XXV Workshop on Geometric Methods in Physics 2-8.07.2006

Irina Shchepochkina

Veblen’s problem (1928) “classify invariant differential operators” and Lie superalgebras

All fundamental laws of physics (chemistry, biology, etc.) are invariant with respect to corresponding groups of transformations. In particular, General relativity is invariant wrt all diffeomorphisms, i.e., locally, wrt Lie algebra of vector fields.

The exterior differential is the only unary invariant operator acting in the spaces of tensor fields. All 1st order binary operators determine, miraculously, a Lie superalgebra on their domain; these Lie superalgebras are close to simple ones.

To classify operators invariant wrt symplectomorphisms or Poisson Lie algebra, we need the notion of primitive forms. This notion is most lucid in terms of the Howe duality which most clear from Lie superalgebras point of view.

The talk will be elementary modulo the definition of Lie superalgebras.