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LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART

Part of: Lie algebras and Lie superalgebras Algebraic combinatorics

Published online by Cambridge University Press:  02 August 2021

VOLODYMYR MAZORCHUK Open the ORCID record for VOLODYMYR MAZORCHUK [Opens in a new window]  and
RAFAEL MRÐEN Open the ORCID record for RAFAEL MRÐEN [Opens in a new window]
VOLODYMYR MAZORCHUK
Affiliation:
Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, Swedenmazor@math.uu.se
RAFAEL MRÐEN
Affiliation:
Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, Sweden (On leave from: Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, Zagreb 10000, Croatia) rafael.mrden@math.uu.se
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Abstract

For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, 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.

MSC classification

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 17B30: Solvable, nilpotent (super)algebras 05E10: Combinatorial aspects of representation theory
Type
Article
Information
Nagoya Mathematical Journal , Volume 246 , June 2022 , pp. 430 - 470
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Copyright
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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Footnotes

This research was partially supported by the Swedish Research Council, Göran Gustafsson Stiftelse and Vergstiftelsen. Rafael Mrđen was also partially supported by the QuantiXLie Center of Excellence grant no. KK.01.1.1.01.0004 funded by the European Regional Development Fund.

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