Grozman's solution of O. Veblen's problem: classify invariant operators

  • Dimitry Leites (MPI MiS, Leipzig)
A3 01 (Sophus-Lie room)


In 1928, at the Mathematical Congress in Bologna, Veblen devoted his talk to the problem of classifying invariant operators between "geometric quantities". In modern terms these are tensor fields, connections and similar objects. Eventually, it was conjectured that there is only one (type of) unary invariant differential operators.

I will retell Grozman's list of binary invariant differential operators. The answer is remarkable: there are no operators of order >3, all operators of orders 3 and 2 are compositions of 1st order operators (bar one exception), and the 8 series of first order operators determine a Lie superalgebra structure on their domain. One of these superalgebras is the well-known Poisson algebra. I will explain how to get this result and say what is known about its generalization: differentional operators invariant with respect to the group of symplectomorphisms.

Students and other listeners are most welcome to take up open problems.

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail