Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups

DOI10.1142/S1793557122501935zbMATH Open1510.17017arXiv2105.00536WikidataQ115244506 ScholiaQ115244506MaRDI QIDQ6366655

Publication date: 2 May 2021

Abstract: Let {Lnk} be the class of all

n

-dimensional real solvable Lie algebras having

k

-dimensional derived ideals. In 2020 the authors et al. gave a classification of all non 2-step nilpotent Lie algebras of {Li}. We propose in this paper to study representations of these Lie algebras as well as their corresponding connected and simply connected Lie groups. That is, for each algebra, we give an upper bound of the minimal degree of a faithful representation. Then, we give a geometrical description of coadjoint orbits of corresponding groups. Moreover, we show that the characteristic property of the family of maximal dimensional coadjoint orbits of a MD-group studied by K. P. Shum and the second author et al. is still true for the Lie groups considered here. Namely, we prove that, for each considered group, the family of the maximal dimensional coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

Mathematics Subject Classification ID

Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) (17B10) Solvable, nilpotent (super)algebras (17B30) Representation theory of associative rings and algebras (16G99) Coadjoint orbits; nilpotent varieties (17B08)

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