Noncubic Dirac operators for finite dimensional modules

TL;DR

This paper investigates the structure of kernels of noncubic Dirac operators on finite-dimensional modules, comparing them with cubic Dirac operators and exploring implications for classical and exceptional Lie algebras.

Contribution

It provides a detailed analysis of the irreducible decomposition of noncubic Dirac operator kernels and highlights differences from cubic Dirac operators, including cases where full isotypic components are absent.

Findings

Kernel of noncubic Dirac operators may lack full isotypic components.

Detailed case studies on classical and exceptional Lie algebras.

Insights into kernels of geometric Dirac operators on compact manifolds.

Abstract

We study the decomposition into irreducibles of the kernel of noncubic Dirac operators attached to finite-dimensional modules. We compare this decomposition with features of Kostant's cubic Dirac operator. In particular, we show that the kernel of noncubic Dirac operators need not contain full isotypic components. The cases of classical and exceptional complex Lie algebras are studied in detail. As a by-product, we deduce some information on the kernel of noncubic geometric Dirac operators acting on sections over compact manifolds studied by Slebarski.