论文标题
关于同情谎言代数的不存在，尺寸小于25
On the non-existence of sympathetic Lie algebras with dimension less than 25
论文作者
论文摘要
在本文中，我们调查了同情谎言代数$ \ mathfrak {g} $的最低维度问题，当它的levi subalgebra $ \ mathfrak {g} _l $很简单时。我们建立了完美谎言的nilradical的结构，代数$ \ mathfrak {g} $，作为$ \ mathfrak {g} _l $ -module，并确定可能的谎言结构，即这样一个$ \ mathfrak {g mathfrak {g} $。我们证明，作为$ \ mathfrak {g} _l $ - 模块，nilradical必须将至少分解为至少4个简单的模块。我们明确地计算出完美的谎言代数$ \ mathfrak {g} $的半简单推导，并使用levi subalgebra $ \ mathfrak {g} _l = \ mathfrak {sl} _2 _2 _2 _2 _2（\ mathbb {c}）属于$ \ mathak的必要条件。半神经衍生物。我们表明，没有比尼尔自由基的分解的同情谎言代数低于15。如果Nilradical具有4个简单的模块，我们表明一个交感神经代数的尺寸大于或等于25。
In this article we investigate the question of the lowest possible dimension that a sympathetic Lie algebra $\mathfrak{g}$ can attain, when its Levi subalgebra $\mathfrak{g}_L$ is simple. We establish the structure of the nilradical of a perfect Lie algebra $\mathfrak{g}$, as a $\mathfrak{g}_L$-module, and determine the possible Lie algebra structures that one such $\mathfrak{g}$ admits. We prove that, as a $\mathfrak{g}_L$-module, the nilradical must decompose into at least 4 simple modules. We explicitly calculate the semisimple derivations of a perfect Lie algebra $\mathfrak{g}$ with Levi subalgebra $\mathfrak{g}_L = \mathfrak{sl}_2(\mathbb{C})$ and give necessary conditions for $\mathfrak{g}$ to be a sympathetic Lie algebra in terms of these semisimple derivations. We show that there is no sympathetic Lie algebra of dimension lower than 15, independently of the nilradical's decomposition. If the nilradical has 4 simple modules, we show that a sympathetic Lie algebra has dimension greater or equal than 25.
