Papers

For a finite field $\F_q$, consider the triangular Cholesky map $\Gamma_{n,q}(U)=U^TU$ from upper triangular matrices to symmetric matrices. Generalized Cholesky theory describes the regular locus on which all leading principal minors are nonzero, but it does not determine the multiplicities or root ranks in a fiber over a singular target. We develop a fixed-target fiber theory for this singular boundary and answer three questions posed by Cooper and Whitlatch. Our principal results concern the zero fiber. Over $\F_2$ we construct an explicit, invertible, rank-preserving recursive bijection between square-zero upper triangular matrices and upper triangular matrices satisfying $U^TU=0$. Over every finite field we determine the entire rank distribution of this zero fiber: in even characteristic the square-zero recurrence, and hence the rank-refined equinumerosity, persists over every $\F_{2^e}$, whereas in odd characteristic its failure is measured by an explicit quadratic-character correction. We place these results in a uniform framework by proving an exact first-pivot recursion for the fiber cardinality over an arbitrary symmetric target. Its regular specialization gives the constant fiber sizes on leading-principal-minor cones, while its zero-pivot branch explains the new singular behavior. For binary diagonal targets we further compress the rank-refined count to $O(n^2)$ integer-arithmetic transitions and show that it depends on the order, not merely the number, of the diagonal entries. Executable implementations and exhaustive low-order checks are collected in an appendix.

We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins–Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided $\Gamma_0$-artinian rings need not be $\Gamma_0$-noetherian, but using our tools we prove that one-sided gr-hereditary $\Gamma_0$-artinian rings, two-sided $\Gamma_0$-artinian rings, and $d$-finitely generated one-sided $\Gamma_0$-artinian rings are $\Gamma_0$-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.

Modern data systems are commonly studied through computational and information-theoretic methods, while their algebraic properties remain largely unexplored. This paper introduces a mathematical framework for modelling data systems using algebraic codes over finite groups, organized by an axiom signature: each algebraic axiom of the data domain is shown to be sufficient for a specific coding-theoretic capability, and each implication is given its precise reading (MDS parameters do not require a field; quotients require normality, not commutativity). Three types of results are presented. Proved: a Gilbert-Varshamov existence bound over arbitrary finite groups, commutative or not; explicit direct-sum codes over the elementary divisors of any finite abelian group; that transactions preserving a validity predicate form a submonoid of the full transformation monoid, which is not a group; that schema-preserving transformations form a finite group whose orbits are counted by Burnside's lemma; and that the constraint-preserving bijections of a data system form a finite group whose orbits partition the valid configurations. Demonstrated: an explicit code construction over $\mathbb{F}_5$, with known codes over $\mathbb{Z}_4$ and $(2^U,\triangle)$ cited to situate the framework. Proposed: an encoding efficiency measure Ef and an optimization functional $\Phi$ with tunable weights, both evaluated at declared coding instances. Weight-independent comparison holds under componentwise dominance; in general the weight simplex decomposes into polyhedral chambers on which the induced order is constant.

Let $F$ be a global field and let $p$ be an odd prime with $p\neq\operatorname{char}F$. Assuming first that $\mu_p\subset F$, we record in a uniform global-field form the length-four common-slot chain for equal degree-$p$ symbol classes and emphasize the signed normalization adapted to explicit norm constructions. Thus, from \[(a,b)_p=(c,d)_p\in\textrm{Br}(F)[p]\] one obtains $x,y\in F^\times$ such that \[(a,b)_p=(x^{-1},b)_p=(x,y)_p=(c^{-1},y)_p=(c,d)_p.\] For number fields, the underlying chain is the length-four chain lemma of Gille–Szamuely, based on Tate's simultaneous local–global theorem. The point developed here is that its signed form yields four compatible norm equations that can be used constructively. For the extraspecial central product $H_{p^3}*H_{p^3}$ over a $C_p^4$-Kummer extension, the central-embedding obstruction is $(a,b)_p-(c,d)_p$, while the four norm equations supplied by the chain assemble, under a natural independence hypothesis on the auxiliary Kummer classes, into an explicit factorized radical realization of the central product. Finally, when the ground field does not contain $\mu_p$, we show by a restriction–corestriction argument that the central-embedding obstruction is detected after passage to the cyclotomic extension $F(\mu_p)$. Over that field the problem is Kummer, and the obstruction is again the difference of the two symbol classes.

We define the integral profiles of all modules and introduce integral coefficient rings ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)$ for all finite-dimensional complex algebras $A$. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules $M\cong N$ are isomorphic if and only if their integral profiles are similar, i.e., $M\cong N$ if and only if $\displaystyle \int M \sim \int N$. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of $A$ (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of $A$ is finite if and only if the embedded integral profile of $\mathrm{top}(A)$ lies in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$; (6) the finitistic dimension of $A$ is finite if and only if each embedded integral profile of $M$ lying in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$ implies that its degree is less than or equal to a fixing integer $d\in\mathbb{N}^+$; (7) we provide two sufficient conditions, such that if a finite-dimensional complex algebra $A$ satisfies one of them, then its finitistic dimension is finite.

For a row-finite weighted graph $(E,w)$, Preusser showed that the monoid $\mathcal{V}(L_k(E,w))$ of finitely generated projective modules over the weighted Leavitt path algebra $L_k(E,w)$ is isomorphic to a combinatorially defined weighted graph monoid $\mathcal{M}(E,w)$. We study two structural properties of $\mathcal{M}(E,w)$: confluence and cancellativity. We introduce a reduction system on the free commutative monoid presenting $\mathcal{M}(E,w)$, obtain sufficient conditions for non-confluence by constructing explicit non-confluent triples, and provide a complete confluence characterization for certain classes of weighted graphs. Turning to cancellativity, we work within Preusser's class of weighted graphs satisfying Condition (LPA), for which $L_k(E,w)$ is isomorphic to an unweighted Leavitt path algebra $L_k(F)$ via a two-step construction. We introduce an auxiliary graph associated to the intermediate step of this construction and use it to give a graph-theoretic characterization of when $\mathcal{M}(E,w)$ is cancellative. Finally, under Condition (LPA), we show that Preusser's construction upgrades to a graded isomorphism $L_k(E,w) \cong_{\operatorname{gr}} L_k(F)$ with respect to the standard $\mathbb{Z}^{\lambda(E,w)}$-grading of weighted Leavitt path algebras, yielding $\mathcal{V}^{\operatorname{gr}}(L_k(E,w)) \cong \mathcal{V}^{\operatorname{gr}}(L_k(F))$ as $\mathbb{Z}^{\lambda(E,w)}$-monoids.

We prove a correspondence between the Polchinski flow and the Connes-Kreimer approaches to perturbative renormalisation, in the sense that the first yields the same renormalisation as the latter. More precisely, we show that an ansatz based on decorated graphs (Feynman diagrams),with a combinatorial renormalisation procedure solves the Polchinski equation. This result holds for very general Euclidean quantum field theories. We are able to derive from this the form of the renormalised potential which is, to the best of our knowledge, the most general one in the literature. The main ingredients of the proof are multiple morphism properties with respect to the renormalisation, as well as a novel duality formula for forests of decorated graphs.

We study the Hopf algebra $A$ given by the bosonization of the super Jordan plane over a field of characteristic not 2. We determine its centre, classical quotient ring, prime and primitive ideals. Over an algebraically closed base field, we classify all simple $A$-modules. In characteristic zero, simple modules are either finite-dimensional, of dimension 1 or 2, or infinite-dimensional, arising from a localization of the first Weyl algebra. Over an algebraically closed field of characteristic $p>2$, every simple module is finite-dimensional, of dimension 1, 2, or $2p$, and we give explicit representatives for all isomorphism classes. Our approach relies on realizing a suitable localization of $A$ as a matrix algebra.

In this paper, we prove that global monomials on positive spaces are linearly independent, extending the basic fact that Laurent monomials in a Laurent polynomial algebra are linearly independent to a much more general setting. We also establish a global monomial avoidance phenomenon for positive spaces. Our approach is based on the study of Newton polytopes of Laurent expansions. These general results apply to positive spaces arising from cluster algebras (including the totally sign-skew-symmetric case), $Y$-patterns, and Laurent phenomenon algebras whose clusters are related by subtraction-free birational transformations. In particular, we obtain the proper Laurent monomial property and the linear independence of cluster monomials for all cluster algebras and Laurent phenomenon algebras under consideration. Notably, the proper Laurent monomial property follows from the global monomial avoidance phenomenon for positive spaces.

Let $d_1,\ldots,d_r$ be pairwise relatively prime positive square-free integers, with $d_j\geq2$ for all $1\leq j\leq r$. Using elementary matrices and combinatorial mathematics, we give a complete classification of the irreducible $\mathbb{Z}_+$-modules over the domain $\mathbb{Z}[d^{\frac{1}{N_1}}_1,d^{\frac{1}{N_2}}_2,\dots,d^{\frac{1}{N_r}}_r]$, where $N_i \geq2$ for all $ 1\leq i \leq r$. Furthermore, we explicitly construct all of these irreducible $\mathbb{Z}_+$-modules.

The notion of $d$-stable dg categories axiomatizes $d$-cluster tilting subcategories of stable dg categories. We establish an Auslander correspondence for $d$-stable dg categories: we characterize the $d$-stability of an additive connective dg category in terms of coherence, weak global dimension, and a duality on finitely presented modules. This gives a homological characterization of $d$-stability and reveals it as a twisted form of $(d+1)$-Calabi–Yau duality. For locally finite connective dg algebras, this interpretation becomes particularly transparent under Koszul duality, where $d$-stability corresponds to a shifted self-injectivity condition on the Koszul dual. Following the constructions of Amiot, Guo and Keller, for a $d$-stable dg category $M$, we introduce its $d$-cluster dg category $\mathcal C_{d,{\rm dg}}(M):=\operatorname{per}_{\rm dg}M/^\mathbb{L}\mathcal D^b_{\rm fp, dg}(M)$. Using our Auslander correspondence, we show that $\mathcal C_{d,{\rm dg}}(M)$ contains $M$ as a $d$-cluster tilting subcategory. In particular, every $d$-stable dg category can be realized as a $d$-cluster tilting subcategory of a stable dg category. We then develop cluster Morita theory: a pretriangulated dg category equipped with a $d$-cluster tilting subcategory $M$ is quasi-equivalent to $\mathcal C_{d,{\rm dg}}(M)$. Thus the connective dg structure of a cluster tilting subcategory determines its ambient dg category up to quasi-equivalence. As an application of cluster Morita theory, we prove a Morita-theoretic variant of Amiot's conjecture. More precisely, we establish a Calabi–Yau correspondence: for a locally finite $d$-stable dg category $M$ over a field, right $(d+1)$-Calabi–Yau structures on $\mathcal D^b_{\rm fp, dg}(M)$ are in bijection with right $d$-Calabi–Yau structures on $\mathcal C_{d,{\rm dg}}(M)$.

We introduce and study the so-termed {\it $\sqrt{\Delta}$-fine rings}, a new class of rings that generalizes the classical fine rings introduced by Călugăreanu-Lam in J. Algebra & Appl. (2016) by requiring that every nonzero element $r \in R$ can be written as $r = u + a$, where $u$ is a unit and $a \in \sqrt{\Delta(R)}$. We establish that every such ring is simple, every abelian $\sqrt{\Delta}$-fine ring is indecomposable, and most notably, the matrix ring $M_n(R)$ over a $\sqrt{\Delta}$-fine ring $R$ is again $\sqrt{\Delta}$-fine for every $n \ge 1$. As a consequence, we characterize all semi-local $\sqrt{\Delta}$-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either $\sqrt{\Delta}$-fine or generalized fine, where the latter class was introduced by Zhou in J. Algebra & Appl. (2022), and conclude our work with the difficult open question asking of whether each $\sqrt{\Delta}$-fine ring is necessarily fine.

We develop a purely algebraic framework for index-type theorems based on the Rees construction for filtered differential graded algebras (FDGAs). Alongside the classical Rees module we introduce a smooth variant $C^\omega_{\mathcal{R}}A$, adapted to analytic arguments, and we study traces and their pointwise and coefficient-wise extensions to Rees algebras. The main result (Corollary 3.11) is an abstract index theorem: given a compatible datum of filtered differential graded associative algebras with traces $A$, $B$, $G$ with a morphism $\phi\colon A\to B$, an action $\rho\colon G\otimes A\to A$ and a morphism $i\colon G\to B$, together with a graded-central element, one has $\mathrm{tr}_B(e^{f_0+f_1}) = \mathrm{tr}_{\widehat{\mathrm{gr}}A}(e^{\gamma}h)$, where the left-hand side is the trace in $B$ and the right-hand side the trace in the Laurent series associated graded $\widehat{\mathrm{gr}}A$ of $A$. Here $f_0+f_1$ is a curvature-type element, i.e., an element of the form $d_B\beta+\beta^2$ for some odd-degree element $\beta$, while $\gamma$ and $h$ are certain elements in $\widehat{\mathrm{gr}}A$. The formalism is modelled on the Getzler rescaling technique and on the derivation of the localization formula for the loop space Chern character by Ludewig and Yi.

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of characteristic $p$, where $p$ is an odd prime. Let $*$ be the classical involution of $FG$ and let $FG^-$ be the Lie subalgebra of the skew symmetric elements with respect to $*$. In this paper, we prove that the Lie nilpotency index of $FG$ is determined by its Lie subalgebra $FG^-$, in the case when $G$ is a $p$-group with a commutator subgroup of order $p$. The structure of the terms of the lower Lie central series of $FG^-$ has also been described.

We introduce and study completeness for multiplicative $n$-ary Hom-Nambu superalgebras. Because an $n$-ary Hom-Nambu bracket is not necessarily totally super-skew-symmetric, we define its center as the intersection of its positional centers. A multiplicative $n$-ary Hom-Nambu superalgebra is complete when its center is trivial, and every $\alpha^{k+1}$-derivation is inner for all $k \geq 0$. We show that this notion reduces to the usual completeness for multiplicative $n$-Hom-Lie superalgebras. Furthermore, we establish a completeness criterion for surjective brackets with a zero twisting map and provide a complete $n$-ary Hom-Nambu superalgebra that is not an $n$-Hom-Lie superalgebra. We also study the direct sums of complete $n$-Hom-Lie superalgebras and the behavior of centers under twisting. Finally, starting from a multiplicative Hom-Lie superalgebra, we consider recursively induced multiplicative $n$-ary Hom-Nambu superalgebras. When the twisting map is surjective, we prove that triviality of the center is preserved and reflected in this construction. We also observe that every binary $\alpha^k$-derivation satisfies the corresponding relative $\alpha^k$-derivation identity for the induced bracket. Finally, we determine the complete members in the selected low-dimensional Hom-Lie and $3$-Hom-Lie superalgebra classifications.

Let $\iota: R\to A$ be a Frobenius extension and let $T$ be a Wakamatsu tilting left $R$-module. We give sufficient conditions for the induced module $A\otimes_R T$ to remain Wakamatsu tilting and establish an ascent–descent result for split centrally projective Frobenius extensions. We also characterize the tilting case by the vanishing of $\mathrm{Ext}_R^i(T,A\otimes_R T)$ for all $i>0$. Moreover, if $A\otimes_R T\in\mathrm{add}_R(T)$, then the natural map $S=\mathrm{End}_R(T)\to B=\mathrm{End}_A(A\otimes_R T)$ of endomorphism rings is a Frobenius extension and $T\otimes_S B\cong A\otimes_R T$ as $R$-$B$-bimodules. Applications to Brenner–Butler–Miyashita equivalences and some specific classes of Frobenius extensions are also discussed.

We study polynomial absolute and relative joint first-order differential projective invariants for configurations of $n$ points in the plane. The main approach is based on passing to a homogeneous vector–covector representation, which reduces the problem to the classical invariant theory of the group $SL(3,\mathbb C)$. The algebra of polynomial absolute invariants is described and shown to be generated by cyclic invariants. For polynomial relative invariants of weight $-1$, a graded description is obtained, their finite generation as a module over the algebra of absolute invariants is proved, and an explicit finite generating system is constructed. For $n=3$, the corresponding module is shown to be free of rank $1$. For $n=4$, a minimal homogeneous generating system consisting of $39$ elements is constructed. The obtained results complement the rational theory of joint projective differential invariants by its polynomial counterpart and provide an algebraic foundation for the further construction of projectively invariant integral characteristics.

We resolve two conjectures concerning multiplicatively idempotent semirings. First, we prove that every congruence-simple multiplicatively idempotent semiring is finite, and hence belongs to the finite classification of Kepka, Korbelář and Landsmann. Second, we prove that every finitely generated commutative multiplicatively divisible semiring is multiplicatively idempotent.

In this paper, we introduce the controlling $L_{\infty}[1]$-algebra of extended Rota-Baxter Lie algebras. As applications, we obtain the cohomology of extended Rota-Baxter Lie algebras. We also study abelian extensions and formal deformations by the lower cohomology group. Lastly, we get the notion of homotopy extended Rota-Baxter Lie algebras. Specifically, we present the controlling $L_{\infty}[1]$-algebra of modified Rota-Baxter Lie algebras and of Rota-Baxter Lie algebras with weight.

Poisson algebras are the quasiclassical limits of associative algebra deformations of commutative associative algebras. This paper extends this process to the level of bialgebras. We derive Poisson bialgebras as the quasiclassical limits of antisymmetric infinitesimal bialgebra deformations of commutative and cocommutative antisymmetric infinitesimal bialgebras. It might be regarded as the “infinitesimal" version of the quantization process of Lie bialgebras in terms of Hopf algebras. Such deformations-quasiclassical limits process for a Poisson bialgebra is equivalently characterized in terms of the introduced notions of deformations of a matched pair of associative algebras as well as a standard Manin triple of associative algebras, whose corresponding quasiclassical limits are a matched pair of Poisson algebras and a standard Manin triple of Poisson algebras, respectively. We illustrate these equivalent deformations-quasiclassical limits processes via coherent derivations.

In the spirit of Conway and Coxeter, we classify all $\mathbf{Y}$-frieze patterns of type $A_n$. In particular, we settle a conjecture made by de Saint Germain that all such $\mathbf{Y}$-frieze patterns arise from Conway-Coxeter frieze patterns. Moreover, our approach naturally leads to the enumeration of these $\mathbf{Y}$-frieze patterns in terms of Fuss-Catalan numbers.

Gélinas introduced the delooping level of an Artin algebra as a homological invariant that bounds the big finitistic dimension of the opposite algebra. A natural question is whether such invariants are preserved under derived equivalences. Chen recently showed that the finiteness of the classical delooping level and its sub-derived variant is not derived-invariant, but the case of the finer derived delooping level of Guo and Igusa remained open. In this paper, we prove that the finiteness of the derived delooping level is degree-shift invariant under derived equivalences: if two algebras are derived equivalent via a tilting complex of width $k_T$, then finiteness of the $(k+k_T)$-derived delooping level on one side implies finiteness of the $k$-derived delooping level on the other. Consequently, the finiteness of the derived $\infty$-delooping level is invariant under derived equivalences. Furthermore, we introduce the global derived delooping level and prove that, under a derived equivalence, its $\infty$-version changes by at most $k_T$. In particular, its finiteness is a derived invariant.

It was conjectured that the nonzero idempotents of the integral quandle ring of a finite latin quandle are trivial. We prove this holds for latin dihedral quandles. Furthermore, we discuss counterexamples against this conjecture.

We develop a method for constructing abelian model structures on the category Q,AMod of Q-shaped modules from cotorsion pairs in AMod, where Q is a small preadditive category satisfying certain conditions and AMod denotes the category of left A-modules for any ring A. More precisely, we construct two cotorsion pairs in Q,AMod from a given cotorsion pair in AMod. This leads to a construction of projective model structures on Q,AMod under the condition that Q has no cycles. We further apply this method to the category Dif(A) of differential left A-modules, viewed as a category of Q-shaped modules for a suitable choice of Q. In this case, the induced cotorsion pairs are shown to be compatible, thereby giving rise to abelian model structures on Dif(A).

Strongly bounded objects in a weakly approximable triangulated category are known to relate closely to global dimension and to play a significant role in the uniqueness problem for triangulated enhancements. In this paper, we investigate when the full subcategory of strongly bounded objects admits a bounded $t$-structure. Our main result, under a finiteness condition termed the finite strong finitistic dimension, states that this happens exactly when the subcategory agrees with the full subcategory of bounded objects in the ambient triangulated category. In that case, the bounded $t$-structure is unique up to equivalence; even more, the uniqueness holds unconditionally on the full subcategory of bounded objects.

We introduce the $\mathfrak M$arseille anti-construction $\mathfrak{Ma}$ for varieties of algebras defined by multilinear identities of degree $3$, obtained by changing the sign of the right-nested part of each defining polynomial. This gives, for any variety $\Var$, a new variety $\mathfrak{Ma}$-$\Var$ that generalizes antiassociative algebras and the recently studied anti-Novikov and anti-right-symmetric $\big($pre-Jacobi–Jordan$\big)$ algebras. We describe free objects in several $\mathfrak{Ma}$-varieties. On the operadic level, over a field of characteristic $\mathrm{char}\,\Bbbk\neq 2$, we show $(\mathfrak{Ma}\text{-}\Var)^{(!)}=\mathfrak{Ma}\text{-}(\Var^{(!)})$ and that the Dong property is preserved by $\mathfrak{Ma}.$ Finally, we obtain the necessary and sufficient condition for $\mathfrak{Ma}\text{-}\Var=\Var$. The second part of the paper is dedicated to another type of anti-algebras, called $\mathcal{M}anin$ anti-algebras. We give the description of this type of algebras via calculation of the white Manin product of suitable operads and present various examples of known $\mathcal{M}anin$ anti-algebras. In the end, we proved that the symmetric affinization of an algebra gives a Jacobi-Jordan algebra if and only if the original algebra is a $\mathcal{M}a$-Novikov algebra, that is, an analog of a well-known result about affinization of Novikov algebras given by Balinskii and Novikov.

We introduce a new class of commutative associative algebras whose structural constants are linear operators acting in a linear topological space. An analogue of the theory of monogenic functions is developed for such algebras. It is shown that the components of monogenic functions satisfy second-order operator-differential equations. This approach provides a constructive method for obtaining solutions of various partial differential and integro-differential equations. Several illustrative examples are presented.

We introduce a new class of algebras with variable operator-valued structural constants and develop elements of the theory of monogenic functions in such algebras. Compatibility conditions for the structural operators are established. It is shown that the components of monogenic functions satisfy second-order operator-differential equations with variable coefficients, and converse results are proved. The proposed approach provides a constructive method for obtaining solutions of various differential, integro-differential, and functional-differential equations. Several illustrative examples are presented.

We introduce a new class of algebras with variable structural constants and develop elements of the theory of monogenic functions in such algebras. Compatibility conditions for the structural functions are established. It is shown that the components of monogenic functions satisfy certain second-order partial differential equations with variable coefficients, and converse results are proved. This approach provides a constructive method for obtaining explicit solutions of such equations. Several illustrative examples are presented.

Let $q$ be a primitive $n$-th root of unity, $n>1$, and let $f$ be a nonzero polynomial such that $n\nmid(j+1)$ for every $j\in\supp(f)$. Set $e=\gcd(n,\{j+1:j\in\supp(f)\})$. We show that $\Oz(B_q(f))\cong\mu_e\times\mu_e$: the defining relations are homogeneous for a $\mathbb{Z}/e\times\mathbb{Z}/e$ grading, the center sits in degree zero, and the ozone group is the character group of that grading. The determination of the ozone group only requires the central elements $u^n$, $v^n$, and $\Omega$. The regular normal elements modulo the center form the same group, generated by $u^{n/e}$ and $v^{n/e}$, so every normal element is central exactly when $e=1$. For $f=t^2$ we recover a computation of Chan, Gaddis, Won and Zhang, and for $e>1$ we obtain an infinite family of Calabi–Yau algebras with nontrivial ozone group.