Symmetries of differential operators and the Fueter theorem

by Guner Muarem (University of Antwerp)

Location: Campus Sterre, Building S8, Classroom 3.2, Ghent
Time: Friday March 1, 2019 at 14:30

It is well-known that the quaternions discovered by Hamilton in 1843 form a number system that extends the complex numbers. An obvious first reflex in that time was to mimic the techniques and results from complex analysis in order to obtain a similar function theory. The first step was to find the correct analogue of the notion of holomorphic in the quaternionic setting: the so-called Fueter regularity. In this context the Swiss mathematician Fueter [1935] found a way of producing regular functions from holomorphic ones. One can now extend these notions and results to the general context of Clifford analysis: a function theory studying the so-called Dirac operator and its solutions(called monogenics) that is both a generalization of complex and Harmonic analysis (i.e. the function theory associated to the Laplace operator) but is also a refinement.

In this talk we shall explain how the Fueter theorem can be proven in Clifford algebra context by using a representation theory inspired approach. This approach shall be closely related to (generalized) symmetries of given differential operators and their induced Lie algebraic structure. Furthermore, we shall generalize these results even more for symplectic Clifford algebras, something that was still missing in the existing literature and explore the connection with special functions.