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THE PRIME IDEALS AND SIMPLE MODULES OF THE UNIVERSAL ENVELOPING ALGEBRA U(𝔟⋉V2)

Part of: Chain conditions, growth conditions, and other forms of finiteness Modules, bimodules and ideals Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  29 July 2019

VOLODYMYR V. BAVULA  and
TAO LU
VOLODYMYR V. BAVULA
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield, S3 7RH, UK e-mail: v.bavula@sheffield.ac.uk
TAO LU
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, China e-mail: lutao.sdu@gmail.com
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Abstract

Let 𝔟 be the Borel subalgebra of the Lie algebra 𝔰𝔩2 and V2 be the simple two-dimensional 𝔰𝔩2-module. For the universal enveloping algebra $\[{\cal A}: = U(\gb \ltimes {V_2})\]$ of the semi-direct product 𝔟⋉V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence “The sets of completely prime…”.The sets of completely prime ideals of $\[{\cal A}\]$ are described. The simple unfaithful $\[{\cal A}\]$-modules are classified and an explicit description of all prime factor algebras of $\[{\cal A}\]$ is given. The following classes of simple U(𝔟⋉V2)-modules are classified: the Whittaker modules, the 𝕂[X]-torsion modules and the 𝕂[E]-torsion modules.

MSC classification

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 16D25: Ideals 16D60: Simple and semisimple modules, primitive rings and ideals 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation 16P50: Localization and Noetherian rings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Special Issue S1: Workshop on Nonassociative algebras and their applications , December 2020 , pp. S77 - S98
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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