# zbMATH — the first resource for mathematics

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.
 [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 [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 [5] Belţitǎ, I.; Belţitǎ, D., On Kirillov’s lemma for nilpotent Lie algebras, J. Algebra, 427, 85-103 (2015) · Zbl 1360.17011 [6] Chevalley, C.; Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. Am. Math. Soc., 63, 85-124 (1948) · Zbl 0031.24803 [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 [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 [11] Seely, C., 7-Dimensional nilpotent Lie algebras, Trans. Am. Math. Soc., 335, 2, 479-496 (1993) · Zbl 0770.17003