On characteristic p Verma modules and subalgebras of the hyperalgebra

Thesis



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 = sl₃ 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(λ₁) is given for certain weights, λ = pⁿ λ₁ + (pⁿ — 1)ρ. Furthermore, a result about exact sequences of Weyl modules is carried over to Verma modules for sl₂.

Finally, the connection between the subalgebra u¯₁ 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¯₁sl₃≈ KG. Furthermore, we determine the centre ofu¯_nsl₃, and we obtain an alternative K-basis of U-.

Authors: Carstensen, V; Erdmann, K; Carstensen, Vivi

• Publication date: 1994
• DOI: 10.5287/ora-8jrv1ejzb
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Modules (Algebra)