J.M. Ancochea Bermúdez

On the nonrationality of rigid Lie algebras

In his thesis, Carles made the following conjecture: Every rigid Lie algebra is defined on the field Q. This was quite an interesting question because a positive answer would give a nice explanation of the fact that simple Lie algebras are defined over Q. The goal of this note is to provide a large number of examples of rigid but nonrational and nonreal Lie algebras.

Rigidity-preserving and cohomology-decreasing extensions of solvable rigid Lie algebras

It is shown that the two-known series of rank one and rank two finite-dimensional solvable rigid Lie algebras with non-vanishing second cohomology can be extended to solvable rigid Lie algebras of arbitrary rank such that the cohomology is preserved exactly. For the second series, it is further proved that an extension decreasing the cohomology exists, hence leading to cohomologically rigid Lie algebras.

New examples of rank one solvable real rigid Lie algebras possessing a nonvanishing Chevalley cohomology

Abstract The class of rank one solvable Lie algebras possessing a maximal torus t with eigenvalue spectrum spec ( t ) = ( 1 , 4 , 5 , … , n + 2 ) is studied in the context of rigidity. It is shown that from the value n ≥ 18, three isomorphism classes of rigid Lie algebras exist, two of them being algebraically rigid, and the third being geometrically rigid with a two-dimensional cohomology space H 2 ( g , g ) .