Department of Mathematics and Statistics,
Dalhousie University,
Halifax, Nova Scotia, B3H 3J5 CANADA;
Department of Applied Mathematics,
University of Waterloo,
Waterloo, Ontario, N2L 3G1 CANADA

Algebraic invariants in pseudo-Riemannian geometry and related questions

Classical invariant theory was conceived in the 19th century as the study of  intrinsic properties of homogeneous polynomials. Since then it has been recognized as a common branch of representation theory, algebraic geometry, commutative algebra and algebraic combinatorics. In the modern mathematical language the basic problem of the classical invariant theory can be characterized as follows:

Let V be a K-vector space on which a group G acts linearly. In the ring of polynomial functions K[V] describe the subring K[V]G consisting of all polynomial functions on V that remain fixed under the action of the group G.

We formulate and solve an analogous problem by extending the underlying ideas of the classical theory of algebraic invariants to the study of Killing tensors in pseudo-Riemannian geometry. The new invariants can be effectively used in various applications arising in Mathematical Physics. As an illustration, we apply the new invariants to classification problems of the Hamilton-Jacobi theory of separation of variables.

This is joint work with Ray McLenaghan and Dennis The.