Recent zbMATH articles in MSC 16https://zbmath.org/atom/cc/162024-06-14T15:52:26.737412ZWerkzeugBook review of: J. H. Silverman, Abstract algebra. An integrated approachhttps://zbmath.org/1534.000062024-06-14T15:52:26.737412Z"Dicker, Math"https://zbmath.org/authors/?q=ai:dicker.mathReview of [Zbl 1487.12001].Injective and projective semimodules over involutive semiringshttps://zbmath.org/1534.060092024-06-14T15:52:26.737412Z"Jipsen, Peter"https://zbmath.org/authors/?q=ai:jipsen.peter"Vannucci, Sara"https://zbmath.org/authors/?q=ai:vannucci.saraSummary: We show that the term equivalence between \(\mathrm{MV}\)-algebras and \(\mathrm{MV}\)-semirings lifts to involutive residuated lattices and a class of semirings called \textit{involutive semirings}. The semiring perspective leads to a necessary and sufficient condition for the interval \([d,1]\) to be a subalgebra of an involutive residuated lattice, where \(d\) is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective \(\mathrm{MV}\)-semimodules. Indeed, we note that the involution plays a crucial role and that the results for \(\mathrm{MV}\)-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.Explicit isomorphisms of quaternion algebras over quadratic global fieldshttps://zbmath.org/1534.111362024-06-14T15:52:26.737412Z"Csahók, Tímea"https://zbmath.org/authors/?q=ai:csahok.timea"Kutas, Péter"https://zbmath.org/authors/?q=ai:kutas.peter"Montessinos, Mickaël"https://zbmath.org/authors/?q=ai:montessinos.mickael"Zábrádi, Gergely"https://zbmath.org/authors/?q=ai:zabradi.gergelySummary: Let \(L\) be a separable quadratic extension of either \(\mathbb{Q}\) or \(\mathbb{F}_q(t)\). We exhibit efficient algorithms for finding isomorphisms between quaternion algebras over \(L\). Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple \(L\)-algebra.A comparison of dg algebra resolutions with prime residual characteristichttps://zbmath.org/1534.130092024-06-14T15:52:26.737412Z"DeBellevue, Michael"https://zbmath.org/authors/?q=ai:debellevue.michael"Pollitz, Josh"https://zbmath.org/authors/?q=ai:pollitz.joshSummary: In this article, we fix a prime integer \(p\) and compare certain dg algebra resolutions over a local ring whose residue field has characteristic \(p\). Namely, we show that given a closed surjective map between such algebras there is a precise description for the minimal model in terms of the acyclic closure and that the latter is a quotient of the former. A first application is that the homotopy Lie algebra of a closed surjective map is abelian. We also use these calculations to show deviations enjoy rigidity properties which detect the (quasi-)complete intersection property.Secondary Hochschild homology and differentialshttps://zbmath.org/1534.130102024-06-14T15:52:26.737412Z"Laubacher, Jacob"https://zbmath.org/authors/?q=ai:laubacher.jacobLet \(\Bbbk\) be a field of characteristic 0, and consider a triple \((A,B,\varepsilon)\) where \(B\) is a commutative \(\Bbbk\)-algebra and \(A\) an associative \(\Bbbk\)-algebra with a \(B\)-algebra structure induced by a morphism of \(\Bbbk\)-algebras \(\varepsilon:B\rightarrow A\) with \(\varepsilon(B)\) contained in the center of \(A\). To such a triple one can associate a homology theory known as \emph{secondary Hochschild homology}, which generalizes the classic Hochschild homology since when \(B=\Bbbk\) it reduces to the Hochschild homology of \(A\). It is known that the first Hochschild homology of a \(\Bbbk\)-algebra \(A\) is isomorphic to the first module of Kähler differentials \(\Omega^1_{A|\Bbbk}\). The authors generalize this results by first defining the module of \emph{secondary Kähler differentials} \(\Omega^1_{\mathcal{T}|\Bbbk}\), where \(\mathcal{T}=(A,B,\varepsilon)\), and then proving that this module is isomorphic to \(HH_1(A,B,\varepsilon)\), the first secondary Hochschild homology module, provided \(A\) is also commutative. This constitutes the main result of the paper.
Reviewer: Luigi Ferraro (Edinburg)Bongartz completion via \(c\)-vectorshttps://zbmath.org/1534.130152024-06-14T15:52:26.737412Z"Cao, Peigen"https://zbmath.org/authors/?q=ai:cao.peigen"Gyoda, Yasuaki"https://zbmath.org/authors/?q=ai:gyoda.yasuaki"Yurikusa, Toshiya"https://zbmath.org/authors/?q=ai:yurikusa.toshiyaClassically, Bongartz completion [\textit{K. Bongartz}, Lect. Notes Math. 903, 26--38 (1981; Zbl 0478.16025)] is a method for completing a partial tilting module over a finite-dimensional algebra to a full tilting module, given one tilting module to use as a reference point (a canonical choice being a projective generator). Similar ideas, which realise the completion as a projective generator in a torsion class, appear in work by \textit{I. Assem} [Can. J. Math. 36, 899--913 (1984; Zbl 0539.16022)] and \textit{S. O. Smalø} [Bull. Lond. Math. Soc. 16, 518--522 (1984; Zbl 0519.16016)]. The technique has subsequently been generalised to many related situations in which one wants to complete a set of objects satisfying a ``rigidity'' condition (analogous to the indecomposable summands of a partial tilting module) to a maximal such set.
One such generalisation is to the situation of \(\tau\)-tilting theory [\textit{T. Adachi} et al., Compos. Math. 150, No. 3, 415--452 (2014; Zbl 1330.16004)], in which a version of Bongartz completion can be used to complete a \(\tau\)-rigid pair to a \(\tau\)-tilting pair. For certain finite-dimensional algebras, \(\tau\)-rigid objects have a cluster-theoretic interpretation via associated integer vectors, known as \(g\)-vectors and \(c\)-vectors, and the role of the reference \(\tau\)-tilting object is played by a choice of initial seed. The present paper describes the process of Bongartz completion (and, when possible, the dual notion of co-completion) in terms of \(c\)-vectors, the case of \(g\)-vectors having been treated in earlier work of Peigen Cao and Fang Li [\textit{P. Cao} and \textit{F. Li}, Math. Ann. 377, No. 3--4, 1547--1572 (2020; Zbl 1454.13031)]. These results hold in the context of \(\tau\)-tilting theory for general finite-dimensional algebras, without any need for an associated cluster algebra.
However, when this cluster algebra does exist, there are cluster-theoretic consequences of the results. In particular, the authors show that the collection of seeds in a cluster algebra featuring a particular fixed set of cluster variables is connected in the exchange graph, equivalent to the exchange graph of the cluster algebra defined by the same initial data but with these fixed variables frozen. They additionally show that a seed is uniquely determined by its collection of negative \(y\)-variables (or coefficients).
Reviewer: Matthew Pressland (Glasgow)Root of unity quantum cluster algebras and Cayley-Hamilton algebrashttps://zbmath.org/1534.130162024-06-14T15:52:26.737412Z"Huang, Shengnan"https://zbmath.org/authors/?q=ai:huang.shengnan"Lê, Thang T. Q."https://zbmath.org/authors/?q=ai:le-tu-quoc-thang."Yakimov, Milen"https://zbmath.org/authors/?q=ai:yakimov.milen-tLet \(R\) be an algebra over the commutative ring \(\Bbbk\), \(C\) a central subalgebra of \(R\), and \(\mathrm{tr}:R \to C\) a trace map. A Cayley-Hamilton algebra of degree \(d\) is a triple \((R,C,\mathrm{tr})\) that satisfies the appropriately defined characteristic polynomial of degree \(d\) for each \(a \in R\). This class of algebras includes, for example, maximal orders of PI degree \(d\) whose center has characteristic \(p \notin [1,d]\) with respect to the reduced trace. The formal definition is due to \textit{C. Procesi} [J. Algebra 107, 63--74 (1987; Zbl 0618.16014)], and the structure gives a pleasant environment for studying their representation theory.
In the present article, the authors introduce a generalization of root of unity upper quantum cluster algebras. Such an algebra, denoted \(\mathbf{U}_\varepsilon(M_\varepsilon,\widetilde{B},\mathbf{inv},\Theta)\), has the following data: \(\widetilde{B}\) is an exchange matrix, \(M_\varepsilon\) a toric frame, \(\mathbf{inv}\) is a subset of the set of frozen variables, and \(\Theta\) is a subset of seeds. These are shown to be maximal orders in a central simple algebras, the trace map is given explicitly, and the Azumaya locus is described. The authors define a canonical central subalgebra for \(\mathbf{U}_\varepsilon(M_\varepsilon,\widetilde{B},\mathbf{inv},\Theta)\) and a regular trace map so that the corresponding triple is a Cayley-Hamilton algebra. These results are then lifted to a class of subalgebras of the quantum torus \(\mathcal{T}_\varepsilon(M_\varepsilon)\) generated by monomials with exponents in a submonoid of \(\mathbb{Z}^N\).
Reviewer: Jason Gaddis (Oxford, OH)Ideals generated by differential equationshttps://zbmath.org/1534.130202024-06-14T15:52:26.737412Z"Kaptsov, Oleg V."https://zbmath.org/authors/?q=ai:kaptsov.oleg-viktorovichSummary: We propose a new algebraic approach to study compatibility of partial differential equations. The approach uses concepts from commutative algebra, algebraic geometry and Gröbner bases to clarify crucial notions concerning compatibility such as passivity and reducibility. One obtains sufficient conditions for a differential system to be passive and proves that such systems generate manifolds in the jet space. Some examples of constructions of passive systems associated with the sinh-Cordon equation are given.Symmetric homology and representation homologyhttps://zbmath.org/1534.140032024-06-14T15:52:26.737412Z"Berest, Yuri"https://zbmath.org/authors/?q=ai:berest.yuri-yu"Ramadoss, Ajay C."https://zbmath.org/authors/?q=ai:ramadoss.ajay-cSymmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. Recall that one of the key ideas in \textit{A. Connes}' approach to cyclic homology [C. R. Acad. Sci., Paris, Sér. I 296, 953--958 (1983; Zbl 0534.18009)] is the cyclic category \(\Delta C\), which can be viewed as a ``crossed product'' of the simplicial category \(\Delta\) with the family of cyclic groups \(\{\mathbb{Z}/(n+1)\mathbb{Z}\}_{n\geq 0}\). Similarly, one can define the symmetric category \(\Delta S\) as the categorical crossed product of the simplicial category \(\Delta\) with the family of symmetric groups \(\{\Sigma_{n+1}\}_{n\geq 0}\). For an associative \(k\)-algebra \(A\), \textit{Z. Fiedorowicz} and \textit{J.-L. Loday} introduced the notion of symmetric bar construction \(B_{\mathrm{sym}}A\) of \(A\) [Trans. Am. Math. Soc. 326, No. 1, 57--87 (1991; Zbl 0755.18005)], which is a covariant functor from \(\Delta S\) to the category of \(k\)-modules. The symmetric homology of \(A\) is defined by
\[
\mathrm{HS}_\ast(A)\,=\,\mathrm{Tor}_\ast^{\Delta S}(k,\,B_{\mathrm{sym}}A)\,.
\]
Unlike Hochschild and cyclic homologies, the symmetric homology theory is less accessible to algebraic methods, and \(\mathrm{HS}_\ast(A)\) is unknown when \(A\) is a polynomial algebra with more than one variable.
In a joint work with \textit{G. Khachatryan} [Adv. Math. 245, 625--689 (2013; Zbl 1291.14006)], the authors of the present paper introduce the theory of representation homology using the ideals from homotopical algebra. Let \(M_n\) be the matrix functor that takes a commutative \(k\)-algebra \(C\) to the \(k\)-algebra \(M_n(C)\) of \((n\times n)\)-matrices with entries in \(C\). It is known that \(M_n\) has a left adjoint, denoted by \((\mbox{--})_n\). The \(n\)-dimensional representation homology of an associative \(k\)-algebra \(A\) can be computed as
\[
\mathrm{HR}_\ast(A,\,k^n)\,=\,\mathrm{H}_\ast[(QA)_n]\,,
\]
where \(QA \xrightarrow{\sim} A\) is a(ny) cofibrant resolution of \(A\). In particular, the 1-dimensional representation homology coincides with the derived abelianization. The notion of representation homology can be easily extended to simplicial algebras as well as differential graded (DG) algebras.
The main result of the present paper is showing the equivalence between the symmetric homology and the \(1\)-dimensional representation homology when \(k\) is a field of characteristic zero. In fact the authors prove a slightly general version: there is an isomorphism of graded commutative algebras
\[
\mathrm{HS}_\ast(A)\,\cong\,\mathrm{HR}_\ast(A,\,k),\,
\]
when \(A\) is a simplicial or DG algebra defined over \(k\). Using known results on representation homology, the authors give explicit description of \(\mathrm{HS}_\ast(A)\) when \(A\) is a polynomial algebra, or more general, \(A\) is the universal enveloping algebra of a Lie algebra. As an application, the authors prove two conjectures of \textit{S. Ault} and \textit{Z. Fiedorowicz} [``Symmetric homology of algebras'', Preprint, \url{arXiv:0708.1575}], including a conjecture on topological interpretation of symmetric homology of polynomial algebras.
Reviewer: Yining Zhang (Singapura)Dimer models and group actionshttps://zbmath.org/1534.140082024-06-14T15:52:26.737412Z"Ishii, Akira"https://zbmath.org/authors/?q=ai:ishii.akira"Nolla, Álvaro"https://zbmath.org/authors/?q=ai:nolla.alvaro"Ueda, Kazushi"https://zbmath.org/authors/?q=ai:ueda.kazushiLet \(N\) be a free rank two abelian group and \(M=\Hom(N,{\mathbb Z})\). A \textit{ dimer model} \(G\) is a bicolored graph embedded on the real \(2\)-torus \(T=M \otimes {\mathbb R}/M\) in such a way that there are no univalent nodes (adjacent to just one node), and that every face is simply connected. In a purely combinatorial way one can associate a convex lattice polygon \(\Delta \subset N \otimes {\mathbb R}\) to \(G\), and consequently a Gorenstein affine toric \(3\)-fold denoted \(X_\Delta\).
A \textit{quiver} \(Q\) is a set \(Q_0\) of vertices, a set \(Q_1\) of arrows and a pair \(s,t: Q_1 \to Q_0\), the source and the target. One can produce in a natural way a \textit{ path algebra} on \(Q\), and impose the relations of an ideal in such algebra. It can be shown that there exists a dimer model \(G\) (whose faces are \(Q_0\) and whose edges are \(Q_1\)) which encodes the information of a quiver with relations (see Section 2.1 for details). Under appropriate hypotheses (consistency) on \(G\), the path algebra of a quiver with relations is a non-commutative crepant resolution of \(X_\Delta\).
Let us consider a finite group \(H \subset\) Gl\((2,{\mathbb Z})\) which leaves \(\Delta\) invariant. The first result in the paper under review (see Thm. 1) shows the existence of a (consistent) dimer model \(G\) symmetric with respect to the action of \(H\) (notion introduced in Def. 1), whose characteristic polygonal is \(\Delta\). On the other hand, if \(G\) is symmetric with respect to the \(H\)-action, then \(H\) is shown to act on the associated quiver with relations, and one can associate \(H\)-actions on its path algebra and on \(X_\Delta\). The second result (see Thm.2) shows that the crossed product algebra of the path algebra and \(H\) is a non-commutative crepant resolution of \(X_\Delta/H\).
Reviewer: Roberto Muñoz (Madrid)Decompositions of matrices into a sum of invertible matrices and matrices of fixed nilpotencehttps://zbmath.org/1534.150102024-06-14T15:52:26.737412Z"Danchev, Peter"https://zbmath.org/authors/?q=ai:danchev.peter-vassilev"García, Esther"https://zbmath.org/authors/?q=ai:garcia.esther"Lozano, Miguel Gómez"https://zbmath.org/authors/?q=ai:gomez-lozano.miguel-aInspired by result of \textit{G. Călugăreanu} and \textit{T. Y. Lam} [J. Algebra Appl. 15, No. 9, Article ID 1650173, 18 p. (2016; Zbl 1397.16036)] which states that every nonzero square matrix over a division ring is a sum of an invertible matrix and a nilpotent matrix, the authors investigate when a matrix of order \(n\) defined over an arbitrary field is a sum of an invertible matrix and a nilpotent matrix of fixed nilpotency index \(k\) (\(k\geqslant 1\)). They prove that this happens if and only if the rank of \(A\) is greater than or equal to \(n/k\).
Reviewer: Roksana Słowik (Gliwice)Erratum to: ``The cyclic homology of the group rings''https://zbmath.org/1534.160012024-06-14T15:52:26.737412Z"Burghelea, Dan"https://zbmath.org/authors/?q=ai:burghelea.danSummary: This erratum corrects the statements of Propositions II and IIp of the author's paper [ibid. 60, 354--365 (1985; Zbl 0595.16022)].Algebraic geometry over Heyting algebrashttps://zbmath.org/1534.160022024-06-14T15:52:26.737412Z"Nouri, Mahdiyeh"https://zbmath.org/authors/?q=ai:nouri.mahdiyehSummary: In this article, we study the algebraic geometry over Heyting algebras and we investigate the properties of being equationally Noetherian and \(q_\omega\)-compact over such algebras.Corrigendum and addendum to: ``Structure monoids of set-theoretic solutions of the Yang-Baxter equation''https://zbmath.org/1534.160032024-06-14T15:52:26.737412Z"Cedó, Ferran"https://zbmath.org/authors/?q=ai:cedo.ferran"Jespers, Eric"https://zbmath.org/authors/?q=ai:jespers.eric"Verwimp, Charlotte"https://zbmath.org/authors/?q=ai:verwimp.charlotteSummary: One of the results in our article which appeared in [ibid. 65, No. 2, 499--528 (2021; Zbl 1487.16035)], is that the structure monoid \(M(X,r)\) of a left non-degenerate solution \((X,r)\) of the Yang-Baxter equation is a left semi-truss, in the sense of Brzeziński, with an additive structure monoid that is close to being a normal semigroup. Let \(\eta\) denote the least left cancellative congruence on the additive monoid \(M(X,r)\). It is then shown that \(\eta\) is also a congruence on the multiplicative monoid \(M(X,r)\) and that the left cancellative epimorphic image \(\bar{M}=M(X,r)/\eta\) inherits a semi-truss structure and thus one obtains a natural left non-degenerate solution of the Yang-Baxter equation on \(\bar{M}\). Moreover, it restricts to the original solution \(r\) for some interesting classes, in particular if \((X,r)\) is irretractable. The proof contains a gap. In the first part of the paper we correct this mistake by introducing a new left cancellative congruence \(\mu\) on the additive monoid \(M(X,r)\) and show that it also yields a left cancellative congruence on the multiplicative monoid \(M(X,r)\), and we obtain a semi-truss structure on \(M(X,r)/\mu\) that also yields a natural left non-degenerate solution.
In the second part of the paper we start from the least left cancellative congruence \(\nu\) on the multiplicative monoid \(M(X,r)\) and show that it is also a congruence on the additive monoid \(M(X,r)\) in the case where \(r\) is bijective. If, furthermore, \(r\) is left and right non-degenerate and bijective, then \(\nu =\eta\), the least left cancellative congruence on the additive monoid \(M(X,r)\), extending an earlier result of Jespers, Kubat, and Van Antwerpen to the infinite case.The reduced ring order and lower semi-lattices. IIhttps://zbmath.org/1534.160042024-06-14T15:52:26.737412Z"Burgess, W. D."https://zbmath.org/authors/?q=ai:burgess.walter-d"Raphael, R."https://zbmath.org/authors/?q=ai:raphael.robert-mSummary: This paper continues the study of the reduced ring order (\textbf{rr}-order) in reduced rings where \(a \leq_{\mathrm{rr}}b\) if \(a^2=ab\). A reduced ring is called \textbf{rr}-good if it is a lower semi-lattice in the order. Examples include weakly Baer rings (wB or PP-rings) but many more. Localizations are examined relating to this order as well as the Pierce sheaf. Liftings of \textbf{rr}-orthogonal sets over surjections of reduced rings are studied. A known result about commutative power series rings over wB rings is extended, via methods developed here, to very general, not necessarily commutative, power series rings defined by an ordered monoid, showing that they are wB.
For Part I see [the authors, Contemp. Math. 715, 89--106 (2018; Zbl 1441.16048)].Local superderivations on solvable Lie and Leibniz superalgebrashttps://zbmath.org/1534.170092024-06-14T15:52:26.737412Z"Camacho, Luisa María"https://zbmath.org/authors/?q=ai:camacho.luisa-maria"Navarro, Rosa María"https://zbmath.org/authors/?q=ai:navarro.rosa-maria"Omirov, Bakhrom"https://zbmath.org/authors/?q=ai:omirov.bakhrom-aIn this paper, the concept of local derivation is extended to the case of superalgebras as a local superderivation. It is shown that there exist nilpotent and solvable Lie superalgebras with infinitely many local superderivations which are not (global) superderivations.
Moreover, the authors prove that each local superderivation of the maximal-dimensional solvable Lie and Leibniz superalgebras with model nilpotent nilradical is a (global) superderivation.
Reviewer: Sh. A. Ayupov (Tashkent)Explicit central elements of \(U_q(\mathfrak{gl}(N+1))\)https://zbmath.org/1534.170172024-06-14T15:52:26.737412Z"Kuan, Jeffrey"https://zbmath.org/authors/?q=ai:kuan.jeffrey"Zhang, Keke"https://zbmath.org/authors/?q=ai:zhang.kekeIn this paper the authors obtain explicit expressions for the central elements of the quantum group \( \mathcal{U}_q \left( \mathrm{gl}(N+1) \right) \), in terms of the respective generators. The authors follow Drinfeld's central element construction except that the fused R-matrix is used instead of the universal one.
Reviewer: Nenad Manojlović (Faro)Local automorphisms of nil-triangular subalgebras of classical Lie type Chevalley algebrashttps://zbmath.org/1534.170212024-06-14T15:52:26.737412Z"Zotov, Igor N."https://zbmath.org/authors/?q=ai:zotov.igor-nSummary: We study the problem of describing local automorphisms of nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring with identity.Group algebras of free products of finite groups which are CS-ringshttps://zbmath.org/1534.200032024-06-14T15:52:26.737412Z"Kondo, Shoichi"https://zbmath.org/authors/?q=ai:kondo.shoichiSummary: This paper proves that the infinite dihedral group is only such a free product of two finite groups that its group algebra over a field \(K\) is a CS-ring in case the orders of two groups are not zero in \(K\). Furthermore, it is shown that the group algebra of any free product of two finite cyclic groups does not satisfy the condition \((C_3)\).On the equationally Artinian groupshttps://zbmath.org/1534.200542024-06-14T15:52:26.737412Z"Shahryari, Mohammad"https://zbmath.org/authors/?q=ai:shahryari.mohammad-reza-balooch|shahryari.mohammad"Tayyebi, Javad"https://zbmath.org/authors/?q=ai:tayyebi.javad.1Summary: In this article, we study the property of being equationally Artinian in groups. We define the radical topology corresponding to such groups and investigate the structure of irreducible closed sets of these topologies. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form \(G[t]\) by a normal subgroup which is a finite union of radicals, is again equationally Artnian. A necessary and sufficient condition for an Abelian group to be equationally Artinian will be given as the last result. This will provide a large class of examples of equationally Artinian groups.Harmonic differential forms for pseudo-reflection groups. II: Bi-degree boundshttps://zbmath.org/1534.200582024-06-14T15:52:26.737412Z"Swanson, Joshua P."https://zbmath.org/authors/?q=ai:swanson.joshua-p"Wallach, Nolan R."https://zbmath.org/authors/?q=ai:wallach.nolan-rSummary: This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic \(0\). Our most general result determines the top component in total degree, which we prove for all Shephard-Todd groups \(G(m, p, n)\) with \(m \neq p\) or \(m = 1\). Our strongest result gives tight bi-degree bounds and is proven for all \(G(m, 1, n)\), which includes the Weyl groups of types \(A\) and \(B/C\). For symmetric groups (i.e. type \(A)\), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund-Remmel-Wilson. Finally, we examine analogues of a classic theorem of Steinberg and the Operator Theorem of Haiman.
Our arguments build on the type-independent classification of semi-invariant harmonic differential forms carried out in the first paper in this sequence. In this paper we use concrete constructions including Gröbner and Artin bases for the classical coinvariant algebras of the pseudo-reflection groups \(G(m, p, n)\), which we describe in detail. We also prove that exterior differentiation is exact on the super-coinvariant algebra of a general pseudo-reflection group. Finally, we discuss related conjectures and enumerative consequences.
For Part I see [the authors, J. Comb. Theory, Ser. A 182, Article ID 105474, 30 p. (2021; Zbl 1511.20143)].Spectrally starred advertibly complete \(A\)-\(p\)-normed algebrashttps://zbmath.org/1534.460432024-06-14T15:52:26.737412Z"Ouhmidou, A."https://zbmath.org/authors/?q=ai:ouhmidou.a"El Kinani, A."https://zbmath.org/authors/?q=ai:el-kinani.abdellahAuthors' abstract: We show that an advertibly complete \(A\)-\(p\)-normed algebra \(E\) is isomorphic to the complex field \(\mathbb{C}\), modulo its radical, in any of the following cases: 1)~every element of \(E\) has a star-shaped spectrum, 2)~\(E\) is involutive and every normal element of \(E\) has a star-shaped spectrum; 3)~\(E\) is Hermitian and every unitary element of \(E\) has a star-shaped spectrum.
Reviewer: Mati Abel (Tartu)Universal continuous calculus for \textit{Su}*-algebrashttps://zbmath.org/1534.460452024-06-14T15:52:26.737412Z"Schötz, Matthias"https://zbmath.org/authors/?q=ai:schotz.matthiasAn \(Su^*\)-algebra is a symmetric uniformly complete closed ordered \({}^*\)-algebra; i.e., if \(a+\epsilon b\) belongs to the convex cone \(A_{\mathcal H}^+=\{a\in A, \,a^*=a,\, a\geqslant \theta_A\}\) of positive Hermitian elements of \(A\) for each \(\epsilon>0\), then \(a\in A_{\mathcal H}^+\).
The author constructs a universal continuous calculus for an \(n\)-tuple of pairwise commuting Hermitian elements of an \(Su^*\)-algebra.
Descriptions of the spectrum of a single Hermitian or normal element of a \(Su^*\)-algebra are given.
Several results about so-called \textit{proper} \(Su^*\)-algebras of continuous functions are also obtained and illustrated by examples.
Reviewer: Mart Abel (Tartu)Gaps labeling theorem for the bubble-diamond self-similar graphshttps://zbmath.org/1534.810562024-06-14T15:52:26.737412Z"Melville, Elizabeth"https://zbmath.org/authors/?q=ai:melville.elizabeth"Mograby, Gamal"https://zbmath.org/authors/?q=ai:mograby.gamal"Nagabandi, Nikhil"https://zbmath.org/authors/?q=ai:nagabandi.nikhil"Rogers, Luke G."https://zbmath.org/authors/?q=ai:rogers.luke-g"Teplyaev, Alexander"https://zbmath.org/authors/?q=ai:teplyaev.alexanderSummary: Motivated by the appearance of fractals in several areas of physics, especially in solid state physics and the physics of aperiodic order, and in other sciences, including the quantum information theory, we present a detailed spectral analysis for a new class of fractal-type diamond graphs, referred to as bubble-diamond graphs, and provide a gap-labeling theorem in the sense of Bellissard for the corresponding probabilistic graph Laplacians using the technique of spectral decimation. Labeling the gaps in the Cantor set by the normalized eigenvalue counting function, also known as the integrated density of states, we describe the gap labels as orbits of a second dynamical system that reflects the branching parameter of the bubble construction and the decimation structure. The spectrum of the natural Laplacian on limit graphs is shown generically to be pure point supported on a Cantor set, though one particular graph has a mixture of pure point and singularly continuous components.
{{\copyright} 2023 IOP Publishing Ltd}Optical Schrödinger cats with generalized coherent stateshttps://zbmath.org/1534.810672024-06-14T15:52:26.737412Z"Giraldi, Filippo"https://zbmath.org/authors/?q=ai:giraldi.filippoSummary: Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.Trade-off relations of geometric coherencehttps://zbmath.org/1534.810732024-06-14T15:52:26.737412Z"Hu, Bingyu"https://zbmath.org/authors/?q=ai:hu.bingyu"Zhao, Ming-Jing"https://zbmath.org/authors/?q=ai:zhao.ming-jingSummary: Quantum coherence is an important quantum resource and it is intimately related to various research fields. The geometric coherence is a coherence measure both operationally and geometrically. We study the trade-off relation of geometric coherence in qubit systems. We first derive an upper bound for the geometric coherence by the purity of quantum states. Based on this, a complementarity relation between the quantum coherence and the mixedness is established. We then derive the quantum uncertainty relations of the geometric coherence on two and three general measurement bases in terms of the incompatibility respectively, which turn out to be state-independent for pure states. These trade-off relations provide the limit to the amount of quantum coherence. As a byproduct, the complementarity relation between the minimum error probability for discriminating a pure-states ensemble and the mixedness of quantum states is established.
{{\copyright} 2023 IOP Publishing Ltd}Phase spaces, parity operators, and the Born-Jordan distributionhttps://zbmath.org/1534.810782024-06-14T15:52:26.737412Z"Koczor, Bálint"https://zbmath.org/authors/?q=ai:koczor.balint"vom Ende, Frederik"https://zbmath.org/authors/?q=ai:vom-ende.frederik"de Gosson, Maurice"https://zbmath.org/authors/?q=ai:de-gosson.maurice-a"Glaser, Steffen J."https://zbmath.org/authors/?q=ai:glaser.steffen-j"Zeier, Robert"https://zbmath.org/authors/?q=ai:zeier.robertSummary: Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time-frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (\textit{A. Grossmann} in [Commun. Math. Phys. 48, 191--194 (1976; Zbl 0337.46063)]), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (\textit{L. Cohen} in [J. Math. Phys. 7, No. 5, 781--786, (1966; \url{doi:10.1063/1.1931206})]) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born-Jordan distribution which originates from the Born-Jordan quantization (\textit{M. Born} and \textit{P. Jordan} in [Z. Phys. 34, 858--888 (1925; JFM 51.0728.08)]). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born-Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born-Jordan distribution.Giant atom induced zero modes and localization in the nonreciprocal Su-Schrieffer-Heeger chainhttps://zbmath.org/1534.811842024-06-14T15:52:26.737412Z"Wang, J. J."https://zbmath.org/authors/?q=ai:wang.junjie"Li, Fude"https://zbmath.org/authors/?q=ai:li.fude"Yi, X. X."https://zbmath.org/authors/?q=ai:yi.xuexi|yi.xiangxuanSummary: A notable feature of non-Hermitian systems with skin effects is the sensitivity of their spectra and eigenstates to the boundary conditions. In the literature, three types of boundary conditions-periodic boundary condition, open boundary condition (OBC) and a defect in the system as a boundary, are explored. In this work we introduce the other type of boundary condition provided by a giant atom. The giant atom couples to a nonreciprocal Su-Schrieffer-Heeger (SSH) chain at two points and plays the role of defects. We study the spectrum and localization of eigenstates of the system and find that the giant atom can induce asymmetric zero modes. A remarkable feature is that bulk states might localize at the left or the right chain-atom coupling sites in weak localization regimes. This bipolar localization leads to Bloch-like states, even though translational invariance is broken. Moreover, we find that the localization is obviously weaker than the case with two small atoms or OBCs even in strong coupling regimes. These intriguing results indicate that nonlocal coupling of the giant atom to a nonreciprocal SSH chain weakens the localization of the eigenstates. We also show that the Lyapunov exponent in the long-time dynamics in real space can act as a witness of the localized bulk states.
{{\copyright} 2023 IOP Publishing Ltd}Irreversible Markov dynamics and hydrodynamics for KPZ states in the stochastic six vertex modelhttps://zbmath.org/1534.820102024-06-14T15:52:26.737412Z"Nicoletti, Matthew"https://zbmath.org/authors/?q=ai:nicoletti.matthew"Petrov, Leonid"https://zbmath.org/authors/?q=ai:petrov.leonidSummary: We introduce a family of Markov growth processes on discrete height functions defined on the 2-dimensional square lattice. Each height function corresponds to a configuration of the six vertex model on the infinite square lattice. We focus on the stochastic six vertex model corresponding to a particular two-parameter family of weights within the ferroelectric \((\Delta > 1)\) regime. It is believed (and partially proven, see [\textit{A. Aggarwal}, Proc. Lond. Math. Soc. (3) 124, No. 3, 387--425 (2022; Zbl 1520.82011)]) that the stochastic six vertex model displays nontrivial pure (i.e., translation invariant and ergodic) Gibbs states of two types, KPZ and liquid. These phases have very different long-range correlation structures. The Markov processes we construct preserve the KPZ pure states in the full plane. We also show that the same processes put on the torus preserve arbitrary Gibbs measures for generic six vertex weights (not necessarily in the ferroelectric regime).
Our dynamics arise naturally from the Yang-Baxter equation for the six vertex model. Using the bijectivisation of the Yang-Baxter equation introduced in [\textit{A. Bufetov} and \textit{L. Petrov}, Forum Math. Sigma 7, Paper No. e39, 70 p. (2019; Zbl 1480.60285)], we first construct discrete time dynamics on six vertex configurations with a particular boundary condition, namely with the step initial condition in the quarter plane. Then we take a Poisson-type limit to obtain simpler continuous time dynamics. These dynamics are irreversible; in particular, the height function has a nonzero average drift. In each KPZ pure state, we explicitly compute the average drift (also known as the current) as a function of the slope. We use this to heuristically analyze the hydrodynamics of a non-stationary version of our process acting on quarter plane stochastic six vertex configurations.A note on checkable codes over Frobenius and quasi-Frobenius ringshttps://zbmath.org/1534.941322024-06-14T15:52:26.737412Z"de Araujo, R. R."https://zbmath.org/authors/?q=ai:de-araujo.robson-ricardo"Pereira, A. L. M."https://zbmath.org/authors/?q=ai:pereira.a-l-m"Polcino Milies, C."https://zbmath.org/authors/?q=ai:polcino-milies.cesarThis paper explores checkable codes over finite quasi-Frobenius rings. Some characterizations of code-checkable finite quasi-Frobenius rings are given. The main results are:
\begin{itemize}
\item the Jacobson radical of a quasi-Frobenius ring is checkable if and only if the ring is Frobenius
\item every maximal left ideal of a finite quasi-Frobenius ring is checkable
\item a quasi-Frobenius ring is code-checkable if and only if it is a principal right ideal ring.
\end{itemize}
For the entire collection see [Zbl 1518.16001].
Reviewer: Joël Kabore (Ouagadougou)