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Iterated Hopf Ore extensions in positive characteristic 迭代Hopf - Ore的正特征扩展
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2020-04-30 DOI: 10.4171/jncg/453
K. Brown, James J. Zhang
{"title":"Iterated Hopf Ore extensions in positive characteristic","authors":"K. Brown, James J. Zhang","doi":"10.4171/jncg/453","DOIUrl":"https://doi.org/10.4171/jncg/453","url":null,"abstract":"Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field k of positive characteristic p are studied. We show that every IHOE over k satisfies a polynomial identity, with PI-degree a power of p, and that it is a filtered deformation of a commutative polynomial ring. We classify all 2-step IHOEs over k, thus generalising the classification of 2-dimensional connected unipotent algebraic groups over k. Further properties of 2-step IHOEs are described: for example their simple modules are classified, and every 2-step IHOE is shown to possess a large Hopf center and hence an analog of the restricted enveloping algebra of a Lie k-algebra. As one of a number of questions listed, we propose that such a restricted Hopf algebra may exist for every IHOE over k.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42395207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
The strong homotopy structure of Poisson reduction Poisson约简的强同宗结构
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2020-04-22 DOI: 10.4171/jncg/455
C. Esposito, Andreas Kraft, Jonas Schnitzer
{"title":"The strong homotopy structure of Poisson reduction","authors":"C. Esposito, Andreas Kraft, Jonas Schnitzer","doi":"10.4171/jncg/455","DOIUrl":"https://doi.org/10.4171/jncg/455","url":null,"abstract":"In this paper we propose a reduction scheme for multivector fields phrased in terms of $L_infty$-morphisms. Using well-know geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to built up a suitable deformation retract of DGLA's. We first obtained an explicit formula for the $L_infty$-Projection and -Inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L_infty$-morphism. Finally, we perfom a comparison with other reduction procedures.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41428407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Cosimplicial monoids and deformation theory of tensor categories 张量范畴的同素单胚与变形理论
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2020-03-29 DOI: 10.4171/jncg/512
M. Batanin, A. Davydov
{"title":"Cosimplicial monoids and deformation theory of tensor categories","authors":"M. Batanin, A. Davydov","doi":"10.4171/jncg/512","DOIUrl":"https://doi.org/10.4171/jncg/512","url":null,"abstract":"We introduce a notion of $n$-commutativity ($0le nle infty$) for cosimplicial monoids in a symmetric monoidal category ${bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${bf V,}$ while $n=infty$ corresponds to commutative cosimplicial monoids. If ${bf V}$ has a monoidal model structure we show (under some mild technical conditions) that the total object of an $n$-cosimplicial monoid has a natural $E_{n+1}$-algebra structure. \u0000Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a $1$-commutative cosimplicial monoid and, hence, has an $E_2$-algebra structure similar to the $E_2$-structure on Hochschild complex of an associative algebra provided by Deligne's conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a $2$-commutative cosimplicial monoid and, therefore, is naturally an $E_3$-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47846788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
A variant of Roe algebras for spaces with cylindrical ends with applications in relative higher index theory 具有圆柱形端点空间的Roe代数的一个变体及其在相对高指标理论中的应用
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2020-03-18 DOI: 10.4171/jncg/457
Mehran Seyedhosseini
{"title":"A variant of Roe algebras for spaces with cylindrical ends with applications in relative higher index theory","authors":"Mehran Seyedhosseini","doi":"10.4171/jncg/457","DOIUrl":"https://doi.org/10.4171/jncg/457","url":null,"abstract":"In this paper we define a variant of Roe algebras for spaces with cylindrical ends and use this to study questions regarding existence and classification of metrics of positive scalar curvature on such manifolds which are collared on the cylindrical end. We discuss how our constructions are related to relative higher index theory as developed by Chang, Weinberger, and Yu and use this relationship to define higher rho-invariants for positive scalar curvature metrics on manifolds with boundary. This paves the way for classification of these metrics. Finally, we use the machinery developed here to give a concise proof of a result of Schick and the author, which relates the relative higher index with indices defined in the presence of positive scalar curvature on the boundary.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41822596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Finitely summable $gamma$-elements for word-hyperbolic groups 词-双曲群的有限可和$gamma$-元素
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2020-01-27 DOI: 10.4171/jncg/446
James M Cabrera, M. Puschnigg
{"title":"Finitely summable $gamma$-elements for word-hyperbolic groups","authors":"James M Cabrera, M. Puschnigg","doi":"10.4171/jncg/446","DOIUrl":"https://doi.org/10.4171/jncg/446","url":null,"abstract":"We present two explicit combinatorial constructions of finitely summable reduced \"Gamma\"-elements $gamma_r,in,KK(C^*_r(Gamma),{mathbb C})$ for any word-hyperbolic group $(Gamma,S)$ and obtain summability bounds for them in terms of the cardinality of the generating set $SsubsetGamma$ and the hyperbolicity constant of the associated Cayley graph.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42332592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids clwx2 -代数的同伦泊松代数、Maurer-Cartan元和Dirac结构
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2020-01-06 DOI: 10.4171/JNCG/398
Jiefeng Liu, Y. Sheng
{"title":"Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids","authors":"Jiefeng Liu, Y. Sheng","doi":"10.4171/JNCG/398","DOIUrl":"https://doi.org/10.4171/JNCG/398","url":null,"abstract":"In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin triple for split Lie 2-algebroids to be a CLWX 2-algebroid with two transversal strict Dirac structures. We show that there is a one-to-one correspondence between Manin triples for split Lie 2-algebroids and split Lie 2-bialgebroids. We further introduce the notion of a weak Dirac structure of a CLWX 2-algebroid and show that the graph of a Maurer-Cartan element of the homotopy Poisson algebra of degree 3 associated to a split Lie 2-bialgebroid is a weak Dirac structure. Various examples including the string Lie 2-algebra, split Lie 2-algebroids constructed from integrable distributions and left-symmetric algebroids are given.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47954536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models 计算模糊几何的谱作用:从随机非交换几何到双迹多矩阵模型
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2019-12-31 DOI: 10.4171/JNCG/482
C. I. Pérez-Sánchez
{"title":"Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models","authors":"C. I. Pérez-Sánchez","doi":"10.4171/JNCG/482","DOIUrl":"https://doi.org/10.4171/JNCG/482","url":null,"abstract":"A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion was introduced in [J. Barrett, J. Math. Phys. 56, 082301 (2015)] and accommodates familiar fuzzy spaces like spheres and tori. In the framework of random noncommutative geometry, we use Barrett's characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action $S(D)= mathrm{Tr} f(D)$ for $2n$-dimensional fuzzy geometries. In contrast to the original Chamseddine-Connes spectral action, we take a polynomial $f$ with $f(x)to infty$ as $ |x|toinfty$ in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type $S(D)=N cdot mathrm{tr}, F+textstylesum_i mathrm{tr},A_i cdot mathrm{tr} ,B_i $, being $F,A_i $ and $B_i $ noncommutative polynomials in $2^{2n-1}$ complex $Ntimes N$ matrices that parametrize the Dirac operator $D$. For arbitrary signature---thus for any admissible KO-dimension---formulas for 2-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials $F,A_i $ and $B_i$ are obtained via chord diagrams and satisfy: independence of $N$; self-adjointness of the main polynomial $F$ (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of $A_i $ and $B_i $ simultaneously, for fixed $i$. Collectively, this favors a free probabilistic perspective for the large-$N$ limit we elaborate on.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43324328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
Polarization and deformations of generalized dendriform algebras 广义树状代数的极化与变形
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2019-12-19 DOI: 10.4171/jncg/449
Cyrille Ospel, F. Panaite, P. Vanhaecke
{"title":"Polarization and deformations of generalized dendriform algebras","authors":"Cyrille Ospel, F. Panaite, P. Vanhaecke","doi":"10.4171/jncg/449","DOIUrl":"https://doi.org/10.4171/jncg/449","url":null,"abstract":"We generalize three results of M. Aguiar, which are valid for Loday's dendriform algebras, to arbitrary dendriform algebras, i.e., dendriform algebras associated to algebras satisfying any given set of relations. We define these dendriform algebras using a bimodule property and show how the dendriform relations are easily determined. An important concept which we use is the notion of polarization of an algebra, which we generalize here to (arbitrary) dendriform algebras: it leads to a generalization of two of Aguiar's results, dealing with deformations and filtrations of dendriform algebras. We also introduce weak Rota-Baxter operators for arbitrary algebras, which lead to the construction of generalized dendriform algebras and to a generalization of Aguiar's third result, which provides an interpretation of the natural relation between infinitesimal bialgebras and pre-Lie algebras in terms of dendriform algebras. Throughout the text, we give many examples and show how they are related.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41927183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Quadratic Lie conformal superalgebras related to Novikov superalgebras 与Novikov超代数相关的二次Lie共形超代数
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2019-12-09 DOI: 10.4171/JNCG/445
P. Kolesnikov, R. Kozlov, A. Panasenko
{"title":"Quadratic Lie conformal superalgebras related to Novikov superalgebras","authors":"P. Kolesnikov, R. Kozlov, A. Panasenko","doi":"10.4171/JNCG/445","DOIUrl":"https://doi.org/10.4171/JNCG/445","url":null,"abstract":"We study quadratic Lie conformal superalgebras associated with No-vikov superalgebras. For every Novikov superalgebra $(V,circ)$, we construct an enveloping differential Poisson superalgebra $U(V)$ with a derivation $d$ such that $ucirc v = ud(v)$ and ${u,v} = ucirc v - (-1)^{|u||v|} vcirc u$ for $u,vin V$. The latter means that the commutator Gelfand--Dorfman superalgebra of $V$ is special. Next, we prove that every quadratic Lie conformal superalgebra constructed on a finite-dimensional special Gel'fand--Dorfman superalgebra has a finite faithful conformal representation. This statement is a step toward a solution of the following open problem: whether a finite Lie conformal (super)algebra has a finite faithful conformal representation.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45211250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Doob equivalence and non-commutative peaking for Markov chains 马尔可夫链的Doob等价与非交换峰
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2019-11-23 DOI: 10.4171/jncg/444
Xinxin Chen, Adam Dor-On, Langwen Hui, C. Linden, Yifan Zhang
{"title":"Doob equivalence and non-commutative peaking for Markov chains","authors":"Xinxin Chen, Adam Dor-On, Langwen Hui, C. Linden, Yifan Zhang","doi":"10.4171/jncg/444","DOIUrl":"https://doi.org/10.4171/jncg/444","url":null,"abstract":"In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44903680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
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