Recent zbMATH articles in MSC 18https://zbmath.org/atom/cc/182021-01-08T12:24:00+00:00WerkzeugRealization of rigid \(C^*\)-tensor categories via Tomita bimodules.https://zbmath.org/1449.460482021-01-08T12:24:00+00:00"Giorgetti, Luca"https://zbmath.org/authors/?q=ai:giorgetti.luca"Yuan, Wei"https://zbmath.org/authors/?q=ai:yuan.weiLet \({\mathfrak{A}}\) be a von Neumann algebra, and \(H\) denotes a pre-Hilbert \({\mathfrak{A}}$-${\mathfrak{A}}\) bimodule. \(\Omega \in H\) is a distinguished vacuum vector such that \(\langle \Omega \vert \Omega \rangle_{ {\mathfrak{A}}} = I\) and \(A \cdot \Omega\) \(=\) \(\Omega \cdot A\) for every \(A \in {\mathfrak{A}}\). When \({\mathfrak{A}}\) is a semifinite von Neumann algebra and \(\tau\) is a normal semifinite faithful tracial weight on \({\mathfrak{A}}\), then \[ {\mathfrak{N}}({\mathfrak{A}}, \tau) = \{ A \in {\mathfrak{A}}; \, \tau( A^* A) < \infty \, \} \] is an ideal. \(H\) is said to be a Tomita \({\mathfrak{A}}$-${\mathfrak{A}}\) bimodule if (i) \({\mathfrak{N}}( H, \tau)\) \(=\) \(H\); (ii) \(H\) admits an involution \(S\) such that \(S( A \cdot \zeta \cdot B)\) \(=\) \(B^* \cdot S(\zeta) \cdot A^*\) for every \(\zeta\in H\) and \(A, B \in {\mathfrak{A}}\); (iii) \(H\) admits a complex one-parameter group \(\{ {\mathcal{U}}(\alpha); \, \alpha \in {\mathbb{C}} \, \}\) of linear isomorphisms satisfying some intertwining properties between \(S\) and \({\mathcal{U}}\). \({\mathcal{B}}(H)\) is the set of bounded adjointable linear mappings from \(H\) to \(H\). \({\mathcal{F}}(H)\) is a Fock space in Voiculescu's free probability theory, associated to a Tomita \({\mathfrak{A}}$-${\mathfrak{A}}\) bimodule \(H\). Note that \({\mathcal{F}}(H)\) is a pre-Hilbert \({\mathfrak{A}}$-${\mathfrak{A}}\) bimodule. Let \(\Phi(H)\) be the \(*\)-subalgebra of \({\mathcal{B}}( {\mathcal{F}}(H))\) generated by \({\mathfrak{A}}\) acting on \({\mathcal{F}}(H)\) from the left, and let \(\Phi(H)''\) be the von Neumann algebra generated by \(\pi_{\tau}( \Phi(H))\) on the Hilbert space completion \({\mathcal{F}}(H)_{\tau}\), and let \(\Phi(H)'\) be the commutant of \(\Phi(H)''\). Note that \(\Phi(H) \Omega ={\mathcal{F}}(H)\).
This paper treats a construction of von Neumann algebras \(\Phi(H)''\), starting from a rigid \(C^*\)-tensor category \({\mathcal{C}}\) with simple unit. Actually, these algebras \(\Phi(H)''\) are factors of type II or type III$_{\lambda}\) with \(\lambda \in (0, 1]\). As a matter of fact, the choice of type is tuned by the choice of Tomita structure on certain bimodules of which the authors make use in the construction. In the special case where the spectrum is infinite, the whole tensor category can be realized as endomorphisms of these algebras. Furthermore, it is clarified that, if the Tomita structure is trivial, the algebras obtained in this paper are amplifications of the free group factors with infinitely many generators.
Reviewer: Isamu Dôku (Saitama)Injective hulls of \(L\)-ordered algebras.https://zbmath.org/1449.060292021-01-08T12:24:00+00:00"Tian, Tian"https://zbmath.org/authors/?q=ai:tian.tian.2|tian.tian.1|tian.tian"Zhao, Bin"https://zbmath.org/authors/?q=ai:zhao.bin.1|zhao.binSummary: We discussed the injectivity of the category of \(L\)-ordered algebras, and proved that \(\mathrm{M}^\le\)-injective objects in the category of \(L\)-ordered algebras were exactly \(L\)-sup-algebras, and gave the concrete form of \(\mathrm{M}^\le\)-injective hulls of a special class of \(L\)-ordered algebras in the category of \(L\)-ordered algebras.The category of upper bounded bifinite posets.https://zbmath.org/1449.060122021-01-08T12:24:00+00:00"Li, Jibo"https://zbmath.org/authors/?q=ai:li.jibo"Chen, Yanchang"https://zbmath.org/authors/?q=ai:chen.yanchang"Zhang, Haixia"https://zbmath.org/authors/?q=ai:zhang.haixiaSummary: In this paper, some results about \(\mathcal{D}\)-precontinuous or \(\mathcal{D}\)-prealgebraic posets and \({\mathcal{D}^\Delta}\)-continuous functions are summarized and supplemented. The category BFBP, in which objects are upper bounded bifinite posets and arrows are \({\mathcal{D}^\Delta}\)-continuous functions between them, is shown to be cCartesian closed.Relative and generalized Tate cohomology with respect to balanced pairs.https://zbmath.org/1449.180032021-01-08T12:24:00+00:00"Zhang, Chunxia"https://zbmath.org/authors/?q=ai:zhang.chunxiaSummary: The relative and generalized Tate cohomology with respect to balanced pairs are studied. An Avramov-Martsinkovsky type exact sequence is obtained.Localization of \(n\)-angulated categories.https://zbmath.org/1449.180022021-01-08T12:24:00+00:00"Wang, Minxiong"https://zbmath.org/authors/?q=ai:wang.minxiong"Zheng, Yan"https://zbmath.org/authors/?q=ai:zheng.yanSummary: In this paper, we study localization theory of \(n\)-angulated categories. By using localization method, an \(n\)-angulated category \(K\) and a compatible localizing class \(S\) of \(K\), the quotient category \({S^{-1}}K\) is constructed. In the new category, objects are the same as the original category and the elements of \(S\) become isomorphisms. Furthermore, the quotient category has an \(n\)-angulated structure and satisfies certain universal property, which generalizes the corresponding results in triangulated categories.Some properties of the \(\mathcal{M}_R^l (\Omega)\) category.https://zbmath.org/1449.160142021-01-08T12:24:00+00:00"Geng, Jun"https://zbmath.org/authors/?q=ai:geng.jun"Tang, Jiangang"https://zbmath.org/authors/?q=ai:tang.jiangangSummary: In this paper, we research the coequalizer and the coproduct in the category of \(\Omega\)-left-\(R\)-modules, and give the relationship of coproduct and coequalizer between the \(\Omega\)-left-\(R\)-modules category and the left-\(R\)-modules category. Furthermore, we prove that the category of \(\Omega\)-left-\(R\)-modules is a cocomplete category.The Schur multiplier and stem covers of Leibniz \(n\)-algebras.https://zbmath.org/1449.170072021-01-08T12:24:00+00:00"Casas, José Manuel"https://zbmath.org/authors/?q=ai:casas-miras.jose-manuel"Insua, Manuel Avelino"https://zbmath.org/authors/?q=ai:insua.manuel-avelino"rego, Natália Pacheco"https://zbmath.org/authors/?q=ai:rego.natalia-pachecoLeibniz \(n\)-algebras are the non-skewsymmetric version of \(n\)-Lie algebras. The authors study relations between the Schur multipliers and stem extensions of Leibniz \(n\)-algebras. In particular, they prove that every stem extension of a Leibniz \(n\)-algebra is an epimorphic image of a stem cover and that any two stem covers are isomorphic.
Reviewer: Norbert Knarr (Stuttgart)The \(R\)-matrix of bimonad distributive law.https://zbmath.org/1449.180062021-01-08T12:24:00+00:00"Guo, Shuangjian"https://zbmath.org/authors/?q=ai:guo.shuangjian"Zhang, Xiaohui"https://zbmath.org/authors/?q=ai:zhang.xiaohuiSummary: The aim of this paper is to define and study the \(R\)-matrix of a bimonad distributive law. Assume that \(F\) and \(G\) are bimonads, we give necessary and sufficient conditions for \({\mathcal{C}_{ (F, G)}} (\varphi)\), the category of \(F, G\)-bimodules, to be a braided monoidal category.Clean exactness and derived categories.https://zbmath.org/1449.130142021-01-08T12:24:00+00:00"Li, Ruiting"https://zbmath.org/authors/?q=ai:li.ruiting"Yang, Gang"https://zbmath.org/authors/?q=ai:yang.gang"Wang, Xiaoqing"https://zbmath.org/authors/?q=ai:wang.xiaoqingSummary: Over a commutative ring, the notions of clean exactness and clean derived categories are introduced, the equivalent characterizations of clean exactness for short exact sequences and exact complexes are given, and the properties of clean derived categories are investigated. In particular, it is proved that bounded clean derived categories can be realized as certain homotopy categories.Higher algebraic \(K\)-theory and representations of algebraic groups.https://zbmath.org/1449.190022021-01-08T12:24:00+00:00"Kuku, Aderemi"https://zbmath.org/authors/?q=ai:kuku.aderemi-oSummary: This paper is concerned with Higher Algebraic \(K\)-theory and actions of algebraic groups \(G\) on such `nice' categories as the category of algebraic vector bundles on a scheme \(X\). Such `nice' categories are examples of `exact' categories with the observation that the category of actions on \(G\) on such exact categories also form an exact category called equivariant exact categories on which one can do higher Algebraic \(K\)-theory (of Quillen) called equivariant higher Algebraic \(K\)-theory -- the higher dimensional generalizations of classical equivariant \(K\)-theory which belongs to the field of representation theory. Thus, for an Algebraic group \(G\) over a number field or \(p\)-adic field \(F\), we present constructions and computations of equivariant higher \(K\)-groups as well as `profinite' or `continuous' higher \(K\)-groups for some \(G\)-Scheme \(X\). In particular, we present explicit l-completeness (l a rational prime) and finiteness computations for higher \(K\)-groups and profinite higher \(K\)-groups for twisted flag varieties.Buchweitz theorem in pure singularity category.https://zbmath.org/1449.180042021-01-08T12:24:00+00:00"Cao, Tianya"https://zbmath.org/authors/?q=ai:cao.tianya"Liu, Zhongkui"https://zbmath.org/authors/?q=ai:liu.zhong-kui"Yang, Xiaoyan"https://zbmath.org/authors/?q=ai:yang.xiaoyanSummary: We define the pure singularity category \({\text bf{D}}_{{\mathrm{psg}}}^b\left(R \right)\) as the Verdier quotient of the bounded pure derived category \({\text bf{D}}_{{\mathrm{pur}}}^b\left(R \right)\) by the triangulated subcategory \({\text bf{K}^b}\left({\mathcal{P} \mathcal{P}} \right)\) of the bounded homotopy category consisting of pure projective modules, a sufficient and necessary condition under which \({\text bf{D}}_{{\mathrm{psg}}}^b\left(R \right)\) is equivalent to the stable category of the Gorenstein category \({\mathcal{G}\left({\mathcal{P} \mathcal{P}} \right)}\) of pure projective modules is given. Moreover, we give a sufficient condition for the triangle-equivalence \({\text bf{D}}_{{\mathrm{psg}}}^b\left(R \right) \cong {\text bf{D}}_{{\mathrm{psg}}}^b\left(S \right)\), where \(R\) and \(S\) are rings.The axioms of choice in fuzzy category \(\mathbf{fuz}\).https://zbmath.org/1449.030172021-01-08T12:24:00+00:00"Wang, Xiaoyang"https://zbmath.org/authors/?q=ai:wang.xiaoyang"Wang, Baoshan"https://zbmath.org/authors/?q=ai:wang.baoshan"Wang, Yongjun"https://zbmath.org/authors/?q=ai:wang.yongjun"Zhou, Heng"https://zbmath.org/authors/?q=ai:zhou.hengSummary: Topos structure is a fundamental tool to describe set theory in category theory. There are two forms of axioms, AC1 and AC2, to express the choice axioms in topos. The category \(\mathbf{fuz}\) does not form a topos, it belongs to a structure called weak topos. Two axioms of choice WAC1 and WAC2 are equivalent in \(\mathbf{fuz}\). We strengthen these axioms as WAC1' and WAC2', then give the equivalence of such axioms. Finally, we put forward GWAC1' and GWAC2' as two forms of axioms of choice in weak topos.On the quotient category of the module category of a finite group and its equivalence.https://zbmath.org/1449.160152021-01-08T12:24:00+00:00"Huang, Wenlin"https://zbmath.org/authors/?q=ai:huang.wenlinSummary: In this paper, we find that the class of the homomorphisms which can be decomposed by a \(p\)-divisible module is an ideal of the module category of a finite group. We construct the quotient category of this module category, analyze the zero objects and obtain three equivalence functors of this quotient category.Quillen equivalence of singular model categories.https://zbmath.org/1449.180072021-01-08T12:24:00+00:00"Ren, Wei"https://zbmath.org/authors/?q=ai:ren.wei.3|ren.wei.2|ren.wei.1|ren.wei.5|ren.wei.4Summary: Let \(R\) be a left-Gorenstein ring. We construct a Quillen equivalence between singular contraderived model category and singular coderived model category. As an application, we explicitly give an equivalence \({\text bf{K}}_{\mathrm{ex}} (\mathcal{P}) \simeq {\text bf{K}}_{\mathrm{ex}} (\mathcal{I})\) for the homotopy categories of exact complexes of projective and injective modules.The monad of \(\Omega\)-\(\mathrm{Cat}\) category.https://zbmath.org/1449.180012021-01-08T12:24:00+00:00"Zhao, Na"https://zbmath.org/authors/?q=ai:zhao.na"Lu, Jing"https://zbmath.org/authors/?q=ai:lu.jingSummary: In this paper, we construct a monad of \(\Omega\)-\(\mathrm{Cat}\) category, and prove that \(\Omega\)-\(\mathrm{Poset}\) category is isomorphic to \(\Omega\)-\(\mathrm{Cat}^T\) category.Relative \(d\)-rigid subcategories in \(d\)-cluster category.https://zbmath.org/1449.180052021-01-08T12:24:00+00:00"Chen, Juan"https://zbmath.org/authors/?q=ai:chen.juan"Shuai, Yulin"https://zbmath.org/authors/?q=ai:shuai.yulin"Xie, Yunli"https://zbmath.org/authors/?q=ai:xie.yunliSummary: We defined the relative \(d\)-rigid subcategories in \(d\)-cluster categories, and proved that the corresponding relative \(d\)-rigid subcategories were equivalent to rigid subcategories for given \(d\)-rigid subcategories in \(d\)-cluster categories.