Abstract. The McCool group, denoted P Σn, is the group of pure symmetric automorphisms of a free group of rank n. The cohomology algebra H * (P Σn, Q) was determined by Jensen, McCammond and Meier. We prove that H * (P Σn, Q) is a non-Koszul algebra for n ≥ 4, which answers a question of Cohen and Pruidze. We also study the enveloping algebra of the graded Lie algebra associated to the lower central series of P Σn, and prove that it has two natural decompositions as a smash product of algebras.
Stephenson and Vancliff recently introduced two families of quantum projective 3-spaces (quadratic and Artin-Schelter regular algebras of global dimension 4) which have the property that the associated automorphism of the scheme of point modules is finite order, and yet the algebra is not finite over its center. This is in stark contrast to theorems of Artin, Tate, and Van den Bergh in global dimension 3. We analyze the representation theory of these algebras. We classify all of the finitedimensional simple modules and describe some zero-dimensional elements of Proj, i.e., so called fat point modules. In particular, we observe that the shift functor on zero-dimensional elements of Proj, which is closely related to the above automorphism, actually has infinite order. 2005 Published by Elsevier Inc.
Recently a new construction of rings was introduced by Cassidy, Goetz, and Shelton. Some of these rings, called generalized Laurent polynomial rings, are quadratic Artin-Schelter regular algebras of global dimension 4. We study a family of such algebras which have finite-order point-scheme automorphisms but which are not finitely generated over their centers. Our main result is the classification of all fat point modules for each algebra in the family. We also consider the action of the shift functor τ and prove τ has infinite order on a fat point module F precisely when the center acts trivially on F . The proofs of these facts use the noncommutative geometry of some cubic Artin-Schelter regular algebras of global dimension 3.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.