Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

Abstract

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

Foliations Formed by Generic Coadjoint Orbits of Lie Groups Corresponding to a Class Seven-Dimensional Solvable Lie Algebras

“…First of all, the assertion of Theorem 1 for G ∈ G 2 , G 3 , G 9 , G λ 10 had been proved in [17]. Below we will give detailed calculations for three groups G λ 1 λ 2…”

“…Proof. The proof is analogous to the case of MD-groups in [19,23,24,25] and Lie groups in [17,Theorem 21]. It consists of two steps as follows…”

“…Note that throughout this section the condition for the parameters λ, λ 1 , λ 2 in each Lie group G is the same as the condition for the parameters in each algebra G = Lie(G) in Table 1 if nothing more is said. In this section, we shall continue using the notations as in [17]. Recall that the picture of maximal-dimensional K-orbits of Lie groups G ∈ {G 2 , G 3 , G 00 4 , G 9 , G λ 10 } described in [17,Theorem 19].…”

“…In this section, we shall continue using the notations as in [17]. Recall that the picture of maximal-dimensional K-orbits of Lie groups G ∈ {G 2 , G 3 , G 00 4 , G 9 , G λ 10 } described in [17,Theorem 19]. Here, we will describe the picture of maximal-dimensional K-orbits of the remaining Lie groups.…”

“…Namely, we started to study Lie groups corresponding to Lie algebras in Table 1 which are not too complicated but also not trivial among the groups classified in [26]. In fact, we have solved quite completely in [17] the similar problem of MD-groups for five families of Lie groups corresponding to Lie algebras G 2 , G 3 , G 00 4 , G 9 , G λ 10 in Table 1.…”