Noetherian Hopf Algebras Research Articles

Locally finite representations over Noetherian Hopf algebras

We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products, and affine Hopf algebras of low Gelfand-Kirillov dimension.

Support for Integrable Hopf Algebras via Noncommutative Hypersurfaces

AbstractWe consider finite-dimensional Hopf algebras $u$ that admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $u$, via the singularity categories of the hypersurfaces $U/(f)$ associated with functions $f$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $A$.

Mini-Workshop: Cohomology of Hopf Algebras and Tensor Categories

The mini-workshop featured some open questions about the cohomology of Hopf algebras and tensor categories. Questions included whether the cohomology ring of a finite dimensional Hopf algebra or a finite tensor category is finitely generated, questions about corresponding geometric methods in representation theory, and questions about noetherian Hopf algebras. The workshop brought together mathematicians currently working on these and other open problems.

Connected Hopf algebras and iterated Ore extensions

We investigate when a skew polynomial extension T=R[x;σ,δ] of a Hopf algebra R admits a Hopf algebra structure, substantially generalising a theorem of Panov. When this construction is applied iteratively in characteristic 0 one obtains a large family of connected noetherian Hopf algebras of finite Gelfand–Kirillov dimension, including for example all enveloping algebras of finite dimensional solvable Lie algebras and all coordinate rings of unipotent groups. The properties of these Hopf algebras are investigated.

The Dixmier-Moeglin Equivalence for Cocommutative Hopf Algebras of Finite Gelfand-Kirillov Dimension

Let k be an algebraically closed field of characteristic zero and let H be a noetherian cocommutative Hopf algebra over k. We show that if H has polynomially bounded growth then H satisfies the Dixmier-Moeglin equivalence. That is, for every prime ideal P in Spec(H) we have the equivalences $$P\; \text{primitive}\iff P\; \text{rational}\iff P\; \text{locally closed in}~\text{Spec}(H).$$ We observe that examples due to Lorenz show that this does not hold without the hypothesis that H have polynomially bounded growth. We conjecture, more generally, that the Dixmier-Moeglin equivalence holds for all finitely generated complex noetherian Hopf algebras of polynomially bounded growth.

NOETHERIAN HOPF ALGEBRAS

AbstractA brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.

Noetherian Hopf algebra domains of Gelfand–Kirillov dimension two

We classify all noetherian Hopf algebras H over an algebraically closed field k of characteristic zero which are integral domains of Gelfand–Kirillov dimension two and satisfy the condition Ext H 1 ( k , k ) ≠ 0 . The latter condition is conjecturally redundant, as no examples are known (among noetherian Hopf algebra domains of GK-dimension two) where it fails.

Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras

We show that many noetherian Hopf algebras A have a rigid dualising complex R with R ≅ A 1 ν [ d ] . Here, d is the injective dimension of the algebra and ν is 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 ν the Nakayama automorphism of A. We prove that ν = S 2 ξ , where S is the antipode of A and ξ 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) Poincaré duality holds in this setting, as is deduced from a theorem of Van den Bergh. Calculating ν for 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.

Homological properties of quantized coordinate rings of semisimple groups

We prove that the generic quantized coordinate ring $\mathcal{O}_q(G)$ is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group $G$. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a new set of positive cases for a question of Brown and the first author.

Noetherian Centralizing Hopf Algebra Extensions and Finite Morphisms of Quantum Groups

We study finite centralizing extensions A ⊂ H of noetherian Hopf algebras. Our main results provide necessary and sufficient conditions for the fibres of the surjection spec H [Rarr ]spec A to coincide with the X -orbits in spec H , where X denotes the finite group of characters of H that restrict to the counit of A . In particular, all of the fibres are X -orbits if and only if the fibre over the augmentation ideal of A is an X -orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.

Homological Aspects of Noetherian PI Hopf Algebras and Irreducible Modules of Maximal Dimension

We prove that a Noetherian Hopf algebra of finite global dimension possesses further attractive homological properties, at least when it satisfies a polynomial identity. This applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity, as well as to classical enveloping algebras in positive characteristic. In all three cases we show that these algebras are Auslander-regular and Macaulay. We derive representation theoretic consequences concerning the coincidence of the non-Azumaya and singular loci for each of the above three classes of algebras.

Shorter Notes: A Commutative Noetherian Hopf Algebra Over a Field is Finitely Generated

Shorter Notes: A Commutative Noetherian Hopf Algebra Over a Field is Finitely Generated

A commutative Noetherian Hopf algebra over a field is finitely generated

Let k k be an arbitrary field and H H a commutative Hopf algebra over k k . We give a short proof of the fact that H H is Noetherian if and only if H H is finitely generated as a k k -algebra.