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

Abstract

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.

Keywords:

• Lie algebra
• rigidity
• Chevalley cohomology
• solvable extension
• quadratic Rim map

AMS Subject Classifications:

• 17B30
• 17B56

Acknowledgements

The authors express their gratitude to A. Iósif and T. Raducanu for providing them with a copy of reference [5].

Notes

No potential conflict of interest was reported by the authors.

1 For the basic properties and notations for the Chevalley cohomology, see e.g. [1,4].

2 We recall that a Lie algebra satisfying these two conditions is called complete [27].

Additional information

Funding

The second author (RCS) was financially supported by the MINECO [research project number MTM2016-79422-P].