Classical Lie Groups and Quantum Affine Algebras

Project: Research project (funded)Research

Project Details


This project initially aimed to achieve the following five goals.

1. To provide explicit realizations of all irreducible finite-dimensional representations of affine Hecke algebras as the images of intertwining operators.

2. To establish connections between intertwining operators for affine Hecke algebras, and extremal projectors for the Lie algebras of classical groups.

3. For orbital varieties associated to spherical orbits of simple classical Lie groups, to determine their ideals of definition and to quantize them .

4. For any simple classical Lie group, to find those orbital varieties which are complete intersections in some nilradical, and to quantize them.

5. For any simple classical Lie group, to describe the union of orbital varieties sitting inside intersection of closures of hypersurface orbital varieties.

Layman's description

The classical Lie groups are continuous groups most frequently appearing in various branches of Mathematics. One of them is the group of all invertible linear transformations of a finite-dimensional vector space over the field of real numbers, called the general linear group. If the vector space is equipped with a non-degenerate bilinear form, symmetric or alternating, then the group of all linear transformations preserving this form is called the orthogonal or the symplectic group respectively. Representation Theory is the special branch of Mathematics which investigates, in particular, how these groups appear as "symmetries" of other mathematical objects. It also investigates the analogues of these groups when the basic field of real numbers is replaced by a discontinuous field, such as the field of p-adic numbers that comes from Number Theory. For the two types of basic fields, continuous and not, the corresponding representation theories appear to be far from each other. The principlal aim of the proposed research is to build new bridges between the two theories, hence developing both of them. To achieve this aim, we are going to use recent advances in Quantum Mechanics and Statistical Physics which led to the discovery of new mathematical objects, called quantum affine algebras. It is the technique coming from this new mathematical area, that we will be using to build our bridges.

Key findings

1. We studied the composition of the functor from the category of modules over the Lie algebra gl(m) to the category of modules over the degenerate ane Hecke algebra of GL(N) introduced by Cherednik, with the functor from the latter category to the category of modules over the Yangian Y(gl(n)) due to Drinfeld. We gave a representation theoretic explanation of a link between the intertwining operators on the tensor products of Y(gl(n))-modules, and the extremal cocycle on the Weyl group of gl(m) defined by Zhelobenko. We also establishd a connection between the composition of the functors, and the centralizer construction of the Yangian Y(gl(n)) discovered by Olshanski.

2. We obtained the antismmetric versions of all the results described as Item 1.

3. We introduced the analogue of the composition of the Cherednik and
Drinfeld functors for twisted Yangians Y(sp(n)) and Y(so(n)) corresponding to
the symplecic and the orthogonal Lie algebras. Our construction was based on the Howe duality, and originated from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we established a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.

4. We obtained the antismmetric versions of all the results described as Item 3.

5. We give a proof of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q.

6. For any complex reductive Lie algebra g and any locally finite g-module V , we extended to the tensor product U(g)⊗V the Harish-Chandra description of g-invariants in the universal enveloping algebra U(g).

7. By using our results described as Items 2,4 and 6 we provided explicit realizations of irreducible modules of the Yangians as certain quotients of tensor products of symmetic and exterior powers of the defining vector space. For the Yangian Y(gl(n)) such realizations have been known, but we gave new proofs of these results. For the twisted Yangian Y(sp(n)), we realized all irreducible finite-dimensional modules. For the twisted Yangian Y(so(n)), we realized all those irreducible finite-dimensional modules, where the action of the Lie algebra son integrates to an action of the special orthogonal Lie group SO(n).
Effective start/end date1/05/0531/07/08


  • EPSRC: £137,965.00