2016/04/28 by Stuart Shirrell, Shirrell, Stuart, Raymond Walter +1
Mathematics · #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1604.08647
openalex publication_date 2016/04/28 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
This work reconsiders the holomorphic and anti-holomorphic Dirac operators of\nHermitian Clifford analysis to determine whether or not they are the natural\ngeneralization of the orthogonal Dirac operator to spaces with complex\nstructure. We argue the generalized gradient construction of Stein and Weiss\nbased on representation theory of Lie groups is the natural way to construct\nsuch a Dirac-type operator because applied to a Riemannian spin manifold it\nprovides the Atiyah-Singer Dirac operator. This method, however, does not apply\nto these Hermitian Dirac operators because the representations of the unitary\ngroup used are not irreducible, causing problems in considering invariance\nunder a group larger than U(n). This motivates either the development of\nClifford analysis over a complex vector space with respect to a Hermitian inner\nproduct or the development of Dirac-type operators on Cauchy-Riemann\nstructures.\n