$(Θ_n,sl_n)$-graded Lie algebras $(n=3,4)$
Let $\mathbb{F}$ be a field of characteristic zero and let $\mathfrak{g}$ be a non-zero finite-dimensional split semisimple Lie algebra with root system $Δ$. Let $Γ$ be a finite set of integral weights of $\mathfrak{g}$ containing $Δ$ and $\{0\}$. Following [2,10], we say that a Lie algebra $L$ over $\mathbb{F}$ is \emph{generalized root graded}, or more exactly $(Γ,\mathfrak{g})$-\emph{graded}, if $L$ contains a semisimple subalgebra isomorphic to $\mathfrak{g}$, the $\mathfrak{g}$-module $L$ is the direct sum of its weight subspaces $L_α$ ($α\inΓ$) and $L$ is generated by all $L_α$ with $α\ne0$ as a Lie algebra. Let $\mathfrak{g}\cong sl_{n}$ and \[ Θ_n = \{0,\pm\varepsilon_i \pm\varepsilon_j, \pm\varepsilon_i, \pm2\varepsilon_i \mid1 \leq i \neq j \leq n\} \] where $\{\varepsilon_1, \dots, \varepsilon_n\}$ is the set of weights of the natural $sl_{n}$-module. In [9], we classify $(Θ_{n},sl_{n})$-graded Lie algebras for $n>4$. In this paper we describe the multiplicative structures and the coordinate algebras of $(Θ_{n},sl_{n})$-graded Lie algebras $(n=3,4)$. In $n=3$, we assume that \[ [V(2ω_{1})\otimes C,V(2ω_{1})\otimes C]=[V(2ω_{2})\otimes C',V(2ω_{2})\otimes C']=0 \] where $V(ω)$ is the simple $\mathfrak{g}$-module of highest weight $ω$, $C={\rm Hom_{\mathfrak{g}}}(V(2ω_{1}),L)$ and $C'={\rm Hom_{\mathfrak{g}}}(V(2ω_{2}),L)$ .