In this talk we will explain the role that the Lie algebras play in the construction of 6-dimensional compact quotient spaces M=G/Γ, where Γ is a lattice, endowed with an invariant complex structure with holomorphically trivial canonical bundle.

We will use the approach of stable forms in six dimensions to provide a classification of unimodular Lie algebras ${\displaystyle {\mathfrak {g}}}$ admitting
a complex structure with non-zero closed (3,0)-form.

Some of these spaces can be equipped with balanced Hermitian metrics and they provide invariant solutions of the Hull-Strominger system, even with non-flat connection on the tangent bundle satisfying the Hermitian-Yang-Mills condition.