0. Introduction. 1. Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let J C 5 be the subring of all G-invariant polynomials on X. Now let J be the set of all ƒ £ J having zero constant term and ...

This paper gives a comprehensive analysis of the inverse problem of Lagrangian dynamics for the geodesic equations of the canonical linear connection on Lie groups of dimension four. Starting from the Lie algebra, in every case a faithful four-dimensional representation of the algebra is given as well as one in terms of vector fields and a representation of the linear group of which the given a...

We extend, in the context of connected noncompact semisimple Lie group, two results of Antezana, Massey, and Stojanoff: Given 0 < λ < 1, (a) the limit points of the sequence {∆λ (X)}m∈N are normal, and (b) lim m→∞‖∆ m λ (X)‖ = r(X), where ‖X‖ is the spectral norm and r(X) is the spectral radius of X ∈ Cn×n and ∆λ(X) is the λ-Aluthge transform of X.

This paper is mainly a review concerning singular perturbation methods by means of Lie group analysis which has been presented by the author. We make use of a particular type of approximate Lie symmetries in those methods in order to construct reduced systems which describe the longtime behavior of the original dynamical system. Those methods can be used in analyzing not only ordinary different...

The notion of strict groups encompasses many important infinite dimensional groups in differential geometry. It is shown that important strict groups carry the structure of a regular Lie group in the convenient setting. In the contact case this is related to the integrability of the Poisson algebra of a prequantizable Poisson manifold.

A conjecture is formulated and discussed which provides necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup G of the orthogonal group O(d) (d being the dimension of the billiard in question) be transitive on the unit sphere Sd−1. If C1, . . . , Ck ar...

We study the renormalization group flows of two-dimensional metrics in sigma models using the one-loop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the world-sheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebr...

Let G be a connected Lie group acting by algebra automorphisms on a finite-dimensional complex central simple algebra A. The algebra A is isomorphic to the endomorphism algebra of a projective representation V of G. We study the invariant subalgebras of A. In particular, we show that if V is irreducible, then the invariant subalgebras appear in dual pairs arising from factorizations of V . We a...

We provide an upper bound of the dimension of the maximal projective submodule of the Lie module of the symmetric group of n letters in prime characteristic p, where n = pk with p k.

The Lie algebra of a nuclear group is a locally convex nuclear vector space. Mathematics Subject Classification (2000): 22B05, 22E65, 46Axx 1. The Lie Algebra of Abelian Topological Groups Let G be an abelian topological group. The set of all one–parameter–subgroups L(G) := {λ : R→ G : λ is a continuous homomorphism} is called the Lie algebra of G . We define addition pointwise and scalar multi...