论文标题
$（θ_n，sl_n）$ - 分级谎言代数$（n = 3,4）$
$(Θ_n,sl_n)$-graded Lie algebras $(n=3,4)$
论文作者
论文摘要
令$ \ mathbb {f} $为特征为零的字段，让$ \ mathfrak {g} $为非零有限分式分式拆分半密码lie lie代数，带有root system $Δ$。令$γ$为$ \ mathfrak {g} $包含$δ$和$ \ {0 \} $的有限整体权重。以下[2,10]，我们说，$ \ mathbb {f} $超过\ emph {一般性root rage}或更准确的$（γ，γ，\ mathfrak {G}）$ - $ \ mathfrak {g} $ - module $ l $是其重量子空间的直接总和$l_α$（$α\inγ$），并且$l_α$以$α\ ne0 $作为lie代数生成了所有$l_α$。令$ \ mathfrak {g} \ cong sl_ {n} $和\ [θ_n= \ {0，\ pm \ pm \ varepsilon_i \ pm \ pm \ pm \ varepsilon_j，\ pm \ pm \ pm \ pm \ pm \ varepsilon_i，\ pm2 \ pm2 \ pm2 \ varepsilon_i \ whath whe whing whing whing $ \ {\ varepsilon_1，\ dots，\ varepsilon_n \} $是天然$ sl_ {n} $ - 模块的一组权重。在[9]中，我们对$（θ_{n}，sl_ {n}）$ - 划分的lie代数为$ n> 4 $。在本文中，我们描述了$（θ_{n}，sl_ {n}）$的乘法结构和坐标代数 - 分级lie代数$（n = 3,4）$。在$ n = 3 $中，我们假设\ [[V（2Ω_{1}）\ otimes c，v（2Ω__{1}）\ otimes c] = [v（2Ω__{2}）\ otimes c'，v（2} _ {2}） $ \ mathfrak {g} $ - 最高权重$ω$的模块，$ c = {\ rm hom _ {\ mathfrak {\ mathfrak {g}}}}}}}}}}（v（2Ω_{1}），l）$和$ c'= {\ rm hom _ {\ rm hom _ {\ mathfrak {\ mathfrak}
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 $C'={\rm Hom_{\mathfrak{g}}}(V(2ω_{2}),L)$ .
