Department of Mathematics and Statistics,

Dalhousie University,

Halifax, Nova Scotia, B3H 3J5 CANADA;

and

Department of Applied Mathematics,

University of Waterloo,

Waterloo, Ontario, N2L 3G1 CANADA

E-mail: rsmirnov@sirius.math.uwaterloo.ca

**Algebraic invariants in pseudo-Riemannian geometry
and related questions**

**Abstract:**

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

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.