Universität zu Köln

Backhaus, Teodor (2016). Monomial bases and PBW filtration in representation theory. PhD thesis, Universität zu Köln.

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Abstract

In this thesis we study the Poincaré–Birkhoff–Witt (PBW) filtration on simple finite-dimensional modules of simple complex finite-dimensional Lie algebras. This filtration is induced by the standard degree filtration on the universal enveloping algebra. For modules of certain rectangular highest weights we provide a new de- scription of the associated PBW-graded module in terms of generators and relations. We also construct a new basis parametrized by the lattice points of a normal polytope. If the Lie algebra is of type B3 we construct new bases of PBW-graded modules associated to simple modules of arbitrary highest weight. As an application we find that these modules are favourable modules, implying interesting geometric properties for the degenerate flag varieties. As a side product we state sufficient conditions on convex lattice 0,1-polytopes to be normal. We study the Hilbert–Poincaré polynomials for the associated PBW- graded modules of simple modules. The computation of their degree can be reduced to modules of fundamental highest weight. We provide these degrees explicitly. We extend the framework of the PBW filtration to quantum groups and provide case independent constructions, such as giving a filtration on the negative part of the quantum group, such that the associated graded algebra becomes a q-commutative polynomial algebra. By taking the classical limit we obtain, in some cases new, monomial bases and monomial ideals of the associated graded modules.