Complex product structures on some simple Lie groups

TL;DR

This paper constructs invariant complex product and related structures on certain noncompact simple Lie groups, demonstrating compatibility with metrics and providing examples of Einstein manifolds with special geometric properties.

Contribution

It introduces new invariant complex product, hyperparacomplex, and indefinite quaternion structures on specific Lie groups, some compatible with biinvariant metrics, and presents Einstein manifold examples.

Findings

Constructed invariant complex product structures on $SL(2m-1,\RR)$, $SU(m,m-1)$, and $SL(2m-1,\CC)^\RR$.

Some structures are compatible with biinvariant Killing metrics.

Provided examples of compact hyperparahermitean, non-flat Einstein manifolds.

Abstract

We construct invariant complex product (hyperparacomplex, indefinite quaternion) structures on the manifolds underlying the real noncompact simple Lie groups $SL(2m-1,\RR)$, $SU(m,m-1)$ and $SL(2m-1,\CC)^\RR$. We show that on the last two series of groups some of these structures are compatible with the biinvariant Killing metric. Thus we also provide a class of examples of compact (neutral) hyperparahermitean, non-flat Einstein manifolds.