Parabolic Geometries Related with Several Fueter Operators

摘要

The theory developed for certain geometric structures (the so-called parabolic geometries) was successfully applied in a study of functions of several quaternionic variables satisfying the Fueter (Dirac) equation in each variable separately. The special case of a parabolic geometry used in this application is the quaternionic structure on a manifold with dimension 4n. A homogeneous model for quaternionic geometry is the quaternionic projective space mathbbPn(mathbbH){mathbb{P}}^n({mathbb{H}}). A question discussed in the paper is whether similar applications are possible for dimensions different from four. In the quaternionic case, the Dirac operator is acting on functions defined on R 4n , considered as an open dense subset in mathbbPn(mathbbH){mathbb{P}}^n({mathbb{H}}). This situation has no natural generalization in dimensions different from four. In the paper, we are discussing a possibility to embed R 4n to a homogeneous model for another type of a parabolic geometry in such a way that it would admit generalizations to higher dimensions. There is, indeed, another possibility with such a property but there is a price to pay for it. The embedding is no more onto an open dense subset, there are ‘dummy’ variables necessarily present. It is possible to show that a suitable invariant first order differential operator in the corresponding parabolic geometry, when acting on functions independent of those dummy variables, coincides with the Fueter operators in several quaternionic variables. This scheme can be extended also to other dimensions, different from four.