Totally real and complex subspaces of a right quaternionic vector space with a Hermitian form of signature $(n,1)$

Sungwoon Kim

doi:10.4134/JKMS.j230404

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2024; 61(3): 547-564

Online first article April 16, 2024 Printed May 1, 2024

https://doi.org/10.4134/JKMS.j230404

Totally real and complex subspaces of a right quaternionic vector space with a Hermitian form of signature $(n,1)$

Sungwoon Kim

Jeju National University

Abstract

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Abstract

We study totally real and complex subsets of a right quaternionic vector space of dimension $n+1$ with a Hermitian form of signature $(n,1)$ and extend these notions to right quaternionic projective space. Then we give a necessary and sufficient condition for a subset of a right quaternionic projective space to be totally real or complex in terms of the quaternionic Hermitian triple product. As an application, we show that the limit set of a non-elementary quaternionic Kleinian group $\Gamma$ is totally real (resp.~commutative) with respect to the quaternionic Hermitian triple product if and only if $\Gamma$ leaves a real (resp.~complex) hyperbolic subspace invariant.

Keywords: Quaternionic vector space, quaternionic hyperbolic space, quaternionic Hermitian triple product

MSC numbers: Primary 20G20, 15B33, 51M10

Supported by: This research was supported by the 2023 Scientific Promotion Program funded by Jeju National University.