engleski

Lie algebra modules which are locally finite and with finite multiplicities over the semisimple part

For a finite-dimensional Lie algebra L over C with a fixed Levi decomposition L=g+r where g is semi- simple, we investigate L-modules which decompose, as g-modules, into a direct sum of simple finite- dimensional g-modules with finite multiplicities. We call such modules g-Harish-Chandra modules. We give a complete classification of simple g-Harish- Chandra modules for the Takiff Lie algebra associated to g=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 g- Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff sl_2 and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple g- Harish-Chandra modules.

g-Harish-Chandra module ; Enright-Arkhipov localization ; Takiff Lie algebra ; Schrödinger Lie algebra

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