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On Kirillov’s lemma and the Benson-Ratcliff invariant. (English) Zbl 1465.81054
Summary: In this paper, we study the conjecture of Benson and Ratcliff, which deals with the class of nilpotent Lie algebras of a one-dimensional center. We show that this conjecture is true for any nilpotent Lie algebra \(\mathfrak{g}\) with \(\dim \mathfrak{g} \leq 5\), but it fails for the dimensions greater or equal to 6. To this end, we produce counter-examples to the Benson-Ratcliff conjecture in all dimensions \(n \geq 6\). Finally, we show that this conjecture is true for the class of three-step nilpotent Lie algebras and for some other classes of nilpotent Lie algebras.
©2021 American Institute of Physics
MSC:
81S05 Commutation relations and statistics as related to quantum mechanics (general)
17B30 Solvable, nilpotent (super)algebras
22E25 Nilpotent and solvable Lie groups
17B81 Applications of Lie (super)algebras to physics, etc.
22E70 Applications of Lie groups to the sciences; explicit representations
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