We study finite dimensional almost- and quasi-effective prolongations of nilpotent $${\mathbb {Z}}$$ -graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi–Malcev and Levi–Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we consider the semisimple case and discuss some examples in which the structural algebras are central extensions of orthogonal Lie algebras and their degree $$(-\,1)$$ components arise from spin representations.

中文翻译：

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在 $${\mathbb {Z}}$$-graded Lie 代数的某些类上

我们研究了幂零 $${\mathbb {Z}}$$ - 分级李代数的有限维几乎和准有效扩展，特别关注那些具有可分解还原结构子代数的那些。我们的假设概括了有效性和代数性，并且适用于以非常自然的方式获得 Levi-Malcev 和 Levi-Chevalley 分解以及延伸的高度和其他属性的精度。在最后一节中，我们考虑半简单情况并讨论一些示例，其中结构代数是正交李代数的中心扩展，并且它们的度 $$(-\,1)$$ 分量来自自旋表示。