Totally real and complex subspaces of a right quaternionic vector space with a Hermitian form of signature $(n,1)$
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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.
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.
참고문헌 (6)
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