$(\Theta_n,sl_n)$-graded Lie algebras $(n=3,4)$

Abstract

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 $\Delta$. Let $\Gamma$ be a finite set of integral weights of $\mathfrak{g}$ containing $\Delta$ and $\{0\}$. Following [2,10], we say that a Lie algebra $L$ over $\mathbb{F}$ is \emph{generalized root graded}, or more exactly $(\Gamma,\mathfrak{g})$-\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 $L_{\alpha}$ ($\alpha\in\Gamma$) and $L$ is generated by all $L_{\alpha}$ with $\alpha\ne0$ as a Lie algebra. Let $\mathfrak{g}\cong sl_{n}$ and \[ \Theta_n = \{0,\pm\varepsilon_i \pm\varepsilon_j, \pm\varepsilon_i, \pm2\varepsilon_i \mid1 \leq i \neq j \leq n\} \] where $\{\varepsilon_1, \dots, \varepsilon_n\}$ is the set of weights of the natural $sl_{n}$-module. In [9], we classify $(\Theta_{n},sl_{n})$-graded Lie algebras for $n>4$. In this paper we describe the multiplicative structures and the coordinate algebras of $(\Theta_{n},sl_{n})$-graded Lie algebras $(n=3,4)$. In $n=3$, we assume that \[ [V(2\omega_{1})\otimes C,V(2\omega_{1})\otimes C]=[V(2\omega_{2})\otimes C',V(2\omega_{2})\otimes C']=0 \] where $V(\omega)$ is the simple $\mathfrak{g}$-module of highest weight $\omega$, $C={\rm Hom_{\mathfrak{g}}}(V(2\omega_{1}),L)$ and $C'={\rm Hom_{\mathfrak{g}}}(V(2\omega_{2}),L)$ .