Glider representations of chains of semisimple Lie algebra

ABSTRACT

We start the study of glider representations in the setting of semisimple Lie algebras. A glider representation is defined for some positively filtered ring FR and here we consider the right bounded algebra filtration FU(𝔤) on the universal enveloping algebra U(𝔤) of some semisimple Lie algebra 𝔤 given by a fixed chain of semisimple Lie subalgebras ${𝔤}_1\subset {𝔤}_2\subset \dots \subset {𝔤}_n=𝔤$. Inspired by the classical representation theory, we introduce so called Verma glider representations. Their existence is related to the relations between the root systems of the appearing Lie algebras 𝔤_i. In particular, we consider chains of simple Lie algebras of the same type A,B,C and D.

KEYWORDS:

• Glider representations
• Lie algebra
• representation theory

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B10
• 17B22

Acknowledgements

Acknowledgments The author is grateful to Fred Van Oystaeyen for introducing him to the subject of glider representations and for collaborating with him on many different topics. The author also wishes to thank Jacques Alev for his suggestion to look at nilpotent orbits in particular and for the e-mail correspondence about Lie algebra related topics in general. The author is Aspirant PhD Fellow of FWO.