We construct and study in detail various dual pairs acting on some Fock spaces between a finite dimensional Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to the action of a dual pair, present explicit formulas for the ...

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates bet...

The study of self-duality has attracted some attention over the past decade. A good deal of research in that time has been done on constructing and classifying all self-dual graphs and in particular polyhedra. We will give an overview of the recent research in the first two chapters. In the third chapter, we will show the necessary condition that a self-complementary self-dual graph have n ≡ 0,...

In universal personal telecommunication (UPT) environments, UPT networks retain information related to incall/outcall registration in UPT user service profiles in order to provide incoming UPT calls for UPT users in any location who have registered at a terminal. As UPT networks support incall registration, terminal users can be different from terminal owners, and several UPT users can register...

We use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti–self–dual Yang–Mills equations with a gauge group Diff(S). email [email protected] email [email protected] email [email protected]

We study quantized dual graded graphs, which are graphs equipped with linear operators satisfying the relation DU − qUD = rI. We construct examples based upon: the Fibonacci differential poset, permutations, standard Young tableau, and plane binary trees.

We discuss how, by including gauge singlets in supersymmetric gauge theories, one can construct and test new types of duality. This may help in finding dual theories of supersymmetric GUTs. This talk is based on the recent article Ref.[1].

We discuss a modification of Gopakumar’s prescription for constructing string theory duals of free gauge theories. We use this prescription to construct an unconventional worldsheet dual of the Gaussian matrix model.

I construct an isotropic, gauge invariant monopole condensate in SU(2) QCD. It provides a suitable, gauge-invariant vacuum for the dual-Meissner effect to provide QCD colour confinement without violating the isotropy of real space.

We construct the Nahm transform for Higgs bundles over a Riemann surface of genus at least 2 as hyperholomorphic connections on the total space of the tangent bundle of its dual Jacobian. 2000 MSC: 53C07, 53C26