Odd-dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd-dimensional real Lie algebra $\mathfrak{g}$. We investigate correspondences with even-dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak{g}$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$, obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3-contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.