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

Scientific paper - Original scientific paper

• Author: Volodymyr Mazorchuk
• Author: Rafael Mrđen
• Author's institution: University of Zagreb; Faculty of Civil Engineering
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Civil Engineering

Abstract (english)

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.

• Keywords (english): g-Harish-Chandra module; Enright-Arkhipov localization; Takiff Lie algebra; Schrödinger Lie algebra
• Language: english
• Publication type: Scientific paper - Original scientific paper
• Publication status: Published
• Peer review: Peer review - international
• Publication version: Published version
• Journal title: Nagoya mathematical journal
• Numbering: vol. 246, pp. 430-470
• e-ISSN: 2152-6842
• URN:NBN: urn:nbn:hr:237:520806
• Publication: 2022
• Type of resource: Text
• Access conditions: Open access
• Created on: 2024-09-05 05:18:55