Group actions in Noncommutative Projective Geometry
17 February 2017
UAntwerpen, Campus Middelheim, A.143 - Middelheimlaan 1 - 2020 Antwerpen (route: UAntwerpen, Campus Middelheim
Kevin De Laet
Lieven Le Bruyn
PhD defence Kevin De Laet - Faculty of Science, Department of Mathematics and Computer Science
In this thesis, we consider graded, connected -algebras, finitely generated in degree one such that a reductive group acts as gradation preserving algebra automorphisms. The general construction of such algebras is discussed, together with some interesting examples.
In addition, some extra results on Sklyanin algebras (a two-dimensional family of graded algebras parametrized by an elliptic curve and a point ) of any global dimension are proved using the action of the Heisenberg group of order . Of special interest is the case that is of order two and that is odd, in which case the associated Sklyanin algebra is a graded Clifford algebra. This allows us to calculate the center and the PI-degree of these algebras in this particular case.
In the last chapter, the study of point modules of quantum polynomial rings is discussed. Quantum polynomial rings of global dimension form a -dimensional family of algebras on which acts. It is shown that the point modules are parametrized by unions of coordinate subspaces, with the generic point corresponding to the full graph on points.