Let $G = H\ltimes K$ denote a semidirect product Lie group with Lie algebra $g = h \oplus k$, where $k$ is an ideal and $h$ is a subalgebra of the same dimension as $k$. There exist some natural split isomorphisms $S$ with $S^2 = \pm Id$ on $g$: given any linear isomorphism $j : h \to k$, we get the almost complex structure $J (x, v) = (−j^{−1}v, j x)$ and the almost paracomplex structure $E(x, v) = ( j^{−1}v, j x)$. In this work we show that the integrability of the structures $J$ and $E$ above is equivalent to the existence of a left-invariant torsion-free connection $\nabla$ on $G$ such that$ \nabla J = 0 = \nabla E$ and also to the existence of an affine structure on $H$. Applications include complex, paracomplex and symplectic geometries.

From almost (para)-complex structures to affine structures on Lie groups

Calvaruso, Giovanni;
2018-01-01

Abstract

Let $G = H\ltimes K$ denote a semidirect product Lie group with Lie algebra $g = h \oplus k$, where $k$ is an ideal and $h$ is a subalgebra of the same dimension as $k$. There exist some natural split isomorphisms $S$ with $S^2 = \pm Id$ on $g$: given any linear isomorphism $j : h \to k$, we get the almost complex structure $J (x, v) = (−j^{−1}v, j x)$ and the almost paracomplex structure $E(x, v) = ( j^{−1}v, j x)$. In this work we show that the integrability of the structures $J$ and $E$ above is equivalent to the existence of a left-invariant torsion-free connection $\nabla$ on $G$ such that$ \nabla J = 0 = \nabla E$ and also to the existence of an affine structure on $H$. Applications include complex, paracomplex and symplectic geometries.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11587/426935
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