Rings and Algebras
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Showing new listings for Thursday, 17 April 2025
- [1] arXiv:2504.11940 [pdf, html, other]
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Title: $f$-vectors and $F$-invariant in generalized cluster algebrasComments: 17 pagesSubjects: Rings and Algebras (math.RA)
We establish the initial and final seed mutations of the $f$-vectors in generalized cluster algebras and prove some properties of $f$-vectors. Furthermore, we extend $F$-invariant to generalized cluster algebras without the positivity assumption and prove symmetry property of $f$-vectors using the $F$-invariant.
- [2] arXiv:2504.12138 [pdf, other]
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Title: Exactness of cochain complexes via additive functorsSubjects: Rings and Algebras (math.RA)
We investigate the relation between the notion of $e$-exactness, recently introduced by Akray and Zebary, and some functors naturally related to it, such as the functor $P\colon\operatorname{Mod} R\to \operatorname{Spec}(\operatorname{Mod} R)$, where $\operatorname{Spec}(\operatorname{Mod} R)$ denotes the spectral category of $\operatorname{Mod} R$, and the localization functor with respect to the singular torsion theory.
- [3] arXiv:2504.12155 [pdf, html, other]
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Title: On a category of chains of modules whose endomorphism rings have at most $2n$ maximal idealsSubjects: Rings and Algebras (math.RA)
We describe the endomorphism rings in an additive category whose objects are right $R$-modules $M$ with a fixed chain of submodules $0=M^{(0)}\leq M^{(1)}\leq M^{(2)} \leq \dots \leq M^{(n)}=M$ and the behaviour of these objects as far as their direct sums are concerned.
New submissions (showing 3 of 3 entries)
- [4] arXiv:2504.11639 (cross-list from math.OA) [pdf, html, other]
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Title: Twisted Steinberg algebras, regular inclusions and inductionComments: 55 pages, no figuresSubjects: Operator Algebras (math.OA); Rings and Algebras (math.RA); Representation Theory (math.RT)
Given a field $K$ and an ample (not necessarily Hausdorff) groupoid $G$, we define the concept of a line bundle over $G$ inspired by the well known concept from the theory of C*-algebras. If $E$ is such a line bundle, we construct the associated twisted Steinberg algebra in terms of sections of $E$, which turns out to extend the original construction introduced independently by Steinberg in 2010, and by Clark, Farthing, Sims and Tomforde in a 2014 paper (originally announced in 2011). We also generalize (strictly, in the non-Hausdorff case) the 2023 construction of (cocycle) twisted Steinberg algebras of Armstrong, Clark, Courtney, Lin, Mccormick and Ramagge. We then extend Steinberg's theory of induction of modules, not only to the twisted case, but to the much more general case of regular inclusions of algebras. Our main result shows that every irreducible module is induced by an irreducible module over a certain abstractly defined isotropy algebra.
- [5] arXiv:2504.11960 (cross-list from cs.IT) [pdf, other]
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Title: On Codes from Split Metacyclic GroupsSubjects: Information Theory (cs.IT); Rings and Algebras (math.RA)
The paper presents a comprehensive study of group codes from non-abelian split metacyclic group algebras. We derive an explicit Wedderburn-like decomposition of finite split metacyclic group algebras over fields with characteristic coprime to the group order. Utilizing this decomposition, we develop a systematic theory of metacyclic codes, providing their algebraic description and proving that they can be viewed as generalized concatenated codes with cyclic inner codes and skew quasi-cyclic outer codes. We establish bounds on the minimum distance of metacyclic codes and investigate the class of induced codes. Furthermore, we show the feasibility of constructing a partial key-recovery attack against certain McEliece-type cryptosystems based on metacyclic codes by exploiting their generalized concatenated structure.
- [6] arXiv:2504.12206 (cross-list from math.QA) [pdf, html, other]
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Title: Finite GK-dimensional pre-Nichols algebras and quasi-quantum groupsComments: 45 pagesSubjects: Quantum Algebra (math.QA); Rings and Algebras (math.RA)
In this paper, we study the classification of finite GK-dimensional pre-Nichols algebras in the twisted Yetter-Drinfeld module category $_{\k G}^{\k G} \mathcal{YD}^\Phi$, where $G$ is a finite abelian group and $\Phi$ is a $3$-cocycle on $G$. These algebras naturally arise from quasi-quantum groups over finite abelian groups. We prove that all pre-Nichols algebras of nondiagonal type in $_{\k G}^{\k G} \mathcal{YD}^\Phi$ are infinite GK-dimensional, and every graded pre-Nichols algebra in $_{\k G}^{\k G} \mathcal{YD}^\Phi$ with finite GK-dimension is twist equivalent to a graded pre-Nichols algebra in an ordinary Yetter-Drinfeld module category $_{\k G}^{\k G} \mathcal{YD}^\Phi$, where $\mathbb{G}$ is a finite abelian group determined by $G$. In particular, we obtain a complete classification of finitely generated Nichols algebras with finite GK-dimension in $_{\k G}^{\k G} \mathcal{YD}^\Phi$. We prove that a finitely generated Nichols algebra in $_{\k G}^{\k G} \mathcal{YD}^\Phi$ is finite GK-dimensional if and only if it is of diagonal type and the corresponding root system is finite, i.e., an arithmetic root system. Via bosonization, this yields a large class of infinite quasi-quantum groups over finite abelian groups.
Cross submissions (showing 3 of 3 entries)
- [7] arXiv:2407.09666 (replaced) [pdf, html, other]
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Title: Two-Term Polynomial IdentitiesComments: to appear in Journal of AlgebraSubjects: Rings and Algebras (math.RA); Representation Theory (math.RT)
We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$ not fixing 1 or $n$ are eventually commutative in the sense that the equality $x_1\cdots x_k = x_{\tau(1)} \cdots x_{\tau(k)}$ holds for $k$ large enough and all permutations $\tau \in S_k$. Calling the minimal such $k$ the degree of eventual commutativity, we prove that $k$ is never more than $2n-3$, and that this bound is sharp. For various natural examples, we prove that $k$ can be taken to be $n+1$ or $n+2$. In the case when $q \ne 1$, we establish that the algebra must be nilpotent.
We, moreover, demonstrate that if an algebra is eventually commutative of arbitrary characteristic, then it has a finite basis of its polynomial identities, thus confirming the Specht conjecture in this particular case. - [8] arXiv:2304.09273 (replaced) [pdf, html, other]
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Title: Categories of hypermagmas, hypergroups, and related hyperstructuresComments: 54 pages, 3 figures. Corrections made throughout, but especially to Theorem 1.1, Lemma 2.13, and Theorem 4.19. Added Lemma 4.22 and Proposition 4.23. Final versionSubjects: Category Theory (math.CT); Combinatorics (math.CO); Group Theory (math.GR); Rings and Algebras (math.RA)
In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains categories with desirable features such as completeness and cocompleteness, free functors, regularity, and closed monoidal structures. We show by counterexamples that such constructions cannot be carried out within the category of canonical hypergroups. This suggests that (commutative) unital, reversible hypermagmas -- which we call mosaics -- form a worthwhile generalization of (canonical) hypergroups from the categorical perspective. Notably, mosaics contain pointed simple matroids as a subcategory, and projective geometries as a full subcategory.
- [9] arXiv:2410.17754 (replaced) [pdf, html, other]
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Title: Puncturing Quantum Stabilizer CodesComments: Accepted version for IEEE Journal on Selected Areas in Information TheorySubjects: Information Theory (cs.IT); Rings and Algebras (math.RA); Quantum Physics (quant-ph)
Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. For quantum codes, a form of puncturing is known, but its description is based on the code space rather than its generators. In this work, we generalize the puncturing procedure to allow more freedom in the choice of which coded states are kept and which are removed. We describe this puncturing by focusing on the stabilizer matrix containing the generators of the code. In this way, we are able to explicitly describe the stabilizer matrix of the punctured code given the stabilizer matrix of the original stabilizer code. The additional freedom in the procedure also opens up new ways to construct new codes from old, and we present several ways to utilize this for the search of codes with good or even optimal parameters. In particular, we use the construction to obtain codes whose parameters exceed the best previously known. Lastly, we generalize the proof of the Griesmer bound from the classical setting to stabilizer codes since the proof relies heavily on the puncturing technique.
- [10] arXiv:2503.06354 (replaced) [pdf, html, other]
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Title: Acyclicity test of complexes modulo Serre subcategories using the residue fieldsSubjects: Commutative Algebra (math.AC); Rings and Algebras (math.RA)
Let $R$ be a commutative noetherian ring, and let $\mathscr{S}$(resp. $\mathscr{L}$) be a Serre(resp. localizing) subcategory of the category of $R$-modules. If $\Bbb F$ is an unbounded complex of $R$-modules Tor-perpendicular to $\mathscr{S}$ and $d$ is an integer, then $\HH{i\geqslant d}{S\otimes_R \Bbb F}$ is in $\mathscr{L}$ for each $R$-module $S$ in $\mathscr{S}$ if and only if $\HH{i\geqslant d}{k(\fp)\otimes_R \Bbb F}$ is in $\mathscr{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathscr{S}$, where $k(\fp)$ is the residue field at $\fp$. As an application, we show that for any $R$-module $M$, $\Tor_{i\geqslant 0}^R(k(\fp),M)$ is in $\mathscr{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathscr{S}$ if and only if $\Ext^{i \geqslant 0}_R(S,M)$ is in $\mathscr{L}$ for each cyclic $R$-module $S$ in $\mathscr{S}$. We also obtain some new characterizations of regular and Gorenstein rings in the case of $\mathscr{S}$ consists of finite modules with supports in a specialization-closed subset $V(I)$ of $\Spec R$.
