Allison-Benkart-Gao functor and the cyclicity of free alternative functors

Let $k$ be a field of characteristic $0$. We introduce a pair of adjoint functors, Allison-Benkart-Gao functor $\text{AG}$ and Berman-Moody functor $\text{BM}$, between the category of non-unital alternative algebras over $k$ and the category $\text{LieR}$ of Lie algebras with compatible $s{l}_3(k)$-actions. Surprisingly, when $A$ is an alternative algebra without a unit, the Allison-Benkart-Gao Lie algebra $\text{AG}(A)$ is not isomorphic to the more well-known Steinberg Lie algebra $s{t}_3(A)$ in general. Let $A(D)$ be the free (non-unital) alternative algebra over $D$ generators with the inner derivation algebra $\text{innAD}$. A conjecture on the homology $H_r(\text{AGAD})$ is proposed. Furthermore, consider the degree $n$ component of $A(D{)}_n$(resp. ${\text{innAD}}_n$). The previous conjecture implies another conjecture on the dimensions on $A(D{)}_n$ and $\text{Inner}A(D{)}_n$. Some evidences are given to support these conjectures. Finally, we prove the cyclicity of the alternative structure, namely that the symmetric group $S_{1+D}$ acts on the multilinear part of $A(D)$, which plays an important role to connect the Lie algebra homology of $\text{AGAD}$ and the character of $A(D)$.