successfully completed our long-term project aimed at classifying all finite dimensional simple Lie algebras over algebraically closed fields of characteristic p>3 (for p>7 such a classification was obtained earlier in a series of papers by Block-Wilson and Strade). Our main Classification Theorem reads:
any finite dimensional simple Lie algebra over an algebraically closed field of characteristic p>3 is up to isomorphism either classical or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5.
There are still many interesting open problems in the classification theory of modular Lie algebras. As an example, J.-M. Bois used our Classification Theorem to prove that every finite dimensional simple Lie algebra over an algebraically closed field of characteristic p>3 is generated by two elements. This result is analogous to one of the well-known theorems on finite simple groups.
In 1995 I proved the Kac-Weisfeiler conjecture on p-divisibility of dimensions of simple modules over reductive Lie algebras of good characteristic. Modular representation theory of reductive Lie algebras is now a very active area of research. It is related with the geometry of Springer fibres via the derived version of the Beilinson-Bernstein equivalence discovered by Bezrukavnikov-Mirkovic-Rumynin. There are many open problems here and the interplay between representation theory, geometry of nilpotent orbits and Noetherian ring theory makes this area very attractive for young reserachers.
Around 2004 I started exploring a link between modular representation theory of reductive Lie algebras and the theory of primitive ideals of universal enveloping algebras over complex numbers. I came across this link when studying certain noncommutative deformations of Slodowy slices. These deformations arise as the endomorphism algebras of generalised Gelfand-Graev modules, but it turned out later that they are also isomorphic to finite W-algebras of mathematical physics (this was first rigorously proved in 2007 by D'Andrea-De Concini-De Sole-Heluani-Kac).
Representation theory of finite W-algebras is extremely rich and interesting. For example, it would be important for the theory of primitive ideals to prove that every finite W-algebra admits one-dimensional representations. As a partial confirmation of this, I proved in 2007 that every finite W-algebra has finite-dimensional representations (later this was reproved by Losev, who used a completely different method involving Fedosov quantisation).
One can attach a finite W-algebra to any nilpotent orbit in a complex semisimple Lie algebra, and all finite W-algebras have nice filtrations and PBW bases. However, defining relations of finite W-algebras are difficult to determine. For general linear Lie algebras this was done by Brundan-Kleshchev, who identified the finite W-algebras of type A with truncated shifted Yangians. Outside type A the presentation problem for finite W-algebras remains wide open.
Apart from the above-mentioned topics I am also interested in invariant theory of algebraic groups. This includes various problems involving nilpotent orbits, commuting varieties and symmetric invariants of centralisers in reductive Lie algebras.