Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules. (English) Zbl 1393.17023
Summary: A classification of the simple weight modules is given for the (6-dimensional) Euclidean Lie algebra \(\mathfrak{e}(3) = \mathfrak{sl}_{2} \ltimes V_{3}\). As an intermediate step, a classification of all simple modules is given for the centralizer \(C\) of the Cartan element \(H\) (in the universal enveloping algebra \(\mathcal{U} = \operatorname{U}(\mathfrak{e}(3))\)). Generators and defining relations for the algebra \(C\) are found (there are three quadratic relations and one cubic relation). The algebra \(C\) is a Noetherian domain of Gelfand-Kirillov dimension 5. Classifications of prime, primitive, completely prime, and maximal ideals are given for the algebra \(\mathcal{U}\).{
©2017 American Institute of Physics}
©2017 American Institute of Physics}
MSC:
| 17B35 | Universal enveloping (super)algebras |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
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