×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arnal, D.; Ben Ammar, M.; Currey, B.; Dali, B., Construction of canonical coordinates for completely solvable Lie groups, J. Lie Theory, 2, 521-560 (2005) · Zbl 1074.22003
[2] Baklouti, A.; Tounsi, K., On the Benson-Ratcliff invariant of coadjoint orbits on nilpotent Lie groups, Osaka J. Math., 44, 399-414 (2007) · Zbl 1143.22007
[3] Benson, C.; Ratcliff, G., An invariant for unitary representations of nilpotent Lie groups, Michigan Math. J., 34, 23-30 (1987) · Zbl 0618.22005 · doi:10.1307/mmj/1029003479
[4] Benson, C.; Ratcliff, G., Quantization and invariant for unitary representation of nilpotent Lie groups, Illinois J. Math., 32, 53-64 (1988) · Zbl 0619.22012 · doi:10.1215/ijm/1255989228
[5] Belţitǎ, I.; Belţitǎ, D., On Kirillov’s lemma for nilpotent Lie algebras, J. Algebra, 427, 85-103 (2015) · Zbl 1360.17011 · doi:10.1016/j.jalgebra.2014.12.026
[6] Chevalley, C.; Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. Am. Math. Soc., 63, 85-124 (1948) · Zbl 0031.24803 · doi:10.1090/s0002-9947-1948-0024908-8
[7] Corwin, L. J.; Greenleaf, F. P., Representations of Nilpotent Lie Groups and Their Applications. Part 1: Basic Theory and Examples (1990), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 0704.22007
[8] Gong, M.-P., Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and \(\mathbb{R} ), 165 (1998)\), University of Waterloo: University of Waterloo, Canada
[9] Moore, C. C.; Wolf, J. A., Square integrable representation of nilpotent groups, Trans. Am. Math. Soc., 185, 445-462 (1973) · Zbl 0274.22016 · doi:10.1090/s0002-9947-1973-0338267-9
[10] Pedersen, N. V., Geometric quantization and the universal enveloping algebra of a nilpotent Lie group, Trans. Am. Math. Soc., 315, 511-563 (1989) · Zbl 0684.22004 · doi:10.1090/s0002-9947-1989-0967317-3
[11] Seely, C., 7-Dimensional nilpotent Lie algebras, Trans. Am. Math. Soc., 335, 2, 479-496 (1993) · Zbl 0770.17003 · doi:10.1090/s0002-9947-1993-1068933-46
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.