Nilpotent Lie Groups in Clifford Analysis: Five Directions for Research

TL;DR

This paper advocates for increased research on nilpotent Lie groups like the Heisenberg group within Clifford analysis, highlighting five promising directions to expand understanding and applications in mathematical physics.

Contribution

It introduces five new research directions for exploring nilpotent Lie groups in Clifford analysis and related physical models, emphasizing their underexplored potential.

Findings

Identifies five promising research directions.

Highlights underinvestment in nilpotent Lie groups.

Connects nilpotent Lie groups to mathematical physics applications.

Abstract

The aim of the paper is to popularise nilpotent Lie groups (notably the Heisenberg group and alike) in the context of Clifford analysis and related models of mathematical physics. It is argued that these groups are underinvestigated in comparison with other classical branches of analysis. We list five general directions which seem to be promising for further research. Keywords: Clifford analysis, Heisenberg group, nilpotent Lie group, Segal-Bargmann space, Toeplitz operators, singular integral operators, pseudodifferential operators, functional calculus, joint spectrum, quantum mechanics, spinor field.