Topological Loops with Decomposable Solvable Multiplication Groups

In this paper we deal with the class $$\mathcal {C}$$ C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in $$\mathcal {C}$$ C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

A splitting of local rigidity of Clifford–Klein forms of homogeneous spaces of completely solvable Lie groups

In this paper, we discuss local rigidity of Clifford–Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. We split the property of local rigidity into two conditions: vertical rigidity and horizontal rigidity. By this separation, we discuss local rigidity, in particular, Baklouti’s conjecture.

Two-step solvable SKT shears

We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting a left-invariant SKT structure. We reduce this to a specification of SKT shear data on Abelian Lie algebras, and which then is studied more deeply in different cases. We obtain classifications and structure results for $$\mathfrak {g}$$ g almost Abelian, for derived algebra $$\mathfrak {g}'$$ g ′ of codimension 2 and not J-invariant, for $$\mathfrak {g}'$$ g ′ totally real, and for $$\mathfrak {g}'$$ g ′ of dimension at most 2. This leads to a large part of the full classification for two-step solvable SKT algebras of dimension six.