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 $\AG$ and Berman-Moody functor $\BM$, between the category of non-unital alternative algebras over $k$ and the category $\LieR$ of Lie algebras with compatible $sl_3(k)$-actions. Surprisingly, when $A$ is an alternative algebra without a unit, the Allison-Benkart-Gao Lie algebra $\AG(A)$ is not isomorphic to the more well-known Steinberg Lie algebra $st_3(A)$ in general. Let $A(D)$ be the free (non-unital) alternative algebra over $D$ generators with the inner derivation algebra $\innAD$. A conjecture on the homology $H_r(\AGAD)$ is proposed. Furthermore, consider the degree $n$ component of $A(D)_n$(resp. $\innAD_n$). 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 $\AGAD$ and the character of $A(D)$.