论文标题
LIKE代数模块，在半简单部分上是本地有限的
Lie algebra modules which are locally finite over the semi-simple part
论文作者
论文摘要
对于有限维谎言代数$ \ MATHFRAK {l} $ over $ \ MATHBB {C} $，带有固定的Levi分解$ \ Mathfrak {l} = \ Mathfrak {g} \ oplus \ oplus \ Mathfrak \ Mathfrak \ Mathfrak {r} $ \ mathfrak {l} $ - 分解为$ \ mathfrak {g} $ - 模块的模块，将其直接的简单限时$ \ mathfrak {g} $ - 带有有限多重性的模块的直接总和。我们称此类模块$ \ Mathfrak {G} $ - Harish-Chandra模块。我们提供了简单的$ \ mathfrak {g} $的完整分类 - harish-chandra模块，用于takiff lie代数与$ \ mathfrak {g} = \ mathfrak {slfrak {sl} _2 $相关的代数，以及用于Schrödingerlie algebra，并在其他情况下获得一些部分结果。 Enright和Arkhipov的完整版本的适应版本在我们的论点中起着至关重要的作用。此外，我们计算了第一批无限尺寸简单$ \ mathfrak {g} $ - Harish-Chandra模块及其在通用代数中的歼灭器，用于TAKIFF $ \ MATHFRAK {SL} _2 _2 $ andSchrödingerLieAlgebra。在一般情况下，我们为存在无限维简单$ \ mathfrak {g} $ - Harish-Chandra模块的存在提供了足够的条件。
For a finite-dimensional Lie algebra $\mathfrak{L}$ over $\mathbb{C}$ with a fixed Levi decomposition $\mathfrak{L} = \mathfrak{g} \oplus \mathfrak{r}$ where $\mathfrak{g}$ is semi-simple, we investigate $\mathfrak{L}$-modules which decompose, as $\mathfrak{g}$-modules, into a direct sum of simple finite-dimensional $\mathfrak{g}$-modules with finite multiplicities. We call such modules $\mathfrak{g}$-Harish-Chandra modules. We give a complete classification of simple $\mathfrak{g}$-Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak{g} = \mathfrak{sl}_2$, and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright's and Arkhipov's completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak{g}$-Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak{sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak{g}$-Harish-Chandra modules.
