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THE PRIME IDEALS AND SIMPLE MODULES OF THE UNIVERSAL ENVELOPING ALGEBRA U(๐”Ÿโ‹‰V2)

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

Published online by Cambridge University Press:ย  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: [email protected]
TAO LU
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, China e-mail: [email protected]
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Abstract

Let ๐”Ÿ be the Borel subalgebra of the Lie algebra ๐”ฐ๐”ฉ2 and V2 be the simple two-dimensional ๐”ฐ๐”ฉ2-module. For the universal enveloping algebra $\[{\cal A}: = U(\gb \ltimes {V_2})\]$ of the semi-direct product ๐”Ÿโ‹‰V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence โ€œThe sets of completely primeโ€ฆโ€.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(๐”Ÿโ‹‰V2)-modules are classified: the Whittaker modules, the ๐•‚[X]-torsion modules and the ๐•‚[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 ยฉ Glasgow Mathematical Journal Trust 2019

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