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We study the ozone group of noetherian Artin--Schelter regular algebras satisfying a polynomial identity (or PI for short). The ozone group was shown in previous work by the authors to be an important invariant in the study of PI skew polynomial rings and their centers. In this paper, we show that skew polynomial rings are in fact characterized as those algebras with maximal rank ozone groups. We also classify those with trivial ozone groups, which must necessarily be Calabi--Yau. This class includes most three-dimensional PI Sklyanin algebras. Further examples and applications are given, including applications to the Zariski Cancellation Problem., Comment: Added discussion on Sklyanin algebras of type S3

Automorphism, isomorphism, and embedding problems are investigated for a family of Nambu-Poisson algebras (or $n$-Lie Poisson algebras) using Poisson valuations., Comment: 49 pages. arXiv admin note: text overlap with arXiv:2309.05511

We study Poisson valuations and provide their applications in solving problems related to rigidity, automorphisms, Dixmier property, isomorphisms, and embeddings of Poisson algebras and fields., Comment: 47 pages

We discuss Poisson structures on a weighted polynomial algebra $A:=\Bbbk[x, y, z]$ defined by a homogeneous element $\Omega\in A$, called a potential. We start with classifying potentials $\Omega$ of degree deg$(x)+$deg$(y)+$deg$(z)$ with any positive weight (deg$(x)$, deg$(y)$, deg$(z)$) and list all with isolated singularity. Based on the classification, we study the rigidity of $A$ in terms of graded twistings and classify Poisson fraction fields of $A/(\Omega)$ for irreducible potentials. Using Poisson valuations, we characterize the Poisson automorphism group of $A$ when $\Omega$ has an isolated singularity extending a nice result of Makar-Limanov-Turusbekova-Umirbaev. Finally, Poisson cohomology groups are computed for new classes of Poisson polynomial algebras., Comment: Version 2

We introduce the ozone group of a noncommutative algebra $A$, defined as the group of automorphisms of $A$ which fix every element of its center. In order to initiate the study of ozone groups, we study PI skew polynomial rings, which have long proved to be a fertile testing ground in noncommutative algebra. Using the ozone group and other invariants defined herein, we give explicit conditions for the center of a PI skew polynomial to be Gorenstein (resp. regular) in low dimension., Comment: Some simplifications, clarifications, and corrections throughout. To appear in IMRN

We study and classify the 2-unitary operads of Gelfand-Kirillov dimension three.

We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring $A:=\Bbbk[x_1,\ldots,x_n]$ is a graded twist of a unimodular Poisson structure on $A$, namely, if $\pi$ is a graded Poisson structure on $A$, then $\pi$ has a decomposition $$\pi=\pi_{unim} +\frac{1}{\sum_{i=1}^n {\rm deg} x_i} E\wedge {\mathbf m}$$ where $E$ is the Euler derivation, $\pi_{unim}$ is the unimodular graded Poisson structure on $A$ corresponding to $\pi$, and ${\mathbf m}$ is the modular derivation of $(A,\pi)$. This result is a generalization of the same result in the quadratic setting. The rigidity of graded twisting, $PH^1$-minimality, and $H$-ozoneness are studied. As an application, we compute the Poisson cohomologies of the quadratic Poisson structures on the polynomial ring of three variables when the potential is irreducible, but not necessarily having isolated singularities., Comment: updated version

Let $A$ be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded $A$-modules, providing weighted versions of Castelnuovo--Mumford regularity, Tor-regularity, Artin--Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones., Comment: Accepted in Transactions of the AMS. arXiv admin note: text overlap with arXiv:2107.07474

We study operads with trivial $\mathbb{A}$-actions and prove an equivalence between the category of $\mathbb{A}$-trivial operads and that of pseudo-graded-Perm associative algebras. As a consequence, we show that finitely generated $\mathbb{A}$-trivial operads are right noetherian of integral Gelfand-Kirillov dimension and that every element in a prime $\mathbb{A}$-trivial operad is central., Comment: 28 pages

This paper concerns homological notions of regularity for noncommutative algebras. Properties of an algebra $A$ are reflected in the regularities of certain (complexes of) $A$-modules. We study the classical Tor-regularity and Castelnuovo-Mumford regularity, which were generalized from the commutative setting to the noncommutative setting by J{\o}rgensen and Dong-Wu. We also introduce two new numerical homological invariants: concavity and Artin-Schelter regularity. Artin-Schelter regular algebras occupy a central position in noncommutative algebra and noncommutative algebraic geometry, and we use these invariants to establish criteria which can be used to determine whether a noetherian connected graded algebra is Artin-Schelter regular., Comment: Minor edits and corrections made throughout

We introduce the reflexive hull discriminant as a tool to study noncommutative algebras that are finitely generated, but not necessarily free, over their centers. As an example, we compute the reflexive hull discriminants for quantum generalized Weyl algebras and use them to determine automorphism groups and other properties, recovering results of Su{\'a}rez-Alvarez, Vivas, and others., Comment: 30 pages, to appear in Selecta Mathematica

The paper concerns the Gelfand-Kirillov dimension and the generating series of nonsymmetric operads. An analogue of Bergman's gap theorem is proved, namely, no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov dimension strictly between $1$ and $2$. For every $r\in \{0\}\cup \{1\}\cup [2,\infty)$ or $r=\infty$, we construct a single-element generated nonsymmetric operad with Gelfand-Kirillov dimension $r$. We also provide counterexamples to two expectations of Khoroshkin and Piontkovski about the generating series of operads., Comment: 38 pages, 10 figures. The known typos have been corrected. A detailed proof to Lemma 7.2 has been added. The referee's comments have been incorporated

We study semisimple Hopf algebra actions on Artin-Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds., Comment: To appear in Advances in Mathematics

Let $\Bbbk$ be a base field of characteristic $p>0$ and let $U$ be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful $U$-actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three., Comment: 36 pages

Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field k of positive characteristic p are studied. We show that every IHOE over k satisfies a polynomial identity, with PI-degree a power of p, and that it is a filtered deformation of a commutative polynomial ring. We classify all 2-step IHOEs over k, thus generalising the classification of 2-dimensional connected unipotent algebraic groups over k. Further properties of 2-step IHOEs are described: for example their simple modules are classified, and every 2-step IHOE is shown to possess a large Hopf center and hence an analog of the restricted enveloping algebra of a Lie k-algebra. As one of a number of questions listed, we propose that such a restricted Hopf algebra may exist for every IHOE over k., Comment: Preliminary version; comments welcome. 46 pages

This is a brief survey on the current state of play on the programs to classify infinite dimensional Hopf algebras of Gel'fand-Kirillov dimension one or two. We list a number of open questions and suggest directions for future work., Comment: 19 pages; submitted to Conference on Hopf Algebras and Tensor Categories, Nanjing University, September 2019; comments welcome

We study both Morita cancellative and skew cancellative properties of noncommutative algebras as initiated recently in several papers and explore that which classes of noncommutative algebras are Morita cancellative (respectively, skew cancellative). Several new results concerning these two types of cancellations, as well as the classical cancellation, are proved., Comment: 23 pages, 1 figure

We study various invariants, such as cohomology groups, derivations, automorphisms and infinitesimal deformations, of algebraic operads and show that $\mathcal{A}ss$, $\mathcal{C}com$, $\mathcal{L}ie$ and $\mathcal{P}ois$ are rigid or semirigid.

Let $H$ be a weak Hopf algebra that is a finitely generated module over its affine center. We show that $H$ has finite self-injective dimension and so the Brown--Goodearl Conjecture holds in this special weak Hopf setting., Comment: 24 pages. Comments welcome

We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual) $\mathcal{E}_n^!$ of the Fomin-Kirillov algebras $\mathcal{E}_n$; these algebras are connected $\mathbb{N}$-graded and are defined for $n \geq 2$. We establish that the algebra $\mathcal{E}_n^!$ is module-finite over its center (so, satisfies a polynomial identity), is Noetherian, and has Gelfand-Kirillov dimension $\lfloor n/2 \rfloor$ for each $n \geq 2$. We also observe that $\mathcal{E}_n^!$ is not prime for $n \geq 3$. By a result of Roos, $\mathcal{E}_n$ is not Koszul for $n \geq 3$, so neither is $\mathcal{E}_n^!$ for $n \geq 3$. Nevertheless, we prove that $\mathcal{E}_n^!$ is Artin-Schelter (AS-)regular if and only if $n=2$, and that $\mathcal{E}_n^!$ is both AS-Gorenstein and AS-Cohen-Macaulay if and only if $n=2,3$. We also show that the depth of $\mathcal{E}_n^!$ is $\leq 1$ for each $n \geq 2$, conjecture we have equality, and show this claim holds for $n =2,3$. Several other directions for further examination of $\mathcal{E}_n^!$ are suggested at the end of this article., Comment: v1: 25 pages

We introduce truncation ideals of a $\Bbbk$-linear unitary symmetric operad and use them to study ideal structure, growth property and to classify operads of low Gelfand-Kirillov dimension.

Let $A$ be a Poisson Hopf algebra over an algebraically closed field of characteristic zero. If $A$ is finitely generated and connected graded as an algebra and its Poisson bracket is homogeneous of degree $d \geq 0$, then $A$ is unimodular; that is, the modular derivation of $A$ is zero. This is a Poisson analogue of a recent result concerning Hopf algebras which are connected graded as algebras., Comment: 14 pages; preliminary version, comments welcome

We re-formulate Bezrukavnikov-Kaledin's definition of a restricted Poisson algebra, provide some natural and interesting examples, and discuss connections with other research topics.

We study algebraic and homological properties of two classes of infinite dimensional Hopf algebras over an algebraically closed field k of characteristic zero. The first class consists of those Hopf k-algebras that are connected graded as algebras, and the second class are those Hopf k-algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel'fand-Kirillov dimension. It is shown that if the Hopf algebra H is a connected graded algebra of finite Gel'fand-Kirillov dimension n, then H is a noetherian domain which is Cohen-Macaulay, Artin-Schelter regular and Auslander regular of global dimension n. It has S^2 = Id_H, and is Calabi-Yau. Detailed information is also provided about the Hilbert series of H. Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k-algebras of finite Gel'fand-Kirillov dimension n with connected coalgebra, the underlying coalgebra is shown to be Artin-Schelter regular of global dimension n. Both these classes of Hopf algebra share many features in common with enveloping algebras of finite dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k-algebra H of Gel'fand-Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U(g) for any Lie algebra g., Comment: Comments welcome

A noncommutative analogue of the Zariski cancellation problem asks whether $A[x]\cong B[x]$ implies $A\cong B$ when $A$ and $B$ are noncommutative algebras. We resolve this affirmatively in the case when $A$ is a noncommutative finitely generated domain over the complex field of Gelfand-Kirillov dimension two. In addition, we resolve the Zariski cancellation problem for several classes of Artin-Schelter regular algebras of higher Gelfand-Kirillov dimension., Comment: 22 pages; to appear in Selecta Mathematica

Let $A$ and $B$ be two connected graded algebras finitely generated in degree one. If $A$ is isomorphic to $B$ as ungraded algebras, then they are also isomorphic to each other as graded algebras., Comment: 6 pages