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We show that if two $m$-homogeneous algebras have Morita equivalent graded module categories, then they are quantum-symmetrically equivalent, that is, there is a monoidal equivalence between the categories of comodules for their associated universal quantum groups (in the sense of Manin) which sends one algebra to the other. As a consequence, any Zhang twist of an $m$-homogeneous algebra is a 2-cocycle twist by some 2-cocycle from its Manin's universal quantum group., Comment: To appear in Proc. AMS

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

Let $G$ be a group acting on a left or right rigid monoidal triangulated category ${\mathbf K}$ which has a Noetherian Balmer spectrum. We prove that the Balmer spectrum of the crossed product category of ${\mathbf K}$ by $G$ is homeomorphic to the space of $G$-prime ideals of ${\mathbf K}$, give a concrete description of this space, and classify the $G$-invariant thick ideals of ${\mathbf K}$. Under some additional technical conditions, we prove that the Balmer spectrum of the equivariantization of ${\mathbf K}$ by $G$ is also homeomorphic to the space of $G$-prime ideals. Examples of stable categories of finite tensor categories and perfect derived categories of coherent sheaves on Noetherian schemes are used to illustrate the theory., Comment: Assumption that the category has a thick generator has been removed throughout, section on the equivariantization has been added, examples have been added, comments from experts in the field have been addressed. 24 pages. Comments welcome!

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

In this article, we discuss some recent developments of the Zariski Cancellation Problem in the setting of noncommutative algebras and Poisson algebras., Comment: to appear in Contemp. Math., 16 pages

In this paper, we present a generalization of well-established results regarding symmetries of $\Bbbk$-algebras, where $\Bbbk$ is a field. Traditionally, for a $\Bbbk$-algebra $A$, the group $\Bbbk$-algebra automorphisms of $A$ captures the symmetries of $A$ via group actions. Similarly, the Lie algebra of derivations of $A$ captures the symmetries of $A$ via Lie algebra actions. In this paper, given a category $\mathcal{C}$ whose objects possess $\Bbbk$-linear monoidal categories of modules, we introduce an object $\operatorname{Sym}_{\mathcal{C}}(A)$ that captures the symmetries of $A$ via actions of objects in $\mathcal{C}$. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected $\Bbbk$-algebra $A$, some of its symmetries are naturally captured within the weak Hopf framework., Comment: v1: 34 pages; comments welcome. v2: 45 pages; name changed, sections reorganized, new examples. v3: 47 pages; minor changes and clarifications added; final version to appear in Algebras and Representation Theory

We introduce the notion of quantum-symmetric equivalence of two connected graded algebras, based on Morita-Takeuchi equivalences of their universal quantum groups, in the sense of Manin. We study homological and algebraic invariants of quantum-symmetric equivalence classes, and prove that numerical $\mathrm{Tor}$-regularity, Castelnuovo-Mumford regularity, Artin-Schelter regularity, and the Frobenius property are invariant under any Morita-Takeuchi equivalence. In particular, by combining our results with the work of Raedschelders and Van den Bergh, we prove that Koszul Artin-Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class. Moreover, we characterize 2-cocycle twists (which arise as a special case of quantum-symmetric equivalence) of Koszul duals, of superpotentials, of superpotential algebras, of Nakayama automorphisms of twisted Frobenius algebras, and of Artin-Schelter regular algebras. We also show that finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists., Comment: To appear in Advances in Mathematics

We construct a family of cogroupoids associated to preregular forms and recover the Morita-Takeuchi equivalence for Artin-Schelter regular algebras of dimension two, observed by Raedschelders and Van den Bergh. Moreover, we study the 2-cocycle twists of pivotal analogues of these cogroupoids, by developing a categorical description of preregularity in any tensor category that has a pivotal structure., Comment: Comments are welcome

Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the Zhang twist of $H$ by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated to quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction., Comment: In this version, we have updated references throughout our paper (to appear)

We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a weak bialgebra that coacts on a (not necessarily connected) graded algebra $A$ universally while preserving grading. We restrict our attention to algebraic structures with a commutative base so that the UQSGds under investigation are face algebras (due to Hayashi). The UQSGd construction generalizes the universal quantum linear semigroups introduced by Manin in 1988, which are bialgebras that coact on a connected graded algebra universally while preserving grading. Our main result is that when $A$ is the path algebra $\Bbbk Q$ of a finite quiver $Q$, each of the various UQSGds introduced here is isomorphic to the face algebra attached to $Q$. The UQSGds of preprojective algebras and of other algebras attached to quivers are also investigated., Comment: v3: To appear in J. Pure Appl. Algebra

Let $k$ be a field and let $A$ be a finitely generated $k$-algebra. The algebra $A$ is said to be cancellative if whenever $B$ is another $k$-algebra with the property that $A[x]\cong B[x]$ then we necessarily have $A\cong B$. An important result of Abhyankar, Eakin, and Heinzer shows that if $A$ is a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one, and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has positive characteristic, giving a counterexample to a conjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings., Comment: 19 pages

Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta $ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension., Comment: 9 pages