Generalization of Contact Elements Bundles and Relevant Properties of Weil Algebras
by
Miroslav Kures
Brno University of Technology

The Weil algebra A is a local R-algebra, the nilpotent ideal n of which has a finite dimension as a vector space and A/n=R. One can assume that Weil algebras are finite dimensional factor R-algebras of the algebra R[t¹, ..., t^k] of real polynomials in several indeterminates. It means, the Weil algebra A has a form R[t¹, ..., t^k]/i, where m^r+1 Ì i Ì m for some r, m=át¹, ..., t^kñ being the maximal ideal of R[t¹, ..., t^k] (i with this property is called the Weil ideal).

Weil algebras play very important role in actual research in differential geometry. Classical higher order contact elements are generalized to contact elements of Weil algebra type. The set of such elements has a fibered manifold structure. It is necessary the study of relevant properties of Weil algebras for the description of natural operations on these bundles. New results in this area are in the center of this contribution, especially, we study the subalgebra SA={a Î A; f(a)=a for all f Î Aut A} of a Weil algebra A and we demonstrate the role of SA in the classification of the natural operators lifting vector fields to bundles of contact elements of Weil algebra type (the result was obtained quite recently by W.M. Mikulski).

Let A be a Weil algebra with the width k ³ 1. If there is an expression of A as A=R[t¹, ..., t^k]/i, where i is a homogeneous Weil ideal, we call A the homogeneous Weil algebra. If A is a homogeneous Weil algebra, then SA is the trivial subalgebra R·1. We prove that Weil algebras of k-dimensional velocities functors of order r as well as their nonholonomic, semiholonomic and some other generalizations are homogeneous. (Such bundles are important in analytical mechanics.) Apart from that, we present that there are Weil algebras the subalgebra of fixed elements of which is nontrivial.

Date received: April 17, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cage-29.