Allison–Benkart–Gao functor and the related cyclic triple (A,ℬA,ΘA)

In this paper, we introduce a pair of adjoint functors, the Allison–Benkart–Gao functor [Formula: see text] and the Berman–Moody functor [Formula: see text], which connects the category of possibly non-unital alternative algebras over a field [Formula: see text] and the category [Formula: see text] of Lie algebras with compatible [Formula: see text]-actions. We show that when an alternative algebra [Formula: see text] does not have a unit, the Allison–Benkart–Gao Lie algebra [Formula: see text] is generally not isomorphic to the better-known Steinberg Lie algebra [Formula: see text]. Let [Formula: see text] be the free (non-unital) alternative algebra over [Formula: see text] generators. We study the [Formula: see text]-module structures of certain homology spaces associated with [Formula: see text]. Furthermore, we prove that the triple [Formula: see text] is cyclic—a result that gives another proof of the cyclicity of the alternative operad.