Recent zbMATH articles in MSC 16https://zbmath.org/atom/cc/162022-11-17T18:59:28.764376ZUnknown authorWerkzeugIntroduction to \(N\)-soft algebraic structureshttps://zbmath.org/1496.032142022-11-17T18:59:28.764376Z"Kamaci, Hüseyin"https://zbmath.org/authors/?q=ai:kamaci.huseyinSummary: This paper is dedicated to two main objectives. The first of these is to develop some new operations on \(N\)-soft set, which is the generalization of soft set. The second is to highlight the concepts of \(N\)-soft group, \(N\)-soft ring, \(N\)-soft ideal, completely semiprime \(N\)-soft ideal, \(N\)-soft field and \(N\)-soft lattice. Moreover, in this study, it is attempted to derive certain properties for these concepts and to analyze the relations between them.On classifying the non-Tits \(P\)-critical posetshttps://zbmath.org/1496.060012022-11-17T18:59:28.764376Z"Bondarenko, V. M."https://zbmath.org/authors/?q=ai:bondarenko.vitalij-m"Styopochkina, M. V."https://zbmath.org/authors/?q=ai:stepochkina.maryna-vSummary: In 2005, the authors [Algebra Discrete Math. 2005, No. 2, 20--35 (2005; Zbl 1091.16007)] described all introduced by them \(P\)-critical posets (minimal finite posets with the quadratic Tits form not being positive); up to isomorphism, their number is 132 (75 if duality is considered). Later (in 2014) \textit{A. Polak} and \textit{D. Simson} [Linear Algebra Appl. 445, 223--255 (2014; Zbl 1290.16014)] offered an alternative way of proving by using computer algebra tools. In doing this, they defined and described the Tits \(P\)-critical posets as a special case of the \(P\)-critical posets. In this paper we classify all the non-Tits \(P\)-critical posets without complex calculations and without using the list of all \(P\)-critical ones.Hamiltonian trace graph of matriceshttps://zbmath.org/1496.130032022-11-17T18:59:28.764376Z"Sivagami, M."https://zbmath.org/authors/?q=ai:sivagami.m"Tamizh Chelvam, T."https://zbmath.org/authors/?q=ai:tamizh-chelvam.thirugnanamOn the nonrigidity of trace moduleshttps://zbmath.org/1496.130142022-11-17T18:59:28.764376Z"Lindo, Haydee"https://zbmath.org/authors/?q=ai:lindo.haydeeLet's start by some ingredients used in this paper. Let \(R\) be a ring and \(M\), \(X\) be \(R\)-modules. The trace module of \(M\) in \(X\), denoted \(\tau_M(X)\), is the \(R\)-module \(\sum\alpha(M)\) as \(\alpha\) ranges over \(\mathrm{Hom}_R(M,X)\). Such a trace module is proper provided \(\tau_M(X)\subsetneq X\). A left \(R\)-module \(M\) is said to be rigid if \(\mathrm{Ext}^1_R(M,M)= 0\) and that a local commutative Noetherian ring is Artinian Gorenstein if and only if it is self-injective that is injective as \(R\)-module. Recall that an \(R\)-module \(M\) is G-projective if the following conditions hold:
\begin{itemize}
\item[(i)] The natural homomorphism \(M \to M^{**}\) is an isomorphism.
\item[(ii)] \(\mathrm{Ext}^i_R(M,R)=0\) for any \(i >0\).
\item[(iii)] \(\mathrm{Ext}^i_R(M^*,R)=0\) for any \(i >0\).
\end{itemize}
\par In this paper, the author investigates a link between trace modules and rigidity in modules over Noetherian rings. First result asserts that if \(M\) is a proper submodule of \(X\) such that \(\mathrm{Hom}_R(M,X/M)=0\), then \(M\) is a trace module in \(X\), and that the converse holds when \(M\) is rigid. In the second result the author shows that over a local commutative Artinian ring, the syzygy of a G-projective proper trace module is not rigid. Finally, the Auslander-Reiten conjeture is discussed. Before, we recall the Auslander-Reiten conjecture;
{Auslander-Reiten conjecture :} Let \(A\) be an Artin algebra and \(M\) a finitely generated \(A\)-module. If \(\mathrm{Ext}^1_R(M,M)=0\), \(\mathrm{Ext}^1_R(M,R)=0\) for all \(i>0\), then \(M\) is projective.
Then the author shows that, if \(R\) is a local Artinian Gorenstein ring. Then the Auslander-Reiten conjecture holds for all positive and negative syzygies of ideals \(I\subseteq R\).
Reviewer: Mohamed Aqalmoun (Fez)Geometry of varieties for graded maximal Cohen-Macaulay moduleshttps://zbmath.org/1496.130152022-11-17T18:59:28.764376Z"Hiramatsu, Naoya"https://zbmath.org/authors/?q=ai:hiramatsu.naoyaLet \(k\) be an algebraically closed field and let \(R\) be a commutative positively graded affine \(k\)-algebra with \(R_0 = k\). Let \(S\) be a graded Noetherian normalization of \(R.\) A finitely generated graded \(R\)-module \(M\) is said to be maximal Cohen-Macaulay if \(M\) is graded free as a graded \(S\)-module. Then \(_SM \simeq _SS \otimes_k V\) for some finite dimensional graded \(k\)-vector space \(V.\) The representation scheme \(\mathrm{Rep}_S(R, V)\) for graded maximal Cohen-Macaulay \(R\)-modules is introduced by \textit{H. Dao} and \textit{I. Shipman} [Sel. Math., New Ser. 23, No. 1, 1--14 (2017; Zbl 1391.13018)]. In the paper under review, the authors study this variety.
{Theorem}. Let \(A\) and \(B\) be graded Cohen-Macaulay \(k\)-algebras with \(A, B \simeq S \otimes_k V.\) Suppose that \(B\) is an integral domain and Gorenstein on the punctured spectrum. Then the following conditions are equivalent:
(a) \(A\) and \(B\) are isomorphic graded \(S\)-algebras.
(b) For the finite dimensional graded \(k\)-vector space \(V\), the algebraic sets \(\mathrm{Rep}_S(A, V)(k)\) and \(\mathrm{Rep}_S(B, V)(k)\) are isomorphic in an \(^*\mathrm{End}_S(S \otimes_k V)\)-equivariant.
A graded Cohen-Macaulay ring \(R\) is of graded countable Cohen-Macaulay representation type if there are countably many isomorphism classes of indecomposable graded maximal Cohen-Macaulay \(R\)-modules up to shift. The main result of the paper is the following finiteness result about graded maximal Cohen-Macaulay modules.
{Theorem}. Let \(k\) be an algebraically closed uncountable field and let \(R\) be of graded countable Cohen-Macaulay representation type. For each finite dimensional \(k\)-vector space \(V\), there are finitely many isomorphism classes of maximal Cohen-Macaulay \(R\)-modules which are isomorphic to \(S \otimes_k V\) as graded \(S\)-modules. In other words, there are only a finite number of isomorphism classes of maximal Cohen-Macaulay \(R\)-modules with fixed Hilbert series.
Reviewer: Kriti Goel (Mumbai)On Gerko's strongly Tor-independent moduleshttps://zbmath.org/1496.130232022-11-17T18:59:28.764376Z"Altmann, Hannah"https://zbmath.org/authors/?q=ai:altmann.hannah"Sather-Wagstaff, Keri"https://zbmath.org/authors/?q=ai:sather-wagstaff.keriSummary: \textit{A. Gerko} [Ill. J. Math. 48, No. 3, 965--976 (2004; Zbl 1080.13009)] proved that if an artinian local ring \((R,\mathfrak{m}_R)\) possesses a sequence of strongly Tor-independent modules of length \(n\), then \(\mathfrak{m}_R^n\neq 0\). This generalizes readily to Cohen-Macaulay rings. We present a complement to this result for non-Cohen-Macaulay rings.
For the entire collection see [Zbl 1485.13001].Noncommutative geometry. A functorial approachhttps://zbmath.org/1496.140022022-11-17T18:59:28.764376Z"Nikolaev, Igor V."https://zbmath.org/authors/?q=ai:nikolaev.igor-vladimirovich|nikolaev.igor-vasilievichPublisher's description: Noncommutative geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. This book covers the key concepts of noncommutative geometry and its applications in topology, algebraic geometry, and number theory. Our presentation is accessible to the graduate students as well as nonexperts in the field. The second edition includes two new chapters on arithmetic topology and quantum arithmetic.
\begin{itemize}
\item An authoritative introductory treatment of noncommutative geometry now in its second edition.
\item A novel approach using functors is presented in detail.
\item Covers applications of the theory in topology, algebraic geometry, and number theory.
\end{itemize}
See the review of the first edition in [Zbl 1388.14001].Manifold expressions of all solutions of the Yang-Baxter-like matrix equation for rank-one matriceshttps://zbmath.org/1496.150112022-11-17T18:59:28.764376Z"Lu, Linzhang"https://zbmath.org/authors/?q=ai:lu.linzhangSummary: Let \(A\) be a complex rank-one matrix, we derive simple sufficient and necessary conditions for a complex matrix \(X\) being a (commuting, non-commuting) solution of the quadratic matrix equation \(A X A = X A X\). On the basis, we construct the manifolds, each of which is a product of one free parameter matrix and another one to two matrices fixed by \(A\), to express concisely all the (commuting) solutions of the matrix equation.On characterization of tripotent matrices in triangular matrix ringshttps://zbmath.org/1496.150222022-11-17T18:59:28.764376Z"Petik, Tuğba"https://zbmath.org/authors/?q=ai:petik.tugbaSummary: Let \(\mathfrak{R}\) be a ring with identity 1 whose tripotents are only \(-1\), 0, and 1. It is characterized the structure of tripotents in \(\mathcal{T}(\mathfrak{R})\) which is the ring of triangular matrices over \(\mathfrak{R}\). In addition, when \(\mathfrak{R}\) is finite, it is given number of the tripotents in \(\mathcal{T}_n(\mathfrak{R})\) which is the ring of \(n\times n\) dimensional triangular matrices over \(\mathfrak{R}\) with \(n\) being a positive integer.Generalizations of \(ss\)-supplemented moduleshttps://zbmath.org/1496.160012022-11-17T18:59:28.764376Z"Soydan, I."https://zbmath.org/authors/?q=ai:soydan.i"Türkmen, E."https://zbmath.org/authors/?q=ai:turkmen.ergul|turkmen.esrefSummary: We introduce the concept of (strongly) \(ss\)-radical supplemented modules. We prove that if a submodule \(N\) of \(M\) is strongly \(ss\)-radical supplemented and \(Rad(M/N)=M/N\), then \(M\) is strongly \(ss\)-radical supplemented. For a left good ring \(R\), we show that \(Rad(R)\subseteq Soc(_RR)\) if and only if every left \(R\)-module is \(ss\)-radical supplemented. We characterize the rings over which all modules are strongly \(ss\)-radical supplemented. We also prove that over a left \(WV\)-ring every supplemented module is \(ss\)-supplemented.Rings with nonzero maps between non-projective moduleshttps://zbmath.org/1496.160022022-11-17T18:59:28.764376Z"Türkoğlu, Zübeyir"https://zbmath.org/authors/?q=ai:turkoglu.zubeyirThe aim of this paper is to study rings \(R\) which admit non-zero homomorphisms between any pair of non-zero right \(R\)-modules. The main theorem concerns rings satisfying Property (T): for any pair \(A,\ B\) of non-projective \(R\)-modules, Hom\((A,\,B)\ne 0\).
Theorem. A ring \(R\) satisfies Property (T) if and only if
\begin{itemize}
\item[1.] \(R\) has a non-zero singular submodule;
\item[2.] every non-zero submodule contains a maximal submodule;
\item[3.] \(R\) has a unique up to isomorphism non-projective simple module \(S\);
\item[4.] the second singular submodule of \(R\) is a semi-artinian direct summand whose complement is isomorphic to a direct product of a semisimple artinian module and a 2 \(\times\) 2 matrix ring over a division ring.
\end{itemize}
The author provides several examples of rings not admitting non-zero homomorphisms between some pair of modules which demonstrate the necessity of these conditions.
Reviewer: Phillip Schultz (Perth)A note on essential Ikeda-Nakayama ringshttps://zbmath.org/1496.160032022-11-17T18:59:28.764376Z"Derakhshan, Mahya"https://zbmath.org/authors/?q=ai:derakhshan.mahya"Sahebi, Shervin"https://zbmath.org/authors/?q=ai:sahebi.shervin"Haj Seyed Javadi, Hamid"https://zbmath.org/authors/?q=ai:haj-seyed-javadi.hamidThis paper is a contribution to the ideal theory of non-commutative rings. A ring \(R\) is called Essential Ikeda-Nakayama (EIN) if for any two ideals \(I\) and \(J\), the sum of the left annihilators \(\ell(I)+\ell(J)\) is essential in \(\ell(I\cup J)\). The authors study the closure of EIN rings under direct products, polynomial rings, left triangular matrices and adjunction of identity, and the properties of non-singular EIN rings. They show that every semiprime ring is EIN.
Reviewer: Phillip Schultz (Perth)\(AB\)5*-modules with the exchange propertyhttps://zbmath.org/1496.160042022-11-17T18:59:28.764376Z"Ibrahim, Yasser"https://zbmath.org/authors/?q=ai:ibrahim.yasser-f"Yousif, Mohamed"https://zbmath.org/authors/?q=ai:yousif.mohamed-fSummary: We prove that every \(AB\)5*-module has an indecomposable decomposition. As an immediate consequence, every \(AB5\)*-module \(M\) with the finite exchange is clean and has the full exchange. Moreover, in this case the module \(M\) admits an indecomposable decomposition \(M = \oplus_{i \in I}M_i\) with each \(M_i\) having local endomorphism ring, and the decomposition complements direct summands.Lifting modules with finite internal exchange property and direct sums of hollow moduleshttps://zbmath.org/1496.160052022-11-17T18:59:28.764376Z"Kuratomi, Yosuke"https://zbmath.org/authors/?q=ai:kuratomi.yosukeArmendariz-like properties in bi-amalgamationshttps://zbmath.org/1496.160062022-11-17T18:59:28.764376Z"Chhiti, Mohamed"https://zbmath.org/authors/?q=ai:chhiti.mohamed"Es-Salhi, Loubna"https://zbmath.org/authors/?q=ai:es-salhi.loubnaSummary: This paper examines the transfer of the Armendariz (resp., weak Armendariz, resp., nil-Armendariz) property to bi-amalgamations. Our results cover previously known results on amalgamations, and provide the construction of various and original examples satisfying the above-mentioned properties.Ding injective moduleshttps://zbmath.org/1496.160072022-11-17T18:59:28.764376Z"Iacob, Alina"https://zbmath.org/authors/?q=ai:iacob.alina-cSummary: We prove that, over any ring R, the class of Ding injective modules, \(\mathcal{DI}\), is the right half of a hereditary cotorsion pair, \((^{\perp}\mathcal{DI},\mathcal{DI})\). We also show that, over a coherent ring R, a module M is Ding injective if and only if it is Gorenstein injective and in the class \(\mathcal{FI}^{\perp_{\infty}}\).The finitistic dimension of an Artin algebra with radical square zerohttps://zbmath.org/1496.160082022-11-17T18:59:28.764376Z"Gélinas, Vincent"https://zbmath.org/authors/?q=ai:gelinas.vincentSummary: We investigate the inequality \(\operatorname{Findim} \Lambda^{op}\leq\operatorname{dell}\Lambda\) between the finitistic dimension and the delooping level of an Artin algebra \(\Lambda\), and whether equality holds in general. We prove that equality \(\operatorname{Findim}\Lambda^{op}=\operatorname{dell}\Lambda\) always holds for Artin algebras with radical square zero.Reduction techniques for the finitistic dimensionhttps://zbmath.org/1496.160092022-11-17T18:59:28.764376Z"Green, Edward L."https://zbmath.org/authors/?q=ai:green.edward-lee"Psaroudakis, Chrysostomos"https://zbmath.org/authors/?q=ai:psaroudakis.chrysostomos"Solberg, Øyvind"https://zbmath.org/authors/?q=ai:solberg.oyvindSummary: In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples.Resolving resolution dimension of recollements of abelian categorieshttps://zbmath.org/1496.160102022-11-17T18:59:28.764376Z"Zhang, Houjun"https://zbmath.org/authors/?q=ai:zhang.houjun"Zhu, Xiaosheng"https://zbmath.org/authors/?q=ai:zhu.xiaoshengInjective stabilization of additive functors. III: Asymptotic stabilization of the tensor producthttps://zbmath.org/1496.160112022-11-17T18:59:28.764376Z"Martsinkovsky, A."https://zbmath.org/authors/?q=ai:martsinkovsky.alex"Russell, J."https://zbmath.org/authors/?q=ai:russell.jeremySummary: The injective stabilization of the tensor product is subjected to an iterative procedure that utilizes its bifunctor property. The limit of this procedure, called the asymptotic stabilization of the tensor product, provides a homological counterpart of Buchweitz's asymptotic construction of stable cohomology. The resulting connected sequence of functors is isomorphic to Triulzi's \(J\)-completion of the Tor functor. A comparison map from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be always epic. The category of finitely presented functors is shown to be complete and cocomplete. As a consequence, the inert injective stabilization of the tensor product with fixed variable a finitely generated module over an artin algebra is shown to be finitely presented. Its defect and consequently all right-derived functors are determined. New notions of asymptotic torsion and cotorsion are introduced and are related to each other.
For Part I, see [the authors, ibid. 530, 429--469 (2019; Zbl 1444.16009)]. For Part II, see [the authors, ibid. 548, 53--95 (2020; Zbl 1462.16031)].From the potential to the first Hochschild cohomology group of a cluster tilted algebrahttps://zbmath.org/1496.160122022-11-17T18:59:28.764376Z"Assem, Ibrahim"https://zbmath.org/authors/?q=ai:assem.ibrahim"Bustamante, Juan Carlos"https://zbmath.org/authors/?q=ai:bustamante.juan-carlos"Trepode, Sonia"https://zbmath.org/authors/?q=ai:trepode.sonia-elisabet"Valdivieso, Yadira"https://zbmath.org/authors/?q=ai:valdivieso.yadiraSummary: The objective of this paper is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.Cohomology and deformations of twisted Rota-Baxter operators and NS-algebrashttps://zbmath.org/1496.160132022-11-17T18:59:28.764376Z"Das, Apurba"https://zbmath.org/authors/?q=ai:das.apurba-narayan|das.apurbaLet us recall that a Rota-Baxter is an algebraic abstraction of the integral operator. In this paper, the author considers twisted Rota-Baxter operators on associative algebras. First, the author recalls H-twisted Rota-Baxter operators, Reynolds operators and NS-algebras. Then, he constructs \(L_\infty\)-algebras where Maurer-Cartan elements are given by twisted Rota-Naxter operators.
Reviewer: Angela Gammella-Mathieu (Metz)Tilting modules and dominant dimension with respect to injective moduleshttps://zbmath.org/1496.160142022-11-17T18:59:28.764376Z"Adachi, Takahide"https://zbmath.org/authors/?q=ai:adachi.takahide"Tsukamoto, Mayu"https://zbmath.org/authors/?q=ai:tsukamoto.mayuSummary: In this paper, we study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey-Sauter, Nguyen-Reiten-Todorov-Zhu and Pressland-Sauter. Moreover, we give characterizations of almost \(n\)-Auslander-Gorenstein algebras and almost \(n\)-Auslander algebras by the existence of tilting modules. As an application, we describe a sufficient condition for almost 1-Auslander algebras to be strongly quasi-hereditary by comparing such tilting modules and characteristic tilting modules.On the correspondence between path algebras and generalized path algebrashttps://zbmath.org/1496.160152022-11-17T18:59:28.764376Z"Chust, Viktor"https://zbmath.org/authors/?q=ai:chust.viktor"Coelho, Flávio U."https://zbmath.org/authors/?q=ai:coelho.flavio-ulhoaGeneralized path algebras were introduced by \textit{F. U. Coelho} and \textit{S. Liu} [Lect. Notes Pure Appl. Math. 210, 53--66 (2000; Zbl 0987.16007)]. The Gabriel quiver and relations of a generalized path algebra was described by \textit{R. M. Ibáñez Cobos} et al. [Rev. Roum. Math. Pures Appl. 53, No. 1, 25--36 (2008; Zbl 1178.16011)].
In the paper under review, the authors generalize generalized path algebra to generalized path algebra with relations which is called \textit{generalized bound quiver algebra}, or \textit{gbq-algebra} for short. Then they determine the Gabriel quiver and relations of a gbq-algebra. Moreover, they study the inverse problem -- when a bound quiver algebra is isomorphic to a gbq-algebra.
Reviewer: Yang Han (Beijing)Simple reflexive modules over finite-dimensional algebrashttps://zbmath.org/1496.160162022-11-17T18:59:28.764376Z"Ringel, Claus Michael"https://zbmath.org/authors/?q=ai:ringel.claus-michaelOn \(n\)-slice algebras and related algebrashttps://zbmath.org/1496.160172022-11-17T18:59:28.764376Z"Guo, Jin-Yun"https://zbmath.org/authors/?q=ai:guo.jinyun"Xiao, Cong"https://zbmath.org/authors/?q=ai:xiao.cong"Lu, Xiaojian"https://zbmath.org/authors/?q=ai:lu.xiaojianSummary: The \(n\)-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of \(n\)-slice algebras via their \( (n+1) \)-preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame \(n\)-slice algebras to the McKay quiver of a finite subgroup of \( \mathrm{GL}(n+1,\mathbb{C}) \). In the case of \( n = 2 \), we describe the relations for the 2-slice algebras related to the McKay quiver of finite abelian subgroups of \( \mathrm{SL}(3,\mathbb{C}) \) and of the finite subgroups obtained from embedding \( \mathrm{SL}(2, \mathbb{C}) \) into \( \mathrm{SL}(3,\mathbb{C}) \).Derived Picard groups of preprojective algebras of Dynkin typehttps://zbmath.org/1496.160182022-11-17T18:59:28.764376Z"Mizuno, Yuya"https://zbmath.org/authors/?q=ai:mizuno.yuyaSummary: In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover, we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we determine the derived Picard group of preprojective algebras for type \(A\) and \(D\).An explicit construction of the universal division ring of fractions of \(E\langle\langle x_1,\ldots, x_d \rangle\rangle\)https://zbmath.org/1496.160192022-11-17T18:59:28.764376Z"Jaikin-Zapirain, Andrei"https://zbmath.org/authors/?q=ai:jaikin-zapirain.andreiSummary: We give a sufficient and necessary condition for a regular Sylvester matrix rank function on a ring \(R\) to be equal to its inner rank \(\rho_R\). We apply it in two different contexts.
In our first application, we reprove a recent result of \textit{T. Mai} et al. [``The free field: realization via unbounded operators and Atiyah property'', Preprint, \url{arXiv:1905.08187}]: if \(X_1,\ldots, X_d\) are operators in a finite von Neumann algebra \(\mathcal{M}\) with a faithful normal trace \(\tau\), then they generate the free division ring on \(X_1, \ldots, X_d\) in the algebra of unbounded operators affiliated to \(\mathcal{M}\) if and only if the space of tuples \((T_1,\ldots, T_d)\) of finite rank operators on \(L^2 (\mathcal{M},\tau)\) satisfying
\[
\sum_{i=1}^d [T_k, X_k]=0,
\]
is trivial.
In our second and main application we construct explicitly the universal division ring of fractions of \(E \langle\langle x_1,\ldots, x_n \rangle\rangle \), where \(E\) is a division ring, and we use it in order to show the following instance of pro-\(p\) Lück approximation.
Let \(F\) be a finitely generated free pro \(p\)-group, \(F=F_1 > F_2 > \cdots\) a chain of normal open subgroups of \(F\) with trivial intersection and \(A\) a matrix over \(\mathbb{F}_p [[F]]\). Denote by \(A_i\) the matrix over \(\mathbb{F}_p [F/F_i]\) obtained from the matrix \(A\) by applying the natural homomorphism \(\mathbb{F}_p [[F]] \to \mathbb{F}_p [F/F_i]\). Then there exists the limit
\[
\displaystyle\lim_{i \to \infty} \frac{\mathrm{rk}_{\mathbb{F}_p} (A_i)}{|F:F_i|}
\]
and it is equal to the inner rank \(\rho_{\mathbb{F}_p [[F]]} (A)\) of the matrix \(A\).Nilpotent polynomials and nilpotent coefficientshttps://zbmath.org/1496.160202022-11-17T18:59:28.764376Z"Draper, Thomas L."https://zbmath.org/authors/?q=ai:draper.thomas-l"Nielsen, Pace P."https://zbmath.org/authors/?q=ai:nielsen.pace-p"Šter, Janez"https://zbmath.org/authors/?q=ai:ster.janezMany mathematicians believe that Köthe conjecture is one of the hardest problem in mathematics. It was started in 1930 and it is still open until now. Some reformulations of the Köthe conjecture were found by many prominent authors. The Köthe conjecture is also equivalent to the condition, for any ring \(R\), the Jacobson radical of \(R[x]\) consists of the polynomials with coefficients from the upper nilradical of \(R\). Hence, research related to nilradical is interesting. On the other hand,
\textit{A. Smoktunowicz} [J. Algebra 233, No. 2, 427--436 (2000; Zbl 0969.16006)] introduced a ring \(R\) such that Nil\((R[x])\) is a proper subset of Nil\((R)[x]\) where Nil\((R)[x]\) is the set of nilpotent elements in \(R[x]\) which is precisely the nilradical of \(R[x]\).
In this paper, the authors provide the converse condition by constructing a ring \(R\) such that Nil\((R)^2=0\) and a polynomial \(f \in R[x] \setminus\mathrm{Nil}(R)[x]\) satisfying \(f^2=0\). However, the smallest possible degree a such polynomial is seven. This example also explains the answer of an open problem asked by
\textit{R. Antoine} [Commun. Algebra 38, No. 11, 4130--4143 (2010; Zbl 1218.16013)] related to Armendariz ring.
Reviewer: Puguh Wahyu Prasetyo (Yogyakarta)Some generalized identities on prime rings and their application for the solution of annihilating and centralizing problemshttps://zbmath.org/1496.160212022-11-17T18:59:28.764376Z"De Filippis, Vincenzo"https://zbmath.org/authors/?q=ai:de-filippis.vincenzo"Prajapati, B."https://zbmath.org/authors/?q=ai:prajapati.balchand"Tiwari, S. K."https://zbmath.org/authors/?q=ai:tiwari.shailesh-kumarSeveral results in literature regard the study of the structure of a prime ring \(R\); in many cases the authors get information about the structure of the ring considering appropriate conditions on the following subset of \(R\): \[P(d,k,S)=\{[d(s),s]_k:s\in S\},\] where \(d\) is an additive map defined on \(R\), \(S\) a suitable subset of \(R\) and \(k\geq1\) a fixed integer.\\
Many authors studied \(P(d,k,S)\) in the case in which \(d\) is a (generalized) derivation and \(S\) a Lie ideal or the set of all evaluations of a multilinear polynomial on \(R\); in other papers have also been studied the annihilator and the centralizer of \(P(d,k,S)\).\\
Since the previous results say that the subset \(P(d,k,S)\) is rather large in \(R\), in this paper the authors decided to study a more general condition: specifically, let \(R\) be a non-commutative prime ring, with \(\operatorname{char}(R)\neq2\), \(F\) and \(G\) two non-zero generalized derivations, suppose that \(R\) does not satisfy the standard polynomial identity \(s_4\) and there exist \(p,q\in R\) such that \[p[F(r),r]+q[G(r),r]p+[F(r),r]q=0\] for all \(r\in f(R)\), the set of all evaluations of a multilinear polynomial \(f(x_1,\ldots,x_n)\).
In the main theorem of the paper the authors give the complete description of all possible forms of the involved maps.
Reviewer: Giovanni Scudo (Messina)Noetherian criteria for dimer algebrashttps://zbmath.org/1496.160222022-11-17T18:59:28.764376Z"Beil, Charlie"https://zbmath.org/authors/?q=ai:beil.charlieDimer algebras were introduced in string theory to describe a class of quiver gauge theories. A dimer algebra is a quiver algebra \(A=kQ/I\) of a quiver \(Q\) whose underlying graph \(\overline{Q}\) embeds in a torus \(T^2\) such that each connected component of \(T^2\setminus \overline{Q}\) is simply connected and bounded by an oriented cycle, called a unit cycle. The relations of \(A\) are given by the ideal \(I=<p-q: \text{ there exists } a\in Q_1\) such that \(pa\) and \(qa\) are unit cycles\(>\subset kQ\), where \(p\) and \(q\) are paths. The main theorem of this paper states that if \(A=kQ/I\) is a nondegenerate dimer algebra on a torus with center \(Z\), then the following are equivalent: (a) \(A\) is noetherian, (b) \(A\) is cancellative, (c) \(Z\) is noetherian, (d) \(A\) is a noncommutative crepant resolution, (e) each arrow of \(A\) is contained in a perfect matching whose complement supports a simple module, and (f) the vertex corner rings \(e_iAe_i\) are pairwise isomorphic.
Reviewer: Ashish K. Srivastava (Saint Louis)Cancellation problem for AS-regular algebras of dimension threehttps://zbmath.org/1496.160232022-11-17T18:59:28.764376Z"Tang, Xin"https://zbmath.org/authors/?q=ai:tang.xin"Venegas Ramírez, Helbert J."https://zbmath.org/authors/?q=ai:venegas-ramirez.helbert-j"Zhang, James J."https://zbmath.org/authors/?q=ai:zhang.james-yiming|zhang.james-jSummary: We study a noncommutative version of the Zariski cancellation problem for some classes of connected graded Artin-Schelter regular algebras of global dimension three.Nil-generated algebras and group algebras whose units satisfy a Laurent polynomial identityhttps://zbmath.org/1496.160242022-11-17T18:59:28.764376Z"Rodrigues, Claudenir Freire"https://zbmath.org/authors/?q=ai:rodrigues.claudenir-freireIn the paper under review, the author studied nil-generated algebras and group algebras whose units satisfy the Laurent polynomial identity (briefly abbreviated by LPI). In fact, the more remarkable results are stated in Theorems 3.3 and 3.8, respectively. Concretely, in Theorem 3.3 a confirmation that the Hartley Conjecture is obtained with arbitrary LPI for the group of units \(U(KG)\) in the group ring \(KG\), where \(G\) is a torsion group and \(K\) is an arbitrary field. It is also showed in Theorem 3.8 the existence of a non-matrix identity for a group algebra of a torsion group over an infinite field of characteristic \(p>0\). The paper is well organized and written.
Reviewer: Peter Danchev (Sofia)Universal enveloping of (modified) \(\lambda\)-differential Lie algebrashttps://zbmath.org/1496.160252022-11-17T18:59:28.764376Z"Peng, Xiao-Song"https://zbmath.org/authors/?q=ai:peng.xiao-song"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.3|zhang.yi.2|zhang.yi.1|zhang.yi.6|zhang.yi.12|zhang.yi.14|zhang.yi.8|zhang.yi.5|zhang.yi.10|zhang.yi.4|zhang.yi"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Luo, Yan-Feng"https://zbmath.org/authors/?q=ai:luo.yan-fengThis paper deals with somewhat natural generalizations of differential (associative or Lie) algebras, namely \(\lambda\)-differential (associative or Lie) algebras and modified \(\lambda\)-differential (associative or Lie) algebras. A \(\lambda\)-differential (associative or Lie) algebra, for a constant \(\lambda\), is roughly an associative or Lie algebra \(A\) with a linear endomorphism \(A\xrightarrow{d}A\) which satisfies a relation similar to the Leibniz rule, namely \(d(xy)=xd(y)+d(x)y+\lambda d(x)d(y)\), \(x,y\in A\) (where, when \(A\) is a Lie algebra, a concatenation such as \(xy\) should be read as a Lie bracket \([x,y]\)). In the ``modified'' version the term \(\lambda d(x)d(y)\) is replaced by the term \(\lambda xy\).
The existence and uniqueness (up to a unique isomorphism) results about free objects are obtained easily by the authors because \(\lambda\)-differential \((\Bbbk,\partial)\)-modules and algebras, and their ``modified'' versions are categories concretely equivalent to varieties of algebras in the sense of universal algebra so that each algebraic functor, that is, a functor which preserves the underlying sets, has a left adjoint. Therefore the authors focus on explicit constructions for these objects, which is far more interesting and more tricky (see, e.g., Theorem 3.5, p.~1114, Theorem~3.8, p.~1116).
Reviewer: Laurent Poinsot (Villetaneuse)On clean, weakly clean and feebly clean commutative group ringshttps://zbmath.org/1496.160262022-11-17T18:59:28.764376Z"Li, Yuanlin"https://zbmath.org/authors/?q=ai:li.yuanlin"Zhong, Qinghai"https://zbmath.org/authors/?q=ai:zhong.qinghaiThe authors of this extremely technical paper successfully investigate when certain commutative group rings are either clean, weakly clean or feebly clean, respectively. The most important results of the paper under review are stated and proved in Theorems 3.1, 3.5 and 3.8. In their proofs the authors used several interesting technicalities from group theory, ring theory and number theory. No open questions are given at the end of the text, however, which could motivate some further intensive research on the present subject. Nevertheless, the paper is well written as well as the exploration is well motivated. The presentation of the results is also of good quality.
Reviewer: Peter Danchev (Sofia)\(\pi\)-Rickart ringshttps://zbmath.org/1496.160272022-11-17T18:59:28.764376Z"Birkenmeier, Gary F."https://zbmath.org/authors/?q=ai:birkenmeier.gary-f"Kara, Yeliz"https://zbmath.org/authors/?q=ai:kara.yeliz"Tercan, Adnan"https://zbmath.org/authors/?q=ai:tercan.adnanCanonical resolutions over Koszul algebrashttps://zbmath.org/1496.160282022-11-17T18:59:28.764376Z"Faber, Eleonore"https://zbmath.org/authors/?q=ai:faber.eleonore"Juhnke-Kubitzke, Martina"https://zbmath.org/authors/?q=ai:juhnke-kubitzke.martina"Lindo, Haydee"https://zbmath.org/authors/?q=ai:lindo.haydee"Miller, Claudia"https://zbmath.org/authors/?q=ai:miller.claudia-m"Rebhuhn-Glanz, Rebecca"https://zbmath.org/authors/?q=ai:rebhuhn-glanz.rebecca"Seceleanu, Alexandra"https://zbmath.org/authors/?q=ai:seceleanu.alexandraSummary: We generalize Buchsbaum and Eisenbud's resolutions for the powers of the maximal ideal of a polynomial ring to resolve powers of the homogeneous maximal ideal over graded Koszul algebras. Our approach has the advantage of producing resolutions that are both more explicit and minimal compared to those previously discovered by \textit{E. L. Green} and \textit{R. Martínez-Villa} [CMS Conf. Proc. 18, 247--297 (1996; Zbl 0860.16009)] or \textit{R. Martínez-Villa} and \textit{D. Zacharia} [J. Algebra 266, No. 2, 671--697 (2003; Zbl 1061.16035)].
For the entire collection see [Zbl 1485.13001].Hamming spaces and locally matrix algebrashttps://zbmath.org/1496.160292022-11-17T18:59:28.764376Z"Bezushchak, Oksana"https://zbmath.org/authors/?q=ai:bezushchak.oksana-o"Oliynyk, Bogdana"https://zbmath.org/authors/?q=ai:oliynyk.bogdanaOn the generalized strongly nil-clean property of matrix ringshttps://zbmath.org/1496.160302022-11-17T18:59:28.764376Z"Kostić, Aleksandra S."https://zbmath.org/authors/?q=ai:kostic.aleksandra-s"Petrović, Zoran Z."https://zbmath.org/authors/?q=ai:petrovic.zoran-z"Pucanović, Zoran S."https://zbmath.org/authors/?q=ai:pucanovic.zoran-s"Roslavcev, Maja"https://zbmath.org/authors/?q=ai:roslavcev.majaThe \textit{nil-clean rings} were introduced in [\textit{A. J. Diesl}, J. Algebra 383, 197--211 (2013; Zbl 1296.16016)]. Those are the rings (with identity) in which every element is expressible as a sum of an idempotent and a nilpotent. In the paper under review, the authors consider some generalizations of the nil-clean property. In particular, they investigate the relations between these properties for a ring \(R\) and its matrix ring \(\mathbb M_n(R)\).
Let \(s\) be a positive integer. A ring \(R\) is \textit{\(s\)-nil-clean} if every \(a\in R\) can be written in the form \(a=e_1+\cdots+e_s+n\), where \(e_1,\ldots,e_s\in R\) are idempotents and \(n\in R\) is a nilpotent. \(R\) is \textit{strongly \(s\)-nil-clean} if every \(a\in R\) can be written in the above form, with additional condition that \(e_1,\ldots,e_s,n\) commute with each other. If one requests that only idempotents \(e_1,\ldots,e_s\) commute with each other, then \(R\) is called \textit{semi-strongly \(s\)-nil-clean}.
The paper contains a number of results establishing some of these properties for matrix rings. For example, the authors use Frobenius normal form of a matrix over a field to prove the following: For every \(n>1\), the matrix ring \(\mathbb M_n(\mathbb Z_p)\) is semi-strongly \((p-1)\)-nil-clean, where \(p\) is an odd prime and \(\mathbb Z_p\) is the field with \(p\) elements. Some results concerning the case when \(R\) is \textit{commutative} read as follows:
\begin{itemize}
\item if \(R\) is \(2\)-nil-clean, then \(\mathbb M_n(R)\) is semi-strongly \(2\)-nil-clean (for every \(n>1\));
\item if \(p>3\) is a prime and \(R\) is \((p-1)\)-nil-clean, then \(\mathbb M_n(R)\) is \(\delta(p,n)\)-nil-clean, where \(\delta(p,n)=\min\{p-1,4+\lfloor\frac{p-1}{n}\rfloor\}\).
\end{itemize}
Reviewer: Branislav Prvulović (Beograd)Every graded ideal of a Leavitt path algebra is \textit{graded} isomorphic to a Leavitt path algebrahttps://zbmath.org/1496.160312022-11-17T18:59:28.764376Z"Vaš, Lia"https://zbmath.org/authors/?q=ai:vas.liaIt is shown that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. As pointed out by the author, it is known that a graded ideal \(I\) of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalised hedgehog graph, which is defined based on certain sets of vertices uniquely determined by \(I\). However, this isomorphism may not be graded. The results in this paper show that replacing the short `spines' of the generalised hedgehog graph with possibly fewer, but then necessarily longer spines, one obtains a graph (which is called the porcupine graph by the author) whose Leavitt path algebra is graded isomorphic to \(I\). To give an idea of the new construction consider an admissible pair \((H, S)\). Keeping the standard definitions of \(F_1(H, S)\) and \(F_2(H, S)\), for each \(e\in (F_1 (H, S) \cup F_2 (H, S))\cap E^1\), let \(w^e\) be a new vertex and let \(f^e\) be a new edge such that \(s( f^e ) = w^e\) and \(r( f^e ) = r(e)\). Continue this process inductively as follows. For each path \(p = eq\), where \(q\in F_1 (H, S) \cup F_2 (H, S)\) and \(\vert q\vert\ge 1\), add a new vertex \(w^p\) and a new edge \(f^p\) such that \(s( f^p ) = w^p\) and \(r( f^p ) = w^q\). The porcupine graph \(P_{(H,S)}\) is defined as follows. The set of vertices of \(P_{(H,S)}\) is \(H\cup S\cup \{w^p\colon p \in F_1 (H, S)\cup F_2 (H, S)\}\). The set of edges of \(P_{(H,S)}\) is \[\{e \in E^1\colon s(e)\in H\} \cup \{e \in E^1\colon s(e)\in S, r(e) \in H\} \cup \{ f^p\colon p\in F_1 (H, S)\cup F_2 (H, S)\}.\] The \(s\) and \(r\) maps of \(P_{(H,S)}\) are the same as in \(E\) for the common edges and they are defined as above for the new edges. The proof that the new construction gives a graded isomorphism can be adapted to show that, for every closed gauge-invariant ideal \(J\) of a graph \(C^*\)-algebra, there is a gauge-invariant \(*\)-isomorphism mapping the graph \(C^*\)-algebra of the porcupine graph of \(J\) onto \(J\).
Reviewer: Candido Martín González (Málaga)Characterizations of certain Morita context ringshttps://zbmath.org/1496.160322022-11-17T18:59:28.764376Z"Chimal-Dzul, Henry"https://zbmath.org/authors/?q=ai:chimal-dzul.henry"López-Permouth, Sergio"https://zbmath.org/authors/?q=ai:lopez-permouth.sergio-r"Szabo, Steve"https://zbmath.org/authors/?q=ai:szabo.steveSummary: Characterizations of Morita context rings satisfying various additional ring properties, including reflexive, semiprime, prime, NI, weakly 2-primal and 2-primal, are given. The connections between the characterizations of reflexive, semiprime and prime Morita context rings are outlined and shown to be related to torsionless and faithful modules. Similarly, the characterizations of NI, weakly 2-primal and 2-primal are shown to be related. In particular, a characterization of NI Morita context rings is determined to be equivalent to Köthe's conjecture.The exponential map for Hopf algebrashttps://zbmath.org/1496.160332022-11-17T18:59:28.764376Z"Alhamzi, Ghaliah"https://zbmath.org/authors/?q=ai:alhamzi.ghaliah"Beggs, Edwin"https://zbmath.org/authors/?q=ai:beggs.edwin-jAuthors' abstract: We give an analogue of the classical exponential map on Lie groups for Hopf \(\star\)-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert \(C^\star\)-bimodule of \(\frac{1}{2}\) densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups \(S_3\) and \({\mathbb Z}\), Woronowicz's matrix quantum group \({\mathbb C}_q[SU_2]\) and the Sweedler-Taft algebra.
Reviewer: Salih Çelik (İstanbul)Twisted pre-Lie algebras of finite topological spaceshttps://zbmath.org/1496.160342022-11-17T18:59:28.764376Z"Ayadi, Mohamed"https://zbmath.org/authors/?q=ai:ayadi.mohamed-aliAs shown by Foissy, Malvenuto and Patras [\textit{L. Foissy} et al., J. Pure Appl. Algebra 220, No. 6, 2434--2458 (2016; Zbl 1390.16032)], finite topologies (or pre-orders) are organized into a combinatorial Hopf algebra, which comes from a bialgebra in the category of species by the application of the bosonic Fock functor. In this paper, more algebraic structures are defined on this species. Two pre-Lie products are firstly defined, and it is proved that they are related by a natural involution on finite topologies. The associated Oudom-Guin products are then described, as well as the dual coproducts (which are different from the one described by Foissy, Malvenuto and Patras).
Reviewer: Loïc Foissy (Calais)Symmetries and the \(u\)-condition in weak monoidal Hom-Yetter-Drinfeld categorieshttps://zbmath.org/1496.160352022-11-17T18:59:28.764376Z"Liu, Linlin"https://zbmath.org/authors/?q=ai:liu.linlin"Wang, Shuanhong"https://zbmath.org/authors/?q=ai:wang.shuanhongHopf orders in \(K [ C_p^3]\) in characteristic \(p\)https://zbmath.org/1496.160362022-11-17T18:59:28.764376Z"Underwood, Robert"https://zbmath.org/authors/?q=ai:underwood.robert-gLet \(K\) be a complete discretely valued field of characteristic \(p>0\) with valuation \(\nu: K\to K\cup \{\infty\}\). Let \(R\) be the valuation ring of \(K\). Given a \(K\)-Hopf algebra \(A\), an \(R\)-Hopf order of \(A\) is a finitely generated \(R\) sub-Hopf algebra \(H\) of \(A\) such that \(K\otimes_R H\cong A\) as \(K\)-Hopf algebras. This work is a classification of the \(R\)-Hopf orders in the group algebra \(K[C_p^3]\), where \(C_p^3=\langle g_1,g_2,g_3\rangle\) is the elementary abelian group of order \(p^3\). While it is possible to get the Hopf orders in any finite abelian \(p\)-group algebra indirectly (see [\textit{A. Koch}, Commun. Algebra 45, No. 6, 2673--2689 (2017; Zbl 1393.16022)]), the classification given here is direct, explicit, and is a natural extension of the author's previous work.
The power of this paper is the elegance in which the author, after extensive technical computations, arises at a classification result which is as clear and concise as one could reasonably hope for. Each \(R\)-Hopf order in \(KC_p^3\) depends on six parameters: \(i_1,i_2,i_3,\mu,\alpha,\beta\) where \(i_1,i_2,i_3\) are nonnegative integers, \(\nu(\mu^p-\mu)\ge i_2-pi_1,\;\mu(\beta^p-\beta)\ge i_3=pi_2\), and \(\nu(\alpha^p-\alpha+\beta\mu^p-\beta\mu)\ge i_3-pi_1\). A given set of such parameters gives rise to the \(R\)-Hopf order
\[
H=R\left[ \frac{g_1-1}{\pi^{i_1}},\frac{g_2g_1^{[\mu]}-1}{\pi^{i_2}},\frac{g_3g_1^{[\alpha]}(g_2g_1^{[\mu]})^{[\beta]}-1}{\pi^{i_3}} \right],\;x^{[y]}=\sum_{m=0}^{p-1}\binom ym (x-1)^m.
\]
This \(R\)-Hopf order is constructed by considering extensions of \(R\)-Hopf orders \(R\to C \to H \to D \to R\) where \(C\) and \(D\) are \(R\)-Hopf orders in \(KC_p^2\), constructed previously in [\textit{G. G. Elder} and \textit{R. G. Underwood}, New York J. Math. 23, 11--39 (2017; Zbl 1384.16020)]; and \(KC_p\), a classic result of [\textit{J. Tate} and \textit{F. Oort}, Ann. Sci. Éc. Norm. Supér. (4) 3, 1--21 (1970; Zbl 0195.50801)] respectively. Both of these previous works are also described thoroughly in this work.
Reviewer: Alan Koch (Decatur)A graph with respect to idempotents of a ringhttps://zbmath.org/1496.160372022-11-17T18:59:28.764376Z"Razaghi, Somayyeh"https://zbmath.org/authors/?q=ai:razaghi.somayyeh"Sahebi, Shervin"https://zbmath.org/authors/?q=ai:sahebi.shervinLet \(R\) be a ring and let \(\mathrm{Id}(R)\) be the set of idempotent elements of \(R\). The \textit{idempotent graph} of \(R\), denoted \(GI_d(R)\), is a graph obtained by setting all elements of \(R\) to be the vertices and defining distinct \(x\) and \(y\) to be adjacent if, and only if, \(x + y\in \mathrm{Id}(R)\).
In the paper under review, the authors study in Section 2 some elementary properties of this graph (see, for instance, Proposition 2.2, Corollary 2.3 and Remark 2.4). In Section 3, they give a survey of the connectivity of idempotent graphs. Also, they focus on some properties of idempotent graphs such as diameter and girth (see for more details, Theorems 3.1,3.2,3.4,3.6,3.7, 3.10, 3.11 as well as their consequences).
Reviewer: Peter Danchev (Sofia)A symmetrization in \(\pi\)-regular ringshttps://zbmath.org/1496.160382022-11-17T18:59:28.764376Z"Danchev, Peter V."https://zbmath.org/authors/?q=ai:danchev.peter-vassilevThis author introduces a class of rings called \((m, n)\)-regularly nil clean rings and shows that these rings are a generalization of the \(\pi\)-regular rings.
Reviewer: Ashish K. Srivastava (Saint Louis)On \(n^{th}\) power *-property in *-rings with applicationshttps://zbmath.org/1496.160392022-11-17T18:59:28.764376Z"Jeelani, Mohd"https://zbmath.org/authors/?q=ai:jeelani.mohd"Alhazmi, Husain"https://zbmath.org/authors/?q=ai:alhazmi.husain"Singh, Kishan Pal"https://zbmath.org/authors/?q=ai:singh.kishan-palThe main idea of this paper came via \textit{N. A. Dar} and \textit{S. Ali}'s elegant work [Commun. Algebra 49, No. 4, 1422--1430 (2021; Zbl 1478.16012)] The results in this paper provide an affirmative answer to the open Problem 3.3 [loc. cit.]. Hence, the target of this article is to study the \(n\)th power \(n^{\text{th}}\)-property in rings with involution and its related problems.
More precisely, the main result of this paper gives conditions which ensure that an additive map \(d : R \to R\) satisfying \(n^{\text{th}}\) power \(\ast\)-property must be X-inner. As an application, the authors characterize the additive map \(F: R \to R\) such that \(F(x^{n})= F(x)(x^{\ast})^{n-1}+\sum_{i=1}^{n-1}x^{i}d(x)(x^{\ast})^{n-i-1}\) for all \(x \in R\) where \(d : R \to R\) is an additive map satisfying \(n^{\text{th}}\) power \(\ast\)-property.
They begin their discussion with the following result.
Theorem 2.1. Let \(n>1\) be a fixed integer, \(R\) be an \((n-1)!\)-torsion free \(\ast\)-ring and \(d: R \to R\) be an additive mapping. If d is satisfying \(d(x^{n})= d(x)(x^{\ast})^{n-1}+\sum_{i=1}^{n-1}x^{i}d(x)(x^{\ast})^{n-i-1}\) for all \(x \in R\), then d is a Jordan \(\ast\)-derivation of \(R\).
The authors close their work with some applications of their results which are proved in another section.
Reviewer: Mehsin Atteya (Leicester)Products and intersections of prime-power ideals in Leavitt path algebrashttps://zbmath.org/1496.160402022-11-17T18:59:28.764376Z"Mesyan, Zachary"https://zbmath.org/authors/?q=ai:mesyan.zachary"Rangaswamy, Kulumani M."https://zbmath.org/authors/?q=ai:rangaswamy.kulumani-mAs mentioned in the work under review, the ideal theory of the Leavitt path algebra \(L = L_K(E)\) of a directed graph \(E\) over a field \(K\) has been an active area of research in recent years. Some of the goals that have been achieved in the last years include the characterization of special types of ideals of \(L\) in terms of graphical properties of \(E\), and the description of the ideals of \(L\) that can be factored into products of ideals of these types. Thus, prime, primitive, semiprime, and irreducible ideals have received such treatment in the literature. The paper continues along this path and can be framed within the multiplicative ideal theory of an arbitrary Leavitt path algebra \(L\). It is shown that factorizations of an ideal in \(L\) into irredundant products or intersections of finitely many prime-power ideals are unique, provided that the ideals involved are powers of distinct prime ideals. They are also characterized the completely irreducible ideals in \(L\), which turn out to be prime-power ideals of a special type, as well as ideals that can be factored into products or intersections of finitely many completely irreducible ideals.
Reviewer: Candido Martín González (Málaga)Identities with generalized derivations in prime ringshttps://zbmath.org/1496.160412022-11-17T18:59:28.764376Z"Tiwari, S. K."https://zbmath.org/authors/?q=ai:tiwari.sanjay-kumar|tiwari.surendra-kumar|tiwari.shailendra-kumar|tiwari.satish-kumar|tiwari.shiv-kant|tiwari.sandeep-k|tiwari.shiv-kumar|tiwari.sarvesh-k|tiwari.sharwan-kumar|tiwari.shailesh-kumar|tiwari.sharad-kumarMany authors characterize the structure of a prime ring \(R\) studying particular subsets of \(R\), involving the evaluations of some additive maps defined on \(R\). In this paper, the author considers the set \[P(F,G,H,S)=\biggl\{F(G(x)x)-H(x^2):x\in S\biggr\},\] where \(F,G\) and \(H\) are additive maps defined on \(R\) and \(S\subseteq R\). More precisely, he considers the case in which \(F,G\) and \(H\) are generalized derivations of \(R\) and \(S=f(R)\), the set of all evaluations of a non-central multilinear polynomial over \(C\) (the extended centroid of \(R\)):\\
If \(\operatorname{char}(R)\neq2\) and \(P(F,G,H,f(R))=\{0\}\), then one of the following holds:
\begin{enumerate}
\item \(F=H=0\);
\item there exist \(a,b\in U\) such that \(F(x)=ax\), \(G(x)=bx\) and \(H(x)=abx\), for all \(x\in R\);
\item there exist \(a,b\in U\), \(\lambda\in C\) such that \(F(x)=ax+xb\), \(G(x)=\lambda x\) and \(H(x)=\lambda(ax+xb)\), for all \(x\in R\);
\item \(f(x_1,\ldots,x_n)\) is central-valued on \(R\) and
\begin{enumerate}
\item[a)] either there exist \(a,b,c,p\in U\) such that \(F(x)=ax+xb\), \(G(x)=c x\) and \(H(x)=[p,x]+x(ac+cb)\), for all \(x\in R\);
\item[b)] or there exist \(a\in U\) such that \(F=0\) and \(H(x)=[a,x]\), for all \(x\in R\).
\end{enumerate}
\end{enumerate}
Reviewer: Giovanni Scudo (Messina)Identities involving generalized derivations in prime ringshttps://zbmath.org/1496.160422022-11-17T18:59:28.764376Z"Yadav, V. K."https://zbmath.org/authors/?q=ai:yadav.vinod-kumar|yadav.vijay-kumar|yadav.vineet-k|yadav.vishal-krLet \(R\) be a prime ring with its Utumi quotient ring \(U\), the extended centroid \(C\), and \(Z(R)\) the center of \(R\). It may be noted that the extended centroid \(C\) of a prime ring \(R\) is always a field and \(C=Z(U)\). An additive mapping \(d:R\to R\) is said to be a derivation of \(R\) if \(d(xy)=d(x)y+xd(y)\) holds for all \(x,y \in R\). An additive mapping \(F:R \to R\) is called a generalized derivation of \(R\) if there exists a derivation \(d:R \to R\) such that \(F(xy)=F(x)y+xd(y)\) holds for all \(x,y \in R\). A polynomial \(f = f (x_1, \cdots, x_n) \in \mathbb{Z} \langle X \rangle\) is said to be multilinear if it is linear in every \(x_i\), \(1 \leq i \leq n\), where \(\mathbb{Z}\) is the set of integers.
In the paper under review, the author studied the identity \(G^2(u)d(u) = 0\), for all \(u \in f(R) = \{f(r_1, r_2, \cdots, r_n)| r_i \in R \}\), where \(G\) is a generalized derivation and \(d\) is a non zero derivation on prime ring \(R\) of characteristic different from \(2\). More precisely, the author proved the following:
Theorem. Let \(R\) be a prime ring of characteristic different from \(2\) with Utumi quotient ring \(U\) and extended centroid \(C\), \(f (x_1, \cdots, x_n)\) be a multilinear polynomial over \(C\), which is not central valued on \(R\). Suppose that \(d\) is a nonzero derivation of \(R\) and \(G\) is a generalized derivation of \(R\). If \(G^2(u)d(u) = 0\) for all \(u \in f(R)\), then one of the following holds:
\begin{enumerate}
\item[\((i)\)] there exists \(a \in U\) such that \(G(x) = ax\) for all \(x \in R\) with \(a^2 = 0\),
\item[\((ii)\)] there exists \(a \in U\) such that \(G(x) = xa\) for all \(x \in R\) with \(a^2 = 0\).
\end{enumerate}
Reviewer: Nadeem ur Rehman (Aligarh)Nonlinear Jordan derivations of incidence algebrashttps://zbmath.org/1496.160432022-11-17T18:59:28.764376Z"Yang, Yuping"https://zbmath.org/authors/?q=ai:yang.yupingSummary: Let \((X, \leq)\) be a locally finite preordered set, \(\mathcal{R}\) a two-torsion-free commutative ring with unity and \(I(X,\mathcal{R})\) the incidence algebra of \(X\) over \(\mathcal{R}\) this paper, all the nonlinear Jordan derivations of \(I(X,\mathcal{R})\) are determined. In particular, if all the connected components of \(X\) are nontrivial, we prove that every nonlinear Jordan derivation of \(I(X,\mathcal{R})\) is proper and can be presented as a sum of an inner derivation, a transitive induced derivation, and an additive induced derivation.Triangulating dimension of skew generalized power series ringshttps://zbmath.org/1496.160442022-11-17T18:59:28.764376Z"Paykan, Kamal"https://zbmath.org/authors/?q=ai:paykan.kamalGiven a ring \(R\), a strictly ordered monoid \(S\), and a monoid homomorphism \(\omega: S \to \text{End}(R)\), the author considers the ring of skew generalized power series \(R[[S,\omega]],\) which consists of all formal series of the form \(\sum_{s \in S} r_s s\), where \(r_s = f(s) \in R\). An idempotent \(e\) is called left semicentral idempotent if \(ere=re\) for all \(r \in R\), and the set of all left semicentral idempotents is denoted \(S_\ell(R)\). A ring \(R\) is said to have a set of left triangulating idempotents if there exists an ordered set \(\{b_1, b_2, \dots, b_n\}\) of nonzero, distinct idempotents such that (i) \(1=b_1+b_2+\cdots+b_n\); (ii) \(b_1 \in S_\ell(R)\); and (iii) \(b_k+1 \in S_\ell(c_kRc_k)\) where \(c_k = 1 - (b_1 + b_2 + \cdots + b_k )\) for \(1\le k \le n-1\). An idempotent \(e\) of \(R\) is semicentral reduced if \(S_\ell(eRe) = \{0,e\}\). A ring \(R\) is semicentral reduced, if 1 is semicentral reduced. A set \(\{b_1, b_2, \dots, b_n\}\) of left triangulating idempotents is said to be complete if each \(b_iRb_i\) is semicentral reduced. In general, a ring \(R\) has triangulating dimension \(n\), if \(R\) has a complete set of left triangulating idempotents with exactly \(n\) elements.
The author investigates when the triangulating dimension of \(R\) and \(R[[S,\omega]]\) coincide. In addition, for a piecewise prime ring, the author determines a class of the skew generalized power series rings that have a generalized triangular matrix representation for which the diagonal rings are prime.
Reviewer: Edward Mosteig (Los Angeles)Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systemshttps://zbmath.org/1496.160452022-11-17T18:59:28.764376Z"Fairon, Maxime"https://zbmath.org/authors/?q=ai:fairon.maximeThis paper is a contribution to the theory of non-commutative Poisson structures, with applications to integrable systems. The main theoretical results concern double (quasi)Poisson structures, as introduced in [\textit{M. Van den Bergh}, Trans. Am. Math. Soc. 360, No. 11, 5711--5769 (2008; Zbl 1157.53046)].
Fusion for algebras generalizes fusion for quiver algebras corresponding to identification of vertices; Van den Bergh established that the behaviour of non-commutative Poisson structures under fusion is of significant interest.
For double Poisson structures, the author first proves that iterated fusions are independent of the choices involved and likewise for Hamiltonian algebras (i.e., in presence of a moment map).
The double quasi-Poisson case is much more delicate, since the passage of such a structure to the fusion algebra involves a correction fusion term, analysed in special cases by Van den Bergh and in full generality in [\textit{M. Fairon}, Algebr. Represent. Theory 24, No. 4, 911--958 (2021; Zbl 1480.16049)].
The main algebraic result (announced in [Zbl 1480.16049]) is a double quasi-Poisson analogue of the above, together with a version for quasi-Hamiltonian algebras (i.e., in presence of a multiplicative moment map).
The results are illustrated by examples constructed from quivers (following Van den Bergh) which give, respectively, a Hamiltonian double Poisson structure and a quasi-Hamiltonian structure. The above theorems imply that, up to isomorphism, these structures only depend upon the underlying graph of the quiver.
These results are applied to give a very conceptual explanation of action-angle duality for several examples of classical integrable systems, notably generalizations of the Calogero-Moser system and of the Ruijsenaars-Schneider systems. The associated phase spaces are constructed as quiver varieties, with Poisson structure induced by a NC-Poisson structure in the sense of [\textit{W. Crawley-Boevey}, J. Algebra 325, No. 1, 205--215 (2011; Zbl 1255.17012)]. The author exhibits action-angle coordinates, building upon [\textit{O. Chalykh} and \textit{M. Fairon}, J. Geom. Phys. 121, 413--437 (2017; Zbl 1418.70026)] and [\textit{O. Chalykh} and \textit{A. Silantyev}, J. Math. Phys. 58, No. 7, 071702, 31 p. (2017; Zbl 1370.37126)]. The action-angle duality corresponds to the reversal of arrows of the quiver.
Reviewer: Geoffrey Powell (Angers)Characterization of bipolar soft ideal in near ringshttps://zbmath.org/1496.160462022-11-17T18:59:28.764376Z"Çitak, Filiz"https://zbmath.org/authors/?q=ai:citak.filiz(no abstract)Fuzzy homomorphism theorems on ringshttps://zbmath.org/1496.160472022-11-17T18:59:28.764376Z"Addis, Gezahagne Mulat"https://zbmath.org/authors/?q=ai:addis.gezahagne-mulat"Kausar, Nasreen"https://zbmath.org/authors/?q=ai:kausar.nasreen"Munir, Mohammad"https://zbmath.org/authors/?q=ai:munir.mohammad(no abstract)New fundamental relation and complete part of fuzzy hypermoduleshttps://zbmath.org/1496.160482022-11-17T18:59:28.764376Z"Firouzkouhi, N."https://zbmath.org/authors/?q=ai:firouzkouhi.narjes"Davvaz, B."https://zbmath.org/authors/?q=ai:davvaz.bijan(no abstract)Hypergeometry, integrability and Lie theory. Virtual conference, Lorentz Center, Leiden, the Netherlands, December 7--11, 2020https://zbmath.org/1496.170012022-11-17T18:59:28.764376ZPublisher's description: This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7--11, 2020, which was dedicated to the 50th birthday of Jasper Stokman.
The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Etingof, Pavel; Kazhdan, David}, Characteristic functions of \(p\)-adic integral operators, 1-27 [Zbl 07602313]
\textit{Garbali, Alexandr; Zinn-Justin, Paul}, Shuffle algebras, lattice paths and the commuting scheme, 29-68 [Zbl 07602314]
\textit{Kolb, Stefan}, The bar involution for quantum symmetric pairs -- hidden in plain sight, 69-77 [Zbl 07602315]
\textit{Koornwinder, Tom H.}, Charting the \(q\)-Askey scheme, 79-94 [Zbl 07602316]
\textit{Rains, Eric M.}, Filtered deformations of elliptic algebras, 95-154 [Zbl 07602317]
\textit{Regelskis, Vidas; Vlaar, Bart}, Pseudo-symmetric pairs for Kac-Moody algebras, 155-203 [Zbl 07602318]
\textit{Reshetikhin, N.; Stokman, J. V.}, Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains, 205-241 [Zbl 07602319]
\textit{Rösler, Margit; Voit, Michael}, Elementary symmetric polynomials and martingales for Heckman-Opdam processes, 243-262 [Zbl 07602320]
\textit{Schomerus, Volker}, Conformal hypergeometry and integrability, 263-285 [Zbl 07602321]
\textit{Varchenko, Alexander}, Determinant of \(\mathbb{F}_p\)-hypergeometric solutions under ample reduction, 287-307 [Zbl 07602322]
\textit{Varchenko, Alexander}, Notes on solutions of KZ equations modulo \(p^s\) and \(p\)-adic limit \(s\to\infty\), 309-347 [Zbl 07602323]Homogeneous bases for Demazure moduleshttps://zbmath.org/1496.170052022-11-17T18:59:28.764376Z"Kambaso, Kunda"https://zbmath.org/authors/?q=ai:kambaso.kundaThis paper studies the PBW filtration on various classes of Demazure modules over classical simple Lie algebras. The main results include a construction of a normal polytope labeling a basis for the associated graded vector space of The Demazure module with respect to the PBW filtration and a proof that the annihilating ideal of this associated graded module is a monomial ideal.
Reviewer: Volodymyr Mazorchuk (Uppsala)Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebrashttps://zbmath.org/1496.170082022-11-17T18:59:28.764376Z"Losev, Ivan"https://zbmath.org/authors/?q=ai:losev.ivan-v"Panin, Ivan"https://zbmath.org/authors/?q=ai:panin.ivanLet \(\mathfrak g\) be a semisimple Lie algebra. The authors establish a new relation between the Goldie rank of a primitive ideal \(J \subset U(\mathfrak g )\) and the dimension of the corresponding irreducible representation \(V\) of an appropriate finite \(W\)-algebra, see on \(W\)-algebras [\textit{A. Premet}, Adv. Math. 170, No. 1, 1--55 (2002; Zbl 1005.17007); \textit{I. Losev}, J. Am. Math. Soc. 23, No. 1, 35--59 (2010; Zbl 1246.17015)].
Namely, they show that \(\mathrm{Grk}(J )\le \dim V/d_V\), where \(d_V\) is the index of a suitable equivariant Azumaya algebra on a homogeneous space. They also compute \(d_V\) in representation theoretic terms.
Reviewer: Victor Petrogradsky (Brasília)On quantum toroidal algebra of type \(A_1\)https://zbmath.org/1496.170092022-11-17T18:59:28.764376Z"Chen, Fulin"https://zbmath.org/authors/?q=ai:chen.fulin"Jing, Naihuan"https://zbmath.org/authors/?q=ai:jing.naihuan"Kong, Fei"https://zbmath.org/authors/?q=ai:kong.fei"Tan, Shaobin"https://zbmath.org/authors/?q=ai:tan.shao-binLet \(\dot{\mathfrak{g}}\) be a finite dimensional simple Lie algebra of type \(A_1\) over \(\mathbb C\). The main result of the paper under review is the construction of a middle quantum algebra \(\mathcal U\) between the quantum affinization algebra \(\mathcal U_{\hbar}(\hat{\mathfrak{g}})\) introduced by \textit{N. Jing} [Lett. Math. Phys. 44, No. 4, 261--271 (1998; Zbl 0911.17006)] and the quantum toroidal algebra \(\mathcal U_{\hbar}(\dot{\mathfrak{g}}_{tor})\) introduced by \textit{V. Ginzburg} et al. [Math. Res. Lett. 2, No. 2, 147--160 (1995; Zbl 0914.11040)].
It is shown that \(\mathcal U\) admits a triangular decomposition and it has a deformed Drinfeld coproduct, which allows to define a (topological) Hopf algebra structure on \(\mathcal U\). A vertex representation for \(\mathcal U\) is also obtained.
Reviewer: Sonia Natale (Córdoba)The surjectivity of the evaluation map of the affine super Yangianhttps://zbmath.org/1496.170122022-11-17T18:59:28.764376Z"Ueda, Mamoru"https://zbmath.org/authors/?q=ai:ueda.mamoruIn his previous work [``Affine super Yangian'', Preprint, \url{arXiv:1911.06666}], the author constructed a homomorphism from the affine super Yangian to the completion of the universal enveloping algebra of \(\widehat{\mathfrak{gl}}(m|n)\). In the paper under review he shows that the image of this homomorphism is dense in the completion of \(U(\widehat{\mathfrak{gl}}(m|n))\), which allows him to obtain irreducible representations of the affine super Yangian.
Reviewer: Sonia Natale (Córdoba)A quantum Mirković-Vybornov isomorphismhttps://zbmath.org/1496.170132022-11-17T18:59:28.764376Z"Webster, Ben"https://zbmath.org/authors/?q=ai:webster.ben"Weekes, Alex"https://zbmath.org/authors/?q=ai:weekes.alex"Yacobi, Oded"https://zbmath.org/authors/?q=ai:yacobi.odedMirković and Vybornov construct an isomorphism between slices to (spherical) Schubert varieties in the affine Grassmannian of \(\mathrm{PGL}_n\) on the one hand, and Slodowy slices in \(\mathfrak{gl}_N\) intersected with nilpotent orbit closures on the other. This isomorphism has important applications, such as it appears in works on the mathematical definition of the Coulomb branch associated to quiver gauge theories, the analog of the geometric Satake isomorphism for affine Kac-Moody groups, and geometric approaches to knot homologies. These varieties have quantizations corresponding to natural Poisson structures on them. The goal of this paper is to show that the Mirković-Vybornov isomorphism is the classical limit of an isomorphism of these quantizations.
B. Webster, A. Weekes, and O. Yacobi present a quantization of an isomorphism of Mirković and Vybornov, which relates the intersection of a Slodowy slice and a nilpotent orbit closure in \(\mathfrak{gl}_N\), to a slice between spherical Schubert varieties in the affine Grassmannian of \(\mathrm{PGL}_n\) (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic \(W\)-algebra and of the latter by a truncated shifted Yangian (Theorem 4.3 (c), page 66). The authors also define an explicit isomorphism between these noncommutative algebras, and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov (Theorem 4.3 (d), page 66). As a corollary, they deduce that the \(W\)-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra.
Reviewer: Mee Seong Im (Annapolis)Typical irreducible characters of generalized quantum groupshttps://zbmath.org/1496.170142022-11-17T18:59:28.764376Z"Yamane, Hiroyuki"https://zbmath.org/authors/?q=ai:yamane.hiroyukiIn the paper under review the author provides a counterpart, for a generalized quantum group over a field of any characteristic, of a result of \textit{V. Kac} [Lect. Notes Math. 676, 597--626 (1978; Zbl 0388.17002); Commun. Algebra 5, 889--897 (1977; Zbl 0359.17010)] on a Weyl-type character formula for the typical finite-dimensional irreducible modules of a basic classical Lie superalgebra. As a by-product, he obtains a Weyl-Kac-type character formula of the typical irreducible modules of the quantum superalgebras associated with the basic classical Lie superalgebras.
Reviewer: Sonia Natale (Córdoba)Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebrashttps://zbmath.org/1496.170152022-11-17T18:59:28.764376Z"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Guo, Li"https://zbmath.org/authors/?q=ai:guo.li"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.5|zhang.yi.2|zhang.yi.14|zhang.yi.3|zhang.yi|zhang.yi.1|zhang.yi.10|zhang.yi.6|zhang.yi.8|zhang.yi.4|zhang.yi.12Fix a unitary commutative associative ring \(\mathbf{k}\). A Rota-Baxter algebra is a commutative associative algebra \(R\) over \(\mathbf{k}\) together with a \(\mathbf{k}\)-linear operator \(P: R \longrightarrow R\) satisfying the so-called \textit{Rota-Baxter} identity for \(f, g\) in \(R\):
\[
P(f)P(g) = P(fP(g)) + P(P(f)g).
\]
This is a special case of a more general definition in the paper. We make this simplification in the review because the main results of the paper deal with the special case. As a pioneering example, the ring \(\mathrm{Cont}(\mathbb{R})\) of continuous functions on \(\mathbb{R}\) is a Rota-Baxter algebra over \(\mathbb{R}\), with the operator \(P\) defined by the Riemann integral for \(f \in \mathrm{Cont}(\mathbb{R})\) and \(x \in \mathbb{R}\):
\[
(P(f))(x) := \int_0^x f(t)\, dt.
\]
Let \(\Omega\) be an index set. A matching Rota-Baxter algebra with respect to \(\Omega\) is a commutative associative algebra \(R\) together with a family of linear operators \(P_{\alpha}: R \longrightarrow R\) indexed by \(\alpha \in \Omega\) such that for \(x, y\) in \(R\) and \(\alpha, \beta\) in \(\Omega\) we have:
\[
P_{\alpha}(x) P_{\beta}(y) = P_{\alpha}(xP_{\beta}(y)) + P_{\beta}(P_{\alpha}(x)y).
\]
Each pair \((R, P_{\alpha})\) for a fixed \(\alpha\) forms an ordinary Rota-Baxter algebra. If \((R, P)\) is a Rota-Baxter algebra and \((g_{\alpha})_{\alpha \in \Omega}\) is a family of elements of \(R\), then we can equip \(R\) with a structure of matching Rota-Baxter algebra by setting \(P_{\alpha}(f) := P(g_{\alpha} f)\) for \(\alpha \in \Omega\) and \(f \in R\).
Let \(\mathbf{CAlg}_{\mathbf{k}}\) denote the category of commutative associative algebras over \(\mathbf{k}\) and \(\mathbf{MRBA}_{\mathbf{k},\Omega}\) denote the category of matching Rota-Baxter algebras. By definition we have a forgetful functor \(\mathcal{F}: \mathbf{MRBA}_{\mathbf{k},\Omega} \longrightarrow \mathbf{CAlg}_{\mathbf{k}}\).
One of the main results of this paper is an explicit construction, via \textit{shuffle product}, of a functor \(\mathcal{G}: \mathbf{CAlg}_{\mathbf{k}} \longrightarrow \mathbf{MRBA}_{\mathbf{k},\Omega}\) left adjoint to \(\mathcal{F}\). In more details, for \(A\) a commutative associative algebra, \(\mathcal{G}(A)\) is the tensor product algebra of \(A\) with the shuffle algebra associated to the \(\mathbf{k}\)-module \(\mathbf{k}\Omega \otimes A\), which is naturally a matching Rota-Baxter algebra. The authors also extend this result to the relative setting. Fix an object \(X\) of \(\mathbf{MRBA}_{\mathbf{k},\Omega}\) and let \(\mathcal{C}_X\) denote the category of morphisms \(X \longrightarrow Y\) in \(\mathbf{MRBA}_{\mathbf{k},\Omega}\). Then the forgetful functor \(\mathcal{C}_X \longrightarrow \mathbf{CAlg}_{\mathbf{k}}\) sending a morphism \(X \longrightarrow Y\) to \(\mathcal{F}(Y)\) is shown to admit an explicit left adjoint.
Reviewer: Huafeng Zhang (Villeneuve d'Ascq)An introduction to pre-Lie algebrashttps://zbmath.org/1496.170232022-11-17T18:59:28.764376Z"Bai, Chengming"https://zbmath.org/authors/?q=ai:bai.chengmingAn pre-Lie algebra is a vector space with a binary operation \((x,y)\mapsto xy\) satisfying
\[
(xy)z-x(yz)=(yx)z-y(xz)
\]
The examples of pre-Lie algebras are left-invariant affine structures on Lie groups, deformation complexes of algebras and right-symmetric algebras, rooted tree algebras (which are free pre-Lie algebra), also pre-Lie algebras are naturally related to symplectic Lie algebras and vertex algebras.
In the paper under review these motivating examples are considered in details. In Section 2, we will introduce some background and the different motivation of introducing the notion of pre-Lie algebra. Then the basic properties and construction are explained. A relation between pre-Lie algebras and classical Yang-Baxter equation is explained. Finally the authors puts ``pre-Lie algebras into a bigger framework as one of the algebraic structures of the Lie analogues of Loday algebras. There is an operadic interpretation of these algebraic structures which is related to Manin black products.''
For the entire collection see [Zbl 1492.17001].
Reviewer: Dmitry Artamonov (Moskva)The algebra of \(2 \times 2\) upper triangular matrices as a commutative algebra: gradings, graded polynomial identities and Specht propertyhttps://zbmath.org/1496.170242022-11-17T18:59:28.764376Z"Morais, Pedro"https://zbmath.org/authors/?q=ai:morais.pedro"da Silva Souza, Manuela"https://zbmath.org/authors/?q=ai:souza.manuela-da-silvaLet \(K\) be a field and let \(UT_2(K)\) be the vector space of the \(2\times 2\) upper triangular matrices over \(K\). The authors consider the new multiplication \(a\circ b=ab+ba\) on \(UT_2(K)\) in the case when \(K\) is infinite and of characteristic 2. Thus \(UT_2(K)\) becomes a Lie algebra. The authors describe all group gradings on this algebra. Furthermore they deduce the graded identities satisfied by each one of the gradings. Moreover they prove that the corresponding ideals of graded identities satisfy the Specht property. The authors also obtain that there are non-isomorphic gradings that satisfy the same graded identities. Recall that in [\textit{P. Koshlukov} and \textit{F. Yukihide Yasumura}, Linear Algebra Appl. 534, 1--12 (2017; Zbl 1416.17023)] it was proved that in characteristic 0, a grading on the Jordan algebra \(UT_n(K)\) is uniquely determined by the graded identities it satisfies. For the associative algebra \(UT_n(K)\) the analogous result was obtained in [\textit{O. M. Di Vincenzo} et al., J. Algebra 275, No. 2, 550--566 (2004; Zbl 1066.16047)].
Reviewer: Plamen Koshlukov (Campinas)Morita theorem for hereditary Calabi-Yau categorieshttps://zbmath.org/1496.180152022-11-17T18:59:28.764376Z"Hanihara, Norihiro"https://zbmath.org/authors/?q=ai:hanihara.norihiroThe paper characterizes the structure of Calabi-Yau triangulated category with a hereditary cluster tilting object. Recall that a triangulated category \(\mathcal T\) is \(d\)-Calabi-Yau if it has finite dimensional morphism spaces over a field \(k\), and satisfies the functorial isomorphism \(\mathcal T(A,B)\cong \Hom_k(\mathcal T(B,A[d]),k)\) for all \(A,B\in \mathcal T\). An example of \(d\)-Calabi-Yau category is the \(d\)-cluster category of a hereditary algebra \(H\), which is the orbit category of derived category \(D^b(H)\) under the functor \(\tau^{-1}[d-1]\). The cluster category has a \(d\)-cluster tilting object.
Morita theory suggests that module categories of two rings are equivalent if and only there exists progengentors inducing the equivalence. The author proves a version of Morita theory for Calabi-Yau categories. The \(d\)-cluster tilting objects play the role of progenerators. More precisely, an algebraic \(d\)-Calabi-Yau category with a hereditary \(n\)-cluster tilting object \(T\) is equivalent the \(d\)-cluster category of the hereditary endomorphism algebra of \(T\oplus T[-1]\oplus \dots\oplus T[2-d]\).
As a result, Morita theorems of Keller-Reiten and Keller-Murfet-Van den Berg are followed by specified \(d\) and \(T\) [\textit{B. Keller} and \textit{I. Reiten}, Compos. Math. 144, No. 5, 1332--1348 (2008; Zbl 1171.18008); \textit{B. Keller} et al., Compos. Math. 147, No. 2, 591--612 (2011; Zbl 1264.13016)].
Reviewer: Zhe Han (Kaifeng)Cyclic Gerstenhaber-Schack cohomologyhttps://zbmath.org/1496.180212022-11-17T18:59:28.764376Z"Fiorenza, Domenico"https://zbmath.org/authors/?q=ai:fiorenza.domenico"Kowalzig, Niels"https://zbmath.org/authors/?q=ai:kowalzig.nielsWe know that deformations of an associative algebra is controlled by its Hochschild cohomology [\textit{M. Gerstenhaber}, Ann. Math. (2) 78, 267--288 (1963; Zbl 0131.27302)]. The same way, deformations of a bialgebra is controlled by Gerstenhaber-Schack cohomology [\textit{M. Gerstenhaber} and \textit{S. D. Schack}, Contemp. Math. 134, 51--92 (1992; Zbl 0788.17009)]. The main problem the authors tackle arises from the question whether Gerstenhaber-Schack cohomology carries a Gerstenhaber bracket analogous to the bracket structure on Hochschild cohomology. The authors show that when the underlying bialgebra is a Hopf algebra the diagonal of the Gerstenhaber-Schack complex carries an operad structure with multiplication (Theorem B). Since bisimplicial objects are homotopic to their diagonals, the Gerstenhaber-Schack cohomology carries a natural Gerstenhaber bracket due to [\textit{M. Gerstenhaber} and \textit{S. D. Schack}, Contemp. Math. 134, 51--92 (1992; Zbl 0788.17009); \textit{J. E. McClure} and \textit{J. H. Smith}, Contemp. Math. 293, 153--193 (2002; Zbl 1009.18009)]. Moreover, they also show that (Theorem C) when the antipode of the underlying Hopf algebra is involutive, or when the Hopf algebra has a modular pair in involution, then operad is cyclic, and therefore, the Gerstenhaber-Schack cohomology supports a Batalin-Vilkovisky algebra structure by [\textit{L. Menichi}, \(K\)-Theory 32, No. 3, 231--251 (2004; Zbl 1101.19003)]. Interestingly, the bracket is trivial when the Hopf algebra is finite dimensional (Theorem D), and therefore, Gerstenhaber-Schack cohomology has a \(e_3\)-algebra structure due to [\textit{D. Fiorenza} and \textit{N. Kowalzig}, Int. Math. Res. Not. 2020, No. 23, 9148--9209 (2020; Zbl 1468.55008)].
Reviewer: Atabey Kaygun (İstanbul)Resolutions of operads via Koszul (bi)algebrashttps://zbmath.org/1496.180222022-11-17T18:59:28.764376Z"Tamaroff, Pedro"https://zbmath.org/authors/?q=ai:tamaroff.pedroFor any bialgebra \(H\), the category of associative algebras in the category of \(H\)-modules is isomorphic to the category of algebras over an operad \(\mathrm{Ass}_H\) which is described in this paper. This operad is generally not quadratic. When \(H\) is Koszul (as an algebra), then a minimal model of \(\mathrm{Ass}_H\) is described, involving only the coproduct of \(H\) and the Koszul model of \(H\). It is also shown how to construct a Gröbner basis of \(\mathrm{Ass}_H\) from a Gröbner basis of \(H\). Links are made with homotopy theory when \(H\) is the mod-2 Steenrod algebra.
Reviewer: Loïc Foissy (Calais)Regularity of spectral stacks and discreteness of weight-heartshttps://zbmath.org/1496.180242022-11-17T18:59:28.764376Z"Sosnilo, Vladimir"https://zbmath.org/authors/?q=ai:sosnilo.vladimir-aThe main motivation of this paper is the study of bounded \(t\)-structures, after the key contributions of Antieau, Gepner and Heller who proved that obstructions to the existence of bounded \(t\)-structures on a stable \(\infty\)-category \(\mathscr{C}\) are controlled by the first negative \(K\)-group \(K_{-1}(\mathscr{C})\) [\textit{B. Antieau} et al., Invent. Math. 216, No. 1, 241--300 (2019; Zbl 1430.18009)]. This work tackles the problem assuming that \(\mathscr{C}\) is endowed with a \textit{weight structure}, a structure introduced in [\textit{M. V. Bondarko}, J. \(K\)-Theory 6, No. 3, 387--504 (2010; Zbl 1303.18019)], similar (but not dual) to a \(t\)-structure, which axiomatizes the properties of naive truncations of chain complexes. This notion is closely related to the concept of \textit{regularity} of \(\mathbb{E}_1\)-ring spectra, introduced in [\textit{C. Barwick} and \textit{T. Lawson}, ``Regularity of structured ring spectra and localization in \(K\)-theory'', Preprint, \url{arXiv:1402.6038}].
In the first part, the author contributes to the topic of regular \(\mathbb{E}_1\)-ring spectra by proving the stability of regularity spectra under localizations (Proposition \(2.8\)) and the discreteness of bounded regular \(\mathbb{E}_1\)-ring spectra which are \textit{quasicommutative} (Theorem \(2.11\)). The paper also provides a counterexample to the latter statement in the non-quasicommutative case (Construction \(2.12\)). The bulk of this section, however, is the (twofold) generalization of the concept of regularity to spectral stacks (Definition \(2.15\)): a spectral stack \(X\) is \textit{regular} if there exists a regular atlas \(\operatorname{Spec}(R) \to X\), while it is \textit{homological regular} if the standard \(t\)-structure on \(\operatorname{QCoh}(X)\) restricts to the subcategory of compact objects. While in general not equivalent, the two definitions agree if \(X\) is an affine spectral scheme or, with some additional hypothesis, if \(X\) is a quotient of a Noetherian connective \(\mathbb{E}_{\infty}\)-\(k\)-algebra under the action of a smooth affine group scheme (Theorem \(2.16\)).
In the second part, the author recalls the main definitions, properties and examples of weight structures on stable \(\infty\)-categories, and introduces the notion of \textit{adjacent structures} (Definition \(3.9\)) - i.e., a weight structure \(\left(\mathscr{C}_{w\geqslant 0},\mathscr{C}_{w\leqslant 0}\right)\) and a \(t\)-structure \(\left(\mathscr{C}_{t\geqslant 0},\mathscr{C}_{t\leqslant 0}\right)\) such that \(\mathscr{C}_{w\geqslant 0}=\mathscr{C}_{t\geqslant 0}\). Arguing that the standard \(t\)-structure and the standard weight structure on \(\operatorname{Mod}^{\operatorname{perf}}_R\) are adjacent if and only if \(R\) is regular, the author conjectures that the existence of adjacent structures on a stable \(\infty\)-category \(\mathscr{C}\) should be a noncommutative analogue of regularity. Hence, it is proposed that if all the mapping spaces in the heart of the weight structure \(\operatorname{Hw}(\mathscr{C})\) are \(N\)-truncated for some fixed \(N\), the \(\infty\)-category \(\operatorname{Hw}(\mathscr{C})\) is actually discrete (Conjecture \(3.12\)). Finally, the author proves that if \(X\) is a quotient spectral stack satisfying the assumptions of Theorem \(2.16\), then Conjecture \(3.12\) holds for \(\operatorname{QCoh}(X)\).
Reviewer: Emanuele Pavia (Trieste)On submultiplicative constants of an algebrahttps://zbmath.org/1496.200052022-11-17T18:59:28.764376Z"Cook, James S."https://zbmath.org/authors/?q=ai:cook.james-s"Nguyen, Khang V."https://zbmath.org/authors/?q=ai:nguyen.khang-vSummary: If \(\mathcal{A}\) is a finite-dimensional commutative associative real algebra with norm \(\| \cdot\|\) then we say that the \(r\)th submultiplicative constant of \(\mathcal{A}\) is the smallest constant \(m_r(\mathcal{A})\) for which \(\| x_1 x_2 \cdots x_r \| \le m_r(\mathcal{A}) \| x_1 \| \| x_2 \| \cdots \| x_r \|\). For a product algebra, we show that there exist zero divisors where equality is attained in the inequality defining \(m_r( \mathcal{A})\). We also study \(\rho_{\mathcal{A}} = \limsup_{r \rightarrow\infty} \sqrt[r]{m_r(\mathcal{A})}\). We explain how \(\rho_{\mathcal{A}}\) appears in the generalization of the Cauchy-Hadamard criterion for hypercomplex power series. We find the submultiplicative constants and \(\rho_{\mathcal{A}}\) for the real group algebra of the cyclic group of order \(n\) as well as the complicated numbers \(\mathcal{C}_n = \{ a_1+a_2k+ \cdots+ a_n k^{n-1}\mid a_i \in\mathbb{R}, k^n = -1 \}\) with Euclidean norm. Submultiplicative constants for the \(n\)-dual numbers \(\Delta_n\) with Euclidean norm are also calculated or conjectured for \(n \le 6\). We show, for \(n \geq2\), \(\rho_{\Delta_n} = 1\) for \(\Delta_n\) given the \(p\)-norm.An analogue of row removal for diagrammatic Cherednik algebrashttps://zbmath.org/1496.200072022-11-17T18:59:28.764376Z"Bowman, Chris"https://zbmath.org/authors/?q=ai:bowman.christopher-david"Speyer, Liron"https://zbmath.org/authors/?q=ai:speyer.lironAssociated to each cyclotomic quiver Hecke algebra \(H_n(\kappa)\), there is a family of diagrammatic Cherednik algebras \(A(n,\theta,\kappa)\) for \(\theta\in\mathbb{Z}^\ell\). In the level \(\ell=1\) case, the algebra \(A(n,\theta,\kappa)\) is Morita equivalent to the classical \(q\)-Schur algebra. These algebras are of central interest in Khovanov homology, knot theory group theory, and higher representation theory.
In this paper, the authors prove an analogue of James-Dokin row removal theorem for diagrammatic Cherednik algebras, which provides the graded decomposition numbers of these algebras over fields of arbitrary characteristic.
Reviewer: Li Luo (Shanghai)Schubert class and cyclotomic nilHecke algebrashttps://zbmath.org/1496.200082022-11-17T18:59:28.764376Z"Zhou, Kai"https://zbmath.org/authors/?q=ai:zhou.kai.3|zhou.kai.1|zhou.kai.2"Hu, Jun"https://zbmath.org/authors/?q=ai:hu.jun.5|hu.jun.6|hu.jun.4|hu.jun.3|hu.jun.1|hu.jun.2On generalized Heisenberg groups: the symmetric casehttps://zbmath.org/1496.200502022-11-17T18:59:28.764376Z"Sangkhanan, Kritsada"https://zbmath.org/authors/?q=ai:sangkhanan.kritsada"Suksumran, Teerapong"https://zbmath.org/authors/?q=ai:suksumran.teerapong(no abstract)Certain properties of Jordan homomorphisms, \(n\)-Jordan homomorphisms and \(n\)-homomorphisms on rings and Banach algebrashttps://zbmath.org/1496.460462022-11-17T18:59:28.764376Z"Honary, Taher Ghasemi"https://zbmath.org/authors/?q=ai:ghasemi-honary.taherSummary: We investigate under what conditions \(n\)-Jordan homomorphisms between rings are \(n\)-homomorphism, or homomorphism; and under what conditions, \(n\)-Jordan homomorphisms are continuous.
One of the main goals in this work is to show that every \(n\)-Jordan homomorphism \(f : A \rightarrow B\), from a unital ring \(A\) into a ring \(B\) with characteristic greater than \(n\), is a multiple of a Jordan homomorphism and hence, it is an \(n\)-homomorphism if every Jordan homomorphism from \(A\) into \(B\) is a homomorphism. In particular, if \(B\) is an integral domain whose characteristic is greater than \(n\), then every \(n\)-Jordan homomorphism \(f : A \rightarrow B\) is an \(n\)-homomorphism.
Along with some other results, we show that if \(A\) and \(B\) are unital rings such that the characteristic of \(B\) is greater than \(n\), then every unital \(n\)-Jordan homomorphism \(f : A \rightarrow B\) is a Jordan homomorphism and hence, it is an \(m\)-Jordan homomorphism for any positive integer \(m \geq 2\).
We also investigate the automatic continuity of \(n\)-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the \(n\)-Jordan homomorphism has dense range.Induced coactions along a homomorphism of locally compact quantum groupshttps://zbmath.org/1496.460702022-11-17T18:59:28.764376Z"Kitamura, Kan"https://zbmath.org/authors/?q=ai:kitamura.kanSummary: We consider induced coactions on \(C^*\)-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group.Homotopy invariance of the cyclic homology of \(A_{\infty}\)-algebras under homotopy equivalences of \(A_{\infty}\)-algebrashttps://zbmath.org/1496.550192022-11-17T18:59:28.764376Z"Lapin, Sergey V."https://zbmath.org/authors/?q=ai:lapin.sergey-vIn [\textit{S. V. Lapin}, Math. Notes 102, No. 6, 806--823 (2017; Zbl 1381.16009); translation from Mat. Zametki 102, No. 6, 874--895 (2017)], Lapin constructed the cyclic bicomplex of an \(A_{\infty}\)-algebra over a commutative unital ring. This generalized the famous cyclic bicomplex for an associative algebra due to \textit{B. L. Tsygan} [Russ. Math. Surv. 38, No. 2, 198--199 (1983; Zbl 0526.17006)] and \textit{J.-L. Loday} and \textit{D. Quillen} [Comment. Math. Helv. 59, 565--591 (1984; Zbl 0565.17006)]. The cyclic homology of an \(A_{\infty}\)-algebra is defined as the homology of the chain complex associated to Lapin's bicomplex.
In the paper under review Lapin proves that the cyclic homology of \(A_{\infty}\)-algebras is homotopy invariant under homotopy equivalences of \(A_{\infty}\)-algebras.
The paper is organized as follows. Section 2 recalls the definition of a cyclic module with \(\infty\)-simplicial faces, or \(CF_{\infty}\)-module, as introduced in [\textit{S. V. Lapin}, loc. cit.]. In particular, as recalled in Section 4 of the paper under review, any \(A_{\infty}\)-algebra gives rise to a \(CF_{\infty}\)-module. The notion of homotopy equivalence of \(CF_{\infty}\)-modules is introduced.
Section 3 recalls the definition of cyclic homology of \(CF_{\infty}\)-modules. It is shown that there is a cyclic homology functor from the category of \(CF_{\infty}\)-modules to the category of graded modules over the ground ring. Furthermore, it is shown that this functor sends homotopy equivalences of \(CF_{\infty}\)-modules to isomorphisms of graded modules.
In Section 4, the author recalls the definition of \(A_{\infty}\)-algebra and how to construct a \(CF_{\infty}\)-module from an \(A_{\infty}\)-algebra. The results of Section 3 are then applied to give a cyclic homology functor from the category of \(A_{\infty}\)-algebras to the category of graded modules, sending homotopy equivalences of \(A_{\infty}\)-algebras to isomorphisms of graded modules.
Reviewer: Daniel Graves (Leeds)On networks over finite ringshttps://zbmath.org/1496.930532022-11-17T18:59:28.764376Z"Cheng, Daizhan"https://zbmath.org/authors/?q=ai:cheng.daizhan"Ji, Zhengping"https://zbmath.org/authors/?q=ai:ji.zhengpingSummary: The (control) networks over finite rings are proposed and their properties are investigated. Based on semi-tensor product (STP) of matrices, a set of algebraic equations are provided to verify whether a finite set with two binary operators is a ring. As an application, all rings with 4 elements are obtained. It is then shown that the STP-based technique developed for logical (control) networks are applicable to (control) networks over finite rings. A (control) sub-network over a proper ideal of the bearing ring is constructed. Certain properties are revealed, showing that a network over the ideal behaves like a subsystem over an invariant subspace. Product rings are then introduced, which provides a tool for both constructing product networks and decomposing complex networks. Finally, the representation of networks over finite rings is considered, which investigates how many finite networks can be expressed as networks over finite rings, showing that the technique developed in this paper is widely applicable.New binary self-dual codes of lengths 80, 84 and 96 from composite matriceshttps://zbmath.org/1496.940822022-11-17T18:59:28.764376Z"Gildea, Joe"https://zbmath.org/authors/?q=ai:gildea.joe"Korban, Adrian"https://zbmath.org/authors/?q=ai:korban.adrian"Roberts, Adam Michael"https://zbmath.org/authors/?q=ai:roberts.adam-michaelOne of the most used techniques for generating binary self-dual codes is the pure double circulant construction using the matrix \(G = (I_n\mid A)\) where \(A\) is a circulant matrix. This technique has since been generalised by assuming a generator matrix of the form \(G = (I_n |\sigma(v))\) where \(\sigma\) is an isomorphism from a group ring. A composite matrix it this work denotes \(G = (I_n |\Omega(v))\) where \(\Omega(v)\) is a matrix that arises from group rings. The code generated by \(G\) is self-dual if and only if \(\Omega(v)\Omega(v)T = -I_n.\)
Using this type of generator matrices for a number of different composite matrices \(\Omega(v)\) over different alphabets \(\mathbb{F}_2,\) \(\mathbb{F}_2 + u\mathbb{F}_2\) and \(\mathbb{F}_4\), the authors find many binary self-dual codes with large length and weight enumerator parameters of previously unknown values. Studying this type of construction the necessary and sufficient conditions needed by each construction to produce a self-dual code are proved. Applying the results, a total of 361 new binary self-dual codes are found, including 28 singly-even binary self-dual \([80, 40, 14]\) codes, 107 binary self-dual \([84, 42, 14]\) codes, 105 singly-even binary self-dual \([96, 48, 16]\) codes and 121 doubly-even binary self-dual \([96, 48, 16]\) codes. All the necessary information for generating these codes as well as the orders of their automorphism group are given in tables.
Reviewer: Nikolay Yankov (Shumen)Sum-rank product codes and bounds on the minimum distancehttps://zbmath.org/1496.940902022-11-17T18:59:28.764376Z"Alfarano, Gianira N."https://zbmath.org/authors/?q=ai:alfarano.gianira-n"Lobillo, F. J."https://zbmath.org/authors/?q=ai:lobillo.francisco-javier"Neri, Alessandro"https://zbmath.org/authors/?q=ai:neri.alessandro"Wachter-Zeh, Antonia"https://zbmath.org/authors/?q=ai:wachter-zeh.antonia\textit{U. Martínez-Peñas} introduced the family of cyclic-skew-cyclic codes endowed with the sum-rank metric [IEEE Trans. Inf. Theory 67, No. 8, 5149--5167 (2021; Zbl 1486.94178)]. Inspired by this work, the authors of the paper under review consider the tensor product of a cyclic code endowed with the Hamming metric with a skew-cyclic code endowed with the rank-metric, both defined over the same field \(\mathbb{F}\). Such a product code turns out to be a cyclic-skew-cyclic code which naturally inherits the sum-rank metric. A group theoretical description of these codes is given, after investigating the semilinear isometries in the sum-rank metric. As a nice application, the authors provide a generalization of the Roos and the Hartmann-Tzeng bounds for cyclic-skew-cyclic codes endowed with the sum-rank metric.
Reviewer: Sami Omar (Sukhair)