On certain graded representations of filiform Lie algebras

Abstract

Let be a connected complex linear algebraic group of the same dimension as V such that the poset of the Zariski closures of the orbits for its action coincides with a full flag of subspaces of V. Using the classification of graded filiform Lie algebras, we determine the isomorphism types of the unipotent radical U of G in case G is not nilpotent and U is of maximal class. In particular, if , there are, up to isomorphism, only two such unipotent groups.

Keywords:

• Filiform Lie algebras
• grading
• graded representations
• linear algebraic groups
• unipotent radical

AMS Subject Classifications:

• 17B30
• 17B70
• 20G20
• 15A30

Acknowledgements

The first named author thanks Pietro Corvaja for discussion on elliptic curves, particularly for pointing out the database [24].

Additional information

Funding

The authors were supported in part by Research and Development Agency of Slovenia [grant number P1-0222].