Flat manifolds with holonomy representation of quaternionic type

Abstract

We are interested in the question of the existence of flat manifolds for which all $\mathbb{R}$-irreducible components of the holonomy representation are either absolutely irreducible, of complex or of quaternionic type. In the first two cases such examples are well known. But the existence of the third type of flat manifolds was unknown to the authors. In this article, we construct such an example. Moreover, we present a list of finite groups for which a construction of manifolds of quaternionic type is impossible.

Keywords:

• Bieberbach group
• flat manifold
• Frobenius-Schur indicator
• hyperkähler manifold
• quaternionic representation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20H15
• Secondary: 20C15
• 53C26
• 57N16

Additional information

Funding

Andrzej Szczepański was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) in Sao Paulo (Brazil) and MPI Bonn.