In this note we study Cartan subalgebras of Lie algebras defined over finite fields. We prove that a possible Lie algebra of minimal dimension without Cartan subalgebras is semisimple. Subsequently, we study Cartan subalgebras of gl(n, F ). AMS classification: 17B50

1.1. Let G be a semisimple complex Lie group with the Cartan datum (I, ·) and the root datum (Y,X, . . . ). Let H ⊂ B = B+,B− ⊂ G be a Cartan subgroup and a pair of opposite Borel subgroups respectively. Let X = G/B be the flag manifold of G. Let C = P ∋ ∞ be the projective line. Let α = ∑ i∈I aii ∈ N[I] ⊂ H2(X,Z). The moduli space of G-monopoles of topological charge α (see e.g. [4]) is natura...

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel of submanifolds. We prove that the submanifolds are anisotropic-minimal obtain a general Cartan-type formula form with vanishing reversible torsion, from which give some classifications on number distinct principal curvatures or their multiplicities.

We show that the characters of all highest weight modules over an affine Lie algebra with the highest weight away from the critical hyperplane are meromorphic functions in the positive half of the Cartan subalgebra, their singularities being at most simple poles at zeros of real roots. We obtain some information about these singularities. 0. Introduction 0.0.1. Let g be a simple finite-dimensio...

An investigation into the Cartan form and nondegeneracy conditions for field-theoretic Lagrangians based on the Cartan equivalence method.

We describe four types of inner involutions of the Cartan-Weyl basis providing (for |q| = 1 and q real) three types of real quantum Lie algebras: Uq(O(3, 2)) (quantum D = 4 anti-de-Sitter), Uq(O(4, 1)) (quantum D = 4 de-Sitter) and Uq(O(5)). We give also two types of inner involutions of the Cartan-Chevalley basis of Uq(Sp(4;C)) which can not be extended to inner involutions of the Cartan-Weyl ...

Without using Gabber’s theorem, the finite-dimensionality of the space of conformal blocks in the Wess-Zumino-Novikov-Witten models is proved. §0 Introduction. Conformal field theory with non-abelian gauge symmetry, called the Wess-ZuminoNovikov-Witten (WZNW) model, has been studied by many physicists and mathematicians. A mathematical formulation of this model over the projective line is given...

We study local automorphisms of holomorphic Cartan geometries. This leads to classification results for compact complex manifolds admitting holomorphic Cartan geometries. We prove that a compact Kähler Calabi-Yau manifold bearing a holomorphic Cartan geometry of algebraic type admits a finite unramified cover which is a complex