We use Cvitanović’s diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation associated with the e6 and e7 families of Lie algebras, and thus explain Westbury’s observations about their uniform spectral decompositions. In doing so we explore the extensions of the Brauer and symmetric group algebras to the centralizer algebras of e7 and e6 on their lowest-dimension...

Graded nilpotent Lie groups, or Carnot Groups are to subRiemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between subRiemannian and Riemannian geometry is a strong one, one might conjecture that the subRiemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture is f...

The Yangian of the Lie algebra gl N has a distinguished family of irreducible finite-dimensional representations, called elementary representations. They are parametrized by pairs, consisting of a skew Young diagram and a complex number. Each of these representations has an explicit realization, it extends the classical realization of the irreducible polynomial representations of gl N by means ...

In [Zh], R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l + 1) and the Lie superalgebra osp(1, 2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebrasà la Drinfeld-Jimbo and to show how this construction can explain the main theorem of [GL2]: the annihilator of a Verma module over the Lie supera...

We use Cvitanović’s diagrammatic techniques to construct the rational solutions of the Yang-Baxter equation associated with the e6 and e7 families of Lie algebras, and thus explain Westbury’s observations about their uniform spectral decompositions. In doing so we explore the extensions of the Brauer and symmetric group algebras to the centralizer algebras of e7 and e6 on their lowest-dimension...

We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gln and sln. We show that these quantum groups can be realized as Drinfel’d doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix. The quantum groups have a natural n-dimensional module V . The R-matrix enables...

Given a complex projective algebraic variety, write H•(X,C) for its cohomology with complex coefficients and IH •(X,C) for its Intersection cohomology. We first show that under some fairly general conditions the canonical map H•(X,C) → IH •(X,C) is injective. Now let Gr := G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. ...

The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizer...

We study cohomological induction for a pair (g, k), g being an infinite dimensional locally reductive Lie algebra and k ⊂ g being of the form k0 + Cg(k0), where k0 ⊂ g is a finite dimensional reductive in g subalgebra and Cg(k0) is the centralizer of k0 in g. We prove a general non-vanishing and k-finiteness theorem for the output. This yields in particular simple (g, k)-modules of finite type ...

for any group g, let c(g) denote the set of centralizers of g.we say that a group g has n centralizers (g is a cn-group) if |c(g)| = n.in this note, we prove that every finite cn-group with n ≤ 21 is soluble andthis estimate is sharp. moreover, we prove that every finite cn-group with|g| < 30n+1519 is non-nilpotent soluble. this result gives a partial answer to aconjecture raised by a. ashrafi in ...