arXiv subjects
Publications and source records attributed to Kenneth A. Brown.
Necessary and sufficient conditions are obtained for an ambiskew polynomial algebra A over a Hopf k-algebra R to possess the structure of a Hopf algebra extending that of R, in which the added variables X+ and X- are skew primitive. The coradical filtration is calculated, many examples are described, and properties determined.
This short survey article reviews current understand- ing of the structure of noetherian Hopf algebras. The focus is on homological properties. A number of open problems are listed.
Basic properties of symplectic reflection algebras over an algebraically closed field k of positive characteristic are laid out. These algebras are always finite modules over their centres, in contrast to the situation in characteristic 0. For the subclass of rational Cherednik algebras, we determine the PI-degree and the Goldie rank, and show that the Azumaya and smooth loci of the centre coincide.
We show that many noetherian Hopf algebras A have a rigid dualising complex R with R isomorphic to ^νA^1 [d]. Here, d is the injective dimension of the algebra and νis a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call νthe Nakayama automorphism of A. We prove that ν= S^2\XXi, where S is the antipode of A and \XXi is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non-zero in the top dimension d, when A is an Artin-Schelter regular noetherian Hopf algebra of global dimension d. (Twisted) Poincare duality holds in this setting, as is deduced from a theorem of Van den Bergh. Calculating νfor A using also the opposite coalgebra structure, we determine a formula for S^4 generalising a 1976 formula of Radford for A finite dimensional. Applications of the results to the cases where A is PI, an enveloping algebra, a quantum group, a quantised function algebra and a group algebra are outlined.
Necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the p-adic integers or the field of p elements to be prime, semiprime and a domain. Necessary and sufficient conditions for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups are found. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al.
A condition on an affine central subalgebra $Z$ of a noetherian algebra $A$ of finite Gelfand-Kirillov dimension, which we call here \emph{unruffledness}, is shown to be equivalent in some circumstances to the flatness of $A$ as a $Z$-module. Unruffledness was studied by Borho and Joseph in work on enveloping algebras of complex semisimple Lie algebras, and we discuss applications of our result to enveloping algebras, as well as beginning the study of this condition for more general algebras.
We introduce a new class of algebras called Poisson orders. This class includes the symplectic reflection algebras of Etingof and Ginzburg, many quantum groups at roots of unity, and enveloping algebras of restricted Lie algebras in positive characteristic. Quite generally, we study this class of algebras from the point of view of Poisson geometry, exhibiting connections between their representation theory and some well-known geometric constructions. As an application, we employ our results in the study of symplectic reflection algebras, completing work of Etingof and Ginzburg on when these algebras are finite over their centres, and providing a framework for the study of their representation theory in the latter case.