  We introduce the notion of abelian almost contact structures on an odd
dimensional real Lie algebra $\mathfrak g$. This a sufficient condition for the
structure to be normal. We investigate correspondences with even dimensional
real Lie algebras endowed with an abelian complex structure, and with K\"ahler
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 and study abelian almost 3-contact structures on
real Lie algebras of dimension $4n+3$. These are given by triples of abelian
almost contact structures, satisfying certain compatibility conditions, which
are equivalent to the existence of a sphere of abelian almost contact
structures. We obtain 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 structure and a compatible left invariant
Riemannian metric. We determine conditions for $G$ to admit a special metric
connection with totally skew-symmetric torsion, called canonical, which plays
the role of the Bismut connection for HKT structures arising from abelian
hypercomplex structures. We provide examples and discuss the parallelism of the
torsion of the canonical connection.
