We introduce a concept of causality in the framework of generalized pseudo-Riemannian Geometry in the sense of Colombeau. This is motivated by a generalized point value characterization of generalized pseudoRiemannian metrics due to M. Kunzinger et al. We prove an appropriate version of the inverse Cauchy-Schwarz inequality. As an application, we establish a dominant energy condition for some e...

For fixed m and n , we consider the vector space of linear differential equations of order n whose coefficients are polynomials of degree at most m . We show that for G in a large class of linear algebraic groups, if we fix the exponents and determining factors at the singular points (but not the singular points themselves) then the set of such differential equations with this fixed data, fixed...

In this paper we investigate the conditions under which a given ring is a sub-ring of a semi-simple ring. For convenience, we say that a ring A is an extension of a ring B, if B is a sub-ring of A. It is found that the existence of a semi-simple extension is equivalent to the vanishing of the extension radical, a two-sided ideal defined analogously to the ordinary radical. In Theorem II we give...

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1, 2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not gene...

Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field k of characteristic p > 0, T a maximal torus of G and B a Borel subgroup containing T . Each weight in X(T ) determines a line bundle on the flag varietyG/B. It turns out that cohomology of the line bundle is isomorphic to the derived functor of the induction functors from the category of B-m...

Let L be a free Lie algebra of finite rank r over an arbitrary field K of characteristic 0, and let Ln denote the homogeneous component of degree n in L . Viewed as a module for the general linear group GL(r, K ), Ln is known to be semisimple with the isomorphism types of the simple summands indexed by partitions of n with at most r parts. Klyachko proved in 1974 that, for n > 6, almost all suc...

This paper is a review of results on generalized Harish-Chandra modules in the framework of cohomological induction. The main results, obtained during the last 10 years, concern the structure of the fundamental series of (g, k)−modules, where g is a semisimple Lie algebra and k is an arbitrary algebraic reductive in g subalgebra. These results lead to a classification of simple (g, k)−modules o...

Let G be a semisimple algebraic group with Lie algebra g. In 1979, J. Dixmier proved that any vector field annihilating all G-invariant polynomials on g lies in the k[g]module generated by the ”adjoint vector fields”, i.e., vector fields ς of the form ς(y)(x) = [x, y], x, y ∈ g. A substantial generalisation of Dixmier’s theorem was found by Levasseur and Stafford. They explicitly described the ...