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Journal of Lie Theory 28 (2018), No. 2, 525--560
Copyright Heldermann Verlag 2018

The Universal Enveloping Algebra U(sl2 sdir V2), its Prime Spectrum and a Classification of its Simple Weight Modules

Vladimir V. Bavula
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, England

Tao Lu
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China


\def\l{{\frak l}} \def\s{{\frak s}} \def\sdir#1{\hbox{$\mathrel\times{\hskip -4.3pt {\vrule height 4.0 pt depth 0 pt}}\hskip 2pt_{#1}$}} For the enveloping algebra $A$ of the Lie algebra $\s\l_2\sdir{}V_2$, explicit descriptions of its prime, primitive, completely prime and maximal spectra are given. A classification of simple weight $\s\l_2\sdir{}V_2$-modules is given. Generators and defining relations are found for the centralizer $C_A(H)$ in $A$ of the Cartan element $H$ of $\s\l_2\sdir{}V_2 $. Explicit descriptions of the prime, primitive, completely prime and maximal spectra of $C_A(H)$ are given. Simple $C_A(H)$-modules are classified.

Keywords: Prime ideal, primitive ideal, weight module, simple module, centralizer.

MSC: 17B10, 16D25, 16D60, 16D70, 16P50

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