Hypercomplex structures on special linear groups

• Andrada, Adrián ; Garrone, Agustín ^[1] ; Tolcachier, Alejandro ^[1]
  1. [1] Universidad Nacional de Córdoba

     Universidad Nacional de Córdoba

     Argentina

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 2, 2026, págs. 433-460
• Idioma: inglés
• DOI: 10.1007/s13348-025-00475-x
• Texto completo no disponible (Saber más ...)
• Resumen

  • The purpose of this article is twofold. First, we prove that the 8-dimensional Lie group \operatorname {SL}(3,{\mathbb {R}}) does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on \operatorname {SL}(3,{\mathbb {R}}) due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on \operatorname {SL}(2n+1,{\mathbb {C}}), which arises from a complex product structure on \operatorname {SL}(2n+1,{\mathbb {R}}), for all n\in {\mathbb {N}}. We then show that there are no left-invariant HKT metrics compatible with this hypercomplex structure. Additionally, we determine the associated Obata connection and we compute explicitly its holonomy group, thus providing a new example of an Obata holonomy group properly contained in \operatorname {GL}(m,{\mathbb {H}}) and not contained in \operatorname {SL}(m,{\mathbb {H}}), where 4m=\dim _{\mathbb {R}}\operatorname {SL}(2n+1,{\mathbb {C}}),

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