Product decompositions of solvable Lie groups
The author studies the structure of real connected solvable Lie groups by decomposing them into a direct or semidirect product of factors introduced by the central or compact embedding property combined with the commutator subgroup. It is well known [cf. \textit{C. Chevalley}, Annals of Math., II. Ser. 42, 668-675 (1941; Zbl 0025.24301)] that any such Lie group is homeomorphic to the product of a torus and a cartesian space. In this paper various product decompositions are studied not only by their topological structure but also by their algebraic structure. Leading notions are Cartan algebras (groups), compact elements in Lie algebras (groups) [cf. \textit{K. H. Hofmann}, Semin. Sophus Lie 2, 41-55 (1992; Zbl 0857.17007)] and compactly embedded subalgebras (subgroups). The author uses these notions combined with those of the center and of the commutator subgroup to introduce product factors and proves several factorization results. The results of \textit{M. Goto} [Annals of Math., II. Ser. 61, 154-169 (1955; Zbl 0064.25802)] are essentially used in some points of the proofs.
scientific article; zbMATH DE number 980688
scientific article