{"title":"The 3-cyclic quantum Weyl algebras, their prime spectra and a classification of simple modules ($q$ is not a root of unity)","authors":"V. Bavula","doi":"10.4171/jncg/547","DOIUrl":"https://doi.org/10.4171/jncg/547","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"62 12","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139381815","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}
We introduce a slight modification of the usual equivariant $KK$-theory. We use this to give a $KK$-theoretical proof of an equivariant index theorem for Dirac-Schrodinger operators on a non-compact manifold of nowhere positive curvature. We incidentally show that the boundary of Dirac is Dirac; generalizing earlier work of Baum and coworkers, and a result of Higson and Roe.
{"title":"A short proof of an index theorem, II","authors":"Yavar Abdolmaleki, Dan Kucerovsky","doi":"10.4171/jncg/524","DOIUrl":"https://doi.org/10.4171/jncg/524","url":null,"abstract":"We introduce a slight modification of the usual equivariant $KK$-theory. We use this to give a $KK$-theoretical proof of an equivariant index theorem for Dirac-Schrodinger operators on a non-compact manifold of nowhere positive curvature. We incidentally show that the boundary of Dirac is Dirac; generalizing earlier work of Baum and coworkers, and a result of Higson and Roe.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"9 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134954579","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}
M. J. H. Al-Kaabi, K. Ebrahimi-Fard, D. Manchon, H. Z. Munthe-Kaas
Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi bracket of vector fields) and a magmatic product without any compatibility relations between them. In this work we will show that an affine connection with curvature and torsion always gives rise to a post-Lie algebra as well as a $D$-algebra. The notions of torsion and curvature together with Gavrilov's special polynomials and double exponential are revisited in this post-Lie algebraic framework. We unfold the relations between the post-Lie Magnus expansion, the Grossman-Larson product and the $K$-map, $alpha$-map and $beta$-map, three particular functions introduced by Gavrilov with the aim of understanding the geometric and algebraic properties of the double-exponential, which can be understood as a geometric variant of the Baker-Campbell-Hausdorff formula. We propose a partial answer to a conjecture by Gavrilov, by showing that a particular class of geometrically special polynomials is generated by torsion and curvature. This approach unlocks many possibilities for further research such as numerical integrators and rough paths on Riemannian manifolds.
理解具有一般仿射连接的流形的代数结构是一个很自然的问题。在这种情况下，a . V. Gavrilov引入了框架李代数的概念，它由一个李括号(向量场通常的雅可比括号)和一个岩浆积组成，两者之间没有任何相容关系。在这项工作中，我们将证明具有曲率和扭转的仿射连接总是产生后李代数以及$D$ -代数。在这个后李代数框架中，我们重新讨论了扭转和曲率的概念以及加夫里洛夫的特殊多项式和二重指数。我们揭示了后lie Magnus展开、Grossman-Larson积和$K$ -map、$alpha$ -map和$beta$ -map之间的关系，这三个特殊的函数是由Gavrilov引入的，目的是理解双指数的几何和代数性质，它可以被理解为Baker-Campbell-Hausdorff公式的几何变化。通过证明一类几何上特殊的多项式是由扭转和曲率产生的，我们给出了对加夫里洛夫猜想的部分回答。这种方法为进一步研究数值积分器和黎曼流形上的粗糙路径开辟了许多可能性。
{"title":"Algebraic aspects of connections: From torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials","authors":"M. J. H. Al-Kaabi, K. Ebrahimi-Fard, D. Manchon, H. Z. Munthe-Kaas","doi":"10.4171/jncg/539","DOIUrl":"https://doi.org/10.4171/jncg/539","url":null,"abstract":"Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi bracket of vector fields) and a magmatic product without any compatibility relations between them. In this work we will show that an affine connection with curvature and torsion always gives rise to a post-Lie algebra as well as a $D$-algebra. The notions of torsion and curvature together with Gavrilov's special polynomials and double exponential are revisited in this post-Lie algebraic framework. We unfold the relations between the post-Lie Magnus expansion, the Grossman-Larson product and the $K$-map, $alpha$-map and $beta$-map, three particular functions introduced by Gavrilov with the aim of understanding the geometric and algebraic properties of the double-exponential, which can be understood as a geometric variant of the Baker-Campbell-Hausdorff formula. We propose a partial answer to a conjecture by Gavrilov, by showing that a particular class of geometrically special polynomials is generated by torsion and curvature. This approach unlocks many possibilities for further research such as numerical integrators and rough paths on Riemannian manifolds.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"105 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136352847","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}
We apply computations of twisted Hodge diamonds to construct an infinite number of non-Fourier-Mukai functors with well behaved target and source spaces. To accomplish this we first study the characteristic morphism in order to control it for tilting bundles. Then we continue by applying twisted Hodge diamonds of hypersurfaces embedded in projective space to compute the Hochschild dimension of these spaces. This allows us to compute the kernel of the embedding into the projective space in Hochschild cohomology. Finally we use the above computations to apply the construction by A. Rizzardo, M. Van den Bergh and A. Neeman of non-Fourier-Mukai functors and verify that the constructed functors indeed cannot be Fourier-Mukai for odd dimensional quadrics. Using this approach we prove that there are a large number of Hochschild cohomology classes that can be used for this type construction. Furthermore, our results allow the application of computer-based calculations to construct candidate functors for arbitrary degree hypersurfaces in arbitrary high dimensions. Verifying that these are not Fourier-Mukai still requires the existence of a tilting bundle. In particular we prove that there is at least one non-Fourier-Mukai functor for every odd dimensional smooth quadric.
{"title":"Twisted Hodge Diamonds give rise to non-Fourier–Mukai functors","authors":"Felix Küng","doi":"10.4171/jncg/543","DOIUrl":"https://doi.org/10.4171/jncg/543","url":null,"abstract":"We apply computations of twisted Hodge diamonds to construct an infinite number of non-Fourier-Mukai functors with well behaved target and source spaces. To accomplish this we first study the characteristic morphism in order to control it for tilting bundles. Then we continue by applying twisted Hodge diamonds of hypersurfaces embedded in projective space to compute the Hochschild dimension of these spaces. This allows us to compute the kernel of the embedding into the projective space in Hochschild cohomology. Finally we use the above computations to apply the construction by A. Rizzardo, M. Van den Bergh and A. Neeman of non-Fourier-Mukai functors and verify that the constructed functors indeed cannot be Fourier-Mukai for odd dimensional quadrics. Using this approach we prove that there are a large number of Hochschild cohomology classes that can be used for this type construction. Furthermore, our results allow the application of computer-based calculations to construct candidate functors for arbitrary degree hypersurfaces in arbitrary high dimensions. Verifying that these are not Fourier-Mukai still requires the existence of a tilting bundle. In particular we prove that there is at least one non-Fourier-Mukai functor for every odd dimensional smooth quadric.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" 47","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135341404","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}
{"title":"Poisson–Lie group structures on semidirect products","authors":"Floris Elzinga, Makoto Yamashita","doi":"10.4171/jncg/538","DOIUrl":"https://doi.org/10.4171/jncg/538","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"33 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432270","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}
{"title":"Representation of commutators on Schatten $p$-classes","authors":"Lixin Cheng, Zhizheng Yu","doi":"10.4171/jncg/542","DOIUrl":"https://doi.org/10.4171/jncg/542","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"34 23‐24","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432528","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}
We examine the structure of two possible candidates of isometry groups for the spectral triples on $AF$-algebras introduced by Christensen and Ivan. In particular, we completely determine the isometry group introduced by Park, and observe that these groups coincide in the case of the Cantor set. We also show that the construction of spectral triples on crossed products given by Hawkins, Skalski, White and Zacharias, is suitable for the purpose of lifting isometries.
{"title":"On isometries of spectral triples associated to $mathrm{AF}$-algebras and crossed products","authors":"Jacopo Bassi, Roberto Conti","doi":"10.4171/jncg/535","DOIUrl":"https://doi.org/10.4171/jncg/535","url":null,"abstract":"We examine the structure of two possible candidates of isometry groups for the spectral triples on $AF$-algebras introduced by Christensen and Ivan. In particular, we completely determine the isometry group introduced by Park, and observe that these groups coincide in the case of the Cantor set. We also show that the construction of spectral triples on crossed products given by Hawkins, Skalski, White and Zacharias, is suitable for the purpose of lifting isometries.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"111 4‐6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818392","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}
We consider Deaconu--Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2-cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions.
{"title":"Simplicity of twisted C*-algebras of Deaconu–Renault groupoids","authors":"Becky Armstrong, Nathan Brownlowe, Aidan Sims","doi":"10.4171/jncg/527","DOIUrl":"https://doi.org/10.4171/jncg/527","url":null,"abstract":"We consider Deaconu--Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2-cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"4 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136017908","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}
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