Let G be a finite dimensional semisimple Lie algebra; we study the class of infinite dimensional representations of Gcalled characteristic p Verma modules. To obtain information about the structure of the Verma module Z(λ) we find primitive weights μ such that a non-zero homomorphism from Z(μ) to Z(λ) exists. For λ + ρ dominant, where ρ is the sum of the fundamental roots, there exist only finitely many primitive weights, and they all appear in a convex, bounded area. In the case of λ + ρ not dominant, and the characteristic p a good prime, there exist infinitely many primitive weights for the Lie algebra. For G = sl<sub>3</sub> we explicitly present a large, but not necessarily complete, set of primitive weights. A method to obtain the Verma module as the tensor product of Steinberg modules and Frobenius twisted Z(λ<sub>1</sub>) is given for certain weights, λ = p<sup>n</sup> λ<sub>1</sub> + (p<sup>n</sup> — 1)ρ. Furthermore, a result about exact sequences of Weyl modules is carried over to Verma modules for sl<sub>2</sub>. Finally, the connection between the subalgebra u¯<sub>1</sub> of the hyperalgebra U for a finite dimensional semisimple Lie algebra, and a group algebra KG for some suitable p-group G is studied. No isomorphism exists, when the characteristic of the field is larger than the Coxeter number. However, in the case of p — 2 we find u¯<sub>1</sub>sl<sub>3</sub>≈ KG. Furthermore, we determine the centre ofu¯<sub>n</sub>sl<sub>3</sub>, and we obtain an alternative K-basis of U-.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:239382 |
| Date | January 1994 |
| Creators | Carstensen, Vivi |
| Contributors | Erdmann, Karin |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:c6f5db9a-db94-4d58-9eb1-dd402c8846c5 |
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