Rupert W. T. Yu

Indice et formes linéaires stables dans les algèbres de Lie

We characterize, in a purely algebraic manner, certain linear forms, called stable, on a Lie algebra. As an application, we determine the index of a Borel subalgebra of a semi-simple Lie algebra. Finally, we give an example of a parabolic subalgebra of a semi-simple Lie algebra which does not admit any stable linear form.

Algébre de lie de type witt

Let ρ be an abelian group and k a commutative field. Inspired by the example of the Witt algebra, we introduce a family of ρ-graded Lie algebras (called Witt type Lie algebras) V = ⊕α∊r Vα. where the dimension of each homogeneous component is 1. We characterise those which are simple and we give a complete classification of these Lie algebras. When P is free and the characteristic of k is zero, we show that the universal central extension of simple Witt type Lie algebras is 1-dimensional, and the corresponding cocycle is similar to the cocycle of the Virasoro algebra.

Lattice Polytopes Associated to Certain Demazure Modules of sln + 1

Let w be an element of the Weyl group of sln + 1. We prove that for a certain class of elements w (which includes the longest element w0 of the Weyl group), there exist a lattice polytope Δ ⊂ Rl(w), for each fundamental weight ωi of sln + 1, such that for any dominant weight λ = ∑i = 1n aiωi, the number of lattice points in the Minkowski sum Δwλ = ∑i = 1naiΔiw is equal to the dimension of the Demazure module Ew(λ). We also define a linear map Aw : Rl(w) → P ⊗Z R where P denotes the weight lattice, such that char Ew(λ) = eλ ∑e−A(x) where the sum runs through the lattice points x of Δλw.