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Journal articles on the topic 'Tannaka duality'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Bhatt, Bhargav, and Daniel Halpern-Leistner. "Tannaka duality revisited." Advances in Mathematics 316 (August 2017): 576–612. http://dx.doi.org/10.1016/j.aim.2016.08.040.

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2

Bhatt, Bhargav. "Algebraization and Tannaka duality." Cambridge Journal of Mathematics 4, no. 4 (2016): 403–61. http://dx.doi.org/10.4310/cjm.2016.v4.n4.a1.

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3

McCrudden, Paddy. "Tannaka duality for Maschkean categories." Journal of Pure and Applied Algebra 168, no. 2-3 (March 2002): 265–307. http://dx.doi.org/10.1016/s0022-4049(01)00099-8.

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4

Savin, Valentin. "Tannaka duality on quotient stacks." manuscripta mathematica 119, no. 3 (January 23, 2006): 287–303. http://dx.doi.org/10.1007/s00229-005-0616-8.

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5

Bui, Huu Hung. "Compact quantum groups and their corepresentations." Bulletin of the Australian Mathematical Society 57, no. 1 (February 1998): 73–91. http://dx.doi.org/10.1017/s0004972700031439.

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A compact quantum group is defined to be a unital Hopf C*–algebra generated by the matrix elements of a family of invertible corepresentations. We present a version of the Tannaka–Krein duality theorem for compact quantum groups in the context of abstract categories; this result encompasses the result of Woronowicz and the classical Tannaka-Krein duality theorem. We construct the orthogonality relations (similar to the case of compact groups). The Plancherel theorem is then established.

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6

Amini, Massoud. "Tannaka-Krein duality for compact groupoids II, duality." Operators and Matrices, no. 4 (2010): 573–92. http://dx.doi.org/10.7153/oam-04-32.

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7

Lu, Di-Ming. "TANNAKA DUALITY AND THE FRT-CONSTRUCTION." Communications in Algebra 29, no. 12 (January 1, 2001): 5717–31. http://dx.doi.org/10.1081/agb-100107955.

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8

Iwanari, Isamu. "Tannaka duality and stable infinity-categories." Journal of Topology 11, no. 2 (April 12, 2018): 469–526. http://dx.doi.org/10.1112/topo.12057.

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9

Hai, Phùng Hô. "Tannaka-Krein duality for Hopf algebroids." Israel Journal of Mathematics 167, no. 1 (October 2008): 193–225. http://dx.doi.org/10.1007/s11856-008-1047-5.

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10

Trentinaglia, Giorgio. "Tannaka duality for proper Lie groupoids." Journal of Pure and Applied Algebra 214, no. 6 (June 2010): 750–68. http://dx.doi.org/10.1016/j.jpaa.2009.08.004.

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11

Hall, Jack, and David Rydh. "Coherent Tannaka duality and algebraicity of Hom-stacks." Algebra & Number Theory 13, no. 7 (September 21, 2019): 1633–75. http://dx.doi.org/10.2140/ant.2019.13.1633.

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12

Pridham, Jonathan. "Tannaka duality for enhanced triangulated categories I: reconstruction." Journal of Noncommutative Geometry 14, no. 2 (July 13, 2020): 591–637. http://dx.doi.org/10.4171/jncg/374.

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13

kDurkdević, Mićo. "Quantum principal bundles and Tannaka-Krein duality theory." Reports on Mathematical Physics 38, no. 3 (December 1996): 313–24. http://dx.doi.org/10.1016/s0034-4877(97)84884-7.

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14

Malacarne, Sara. "Woronowicz Tannaka-Krein duality and free orthogonal quantum groups." MATHEMATICA SCANDINAVICA 122, no. 1 (February 20, 2018): 151. http://dx.doi.org/10.7146/math.scand.a-97320.

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Given a finite-dimensional Hilbert space $H$ and a collection of operators between its tensor powers satisfying certain properties, we give a short proof of the existence of a compact quantum group $G$ with a fundamental representation $U$ on $H$ such that the intertwiners between the tensor powers of $U$ coincide with the given collection of operators. We then explain how the general version of Woronowicz Tannaka-Krein duality can be deduced from this.

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15

Huang, Hua-Lin, Gongxiang Liu, Yuping Yang, and Yu Ye. "Finite quasi-quantum groups of diagonal type." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 759 (February 1, 2020): 201–43. http://dx.doi.org/10.1515/crelle-2017-0058.

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AbstractIn this paper, we give a classification of finite-dimensional radically graded elementary quasi-Hopf algebras of diagonal type, or equivalently, finite-dimensional coradically graded pointed Majid algebras of diagonal type. By a Tannaka–Krein type duality, this determines a big class of pointed finite tensor categories. Some efficient methods of construction are also given.

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16

Lyubinin, Anton. "Tannaka Duality, Coclosed Categories and Reconstruction for Nonarchimedean Bialgebras." Applied Categorical Structures 29, no. 3 (February 20, 2021): 547–71. http://dx.doi.org/10.1007/s10485-021-09632-2.

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17

Amini, Massoud. "Tannaka–Krein duality for compact groupoids I, Representation theory." Advances in Mathematics 214, no. 1 (September 2007): 78–91. http://dx.doi.org/10.1016/j.aim.2006.09.015.

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18

BANICA, TEODOR, and JULIEN BICHON. "HOPF IMAGES AND INNER FAITHFUL REPRESENTATIONS." Glasgow Mathematical Journal 52, no. 3 (August 25, 2010): 677–703. http://dx.doi.org/10.1017/s0017089510000510.

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AbstractWe develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.

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19

Woronowicz, S. L. "Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups." Inventiones Mathematicae 93, no. 1 (February 1988): 35–76. http://dx.doi.org/10.1007/bf01393687.

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20

Schäppi, Daniel. "Which abelian tensor categories are geometric?" Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 734 (January 1, 2018): 145–86. http://dx.doi.org/10.1515/crelle-2014-0053.

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AbstractFor a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding.However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties—expressible entirely within the language of tensor categories—which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne’s result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero.Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.

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21

Kaid, Almar, and Ralf Kasprowitz. "Semistable Vector Bundles and Tannaka Duality from a Computational Point of View." Experimental Mathematics 21, no. 2 (June 2012): 171–88. http://dx.doi.org/10.1080/10586458.2011.594665.

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22

Hofmann, Karl H., and Sidney A. Morris. "Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century." Axioms 10, no. 3 (August 17, 2021): 190. http://dx.doi.org/10.3390/axioms10030190.

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This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to RI×C for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra R[G] of a finite group G to its representation algebra R(G,R), via the natural duality of the topological vector space RI to the vector space R(I), for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups.

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23

Yettera, David N. "Quantum groups and representations of monoidal categories." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 2 (September 1990): 261–90. http://dx.doi.org/10.1017/s0305004100069139.

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This paper is intended to make explicit some aspects of the interactions which have recently come to light between the theory of classical knots and links, the theory of monoidal categories, Hopf-algebra theory, quantum integrable systems, the theory of exactly solvable models in statistical mechanics, and quantum field theories. The main results herein show an intimate relation between representations of certain monoidal categories arising from the study of new knot invariants or from physical considerations and quantum groups (that is, Hopf algebras). In particular categories of modules and comodules over Hopf algebras would seem to be much more fundamental examples of monoidal categories than might at first be apparent. This fundamental role of Hopf algebras in monoidal categories theory is also manifest in the Tannaka duality theory of Deligne and Mime [8a], although the relationship of that result and the present work is less clear than might be hoped.

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24

Daenzer, Calder. "Metric Tannakian duality." Journal of Geometry and Physics 70 (August 2013): 21–29. http://dx.doi.org/10.1016/j.geomphys.2013.03.008.

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25

Banica, Teodor. "Tannakian Duality for Affine Homogeneous Spaces." Canadian Mathematical Bulletin 61, no. 3 (September 1, 2018): 483–94. http://dx.doi.org/10.4153/cmb-2017-084-3.

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AbstractAssociated with any closed quantum subgroup and any index set I ⊂ {1,…,N} is a certain homogeneous space , called an affine homogeneous space. Using Tannakian duality methods, we discuss the abstract axiomatization of the algebraic manifolds that can appear in this way.

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26

Wedhorn, Torsten. "On Tannakian duality over valuation rings." Journal of Algebra 282, no. 2 (December 2004): 575–609. http://dx.doi.org/10.1016/j.jalgebra.2004.07.024.

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27

Duong, Nguyen Dai, and Phùng Hô Hai. "Tannakian duality over Dedekind rings and applications." Mathematische Zeitschrift 288, no. 3-4 (October 14, 2017): 1103–42. http://dx.doi.org/10.1007/s00209-017-1928-6.

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28

Bertrand, Daniel. "Extensions panachées autoduales." Journal of K-Theory 11, no. 2 (March 6, 2013): 393–411. http://dx.doi.org/10.1017/is013001030jkt213.

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AbstractWe study self-duality of Grothendieck's blended extensions in the context of a tannakian category. The set of equivalence classes of symmetric, resp. antisymmetric, blended extensions is naturally endowed with a torsor structure, which enables us to compute the unipotent radical of the associated monodromy groups in various situations.

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29

Phùng, Hô Hai. "On an injectivity lemma in the proof of Tannakian duality." Journal of Algebra and Its Applications 15, no. 09 (August 22, 2016): 1650167. http://dx.doi.org/10.1142/s021949881650167x.

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In this short work we give a very short and elementary proof of the injectivity lemma, which plays an important role in the Tannakian duality for Hopf algebras over a field. Based on this we provide some generalizations of this fact to the case of flat coalgebras over a Noetherian domain.

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30

Papanikolas, Matthew A. "Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms." Inventiones mathematicae 171, no. 1 (September 7, 2007): 123–74. http://dx.doi.org/10.1007/s00222-007-0073-y.

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31

Stalder, Nicolas. "The semisimplicity conjecture for A-motives." Compositio Mathematica 146, no. 3 (March 18, 2010): 561–98. http://dx.doi.org/10.1112/s0010437x09004448.

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AbstractWe prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

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32

Esnault, H., and P. H. Hai. "The Gauss-Manin connection and Tannaka duality." International Mathematics Research Notices, January 1, 2006. http://dx.doi.org/10.1155/imrn/2006/93978.

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33

De Commer, Kenny, and Makoto Yamashita. "Tannaka–Kreĭn duality for compact quantum homogeneous spaces II. Classification of quantum homogeneous spaces for quantum SU(2)." Journal für die reine und angewandte Mathematik (Crelles Journal) 2015, no. 708 (January 1, 2015). http://dx.doi.org/10.1515/crelle-2013-0074.

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AbstractWe apply the Tannaka–Kreĭn duality theory for quantum homogeneous spaces, developed in the first part of this series of papers, to the case of the quantum SU(2) groups. We obtain a classification of their quantum homogeneous spaces in terms of weighted oriented graphs. The equivariant maps between these quantum homogeneous spaces can be characterized by certain quadratic equations associated with the braiding on the representations of

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34

Kalmykov, Artem, and Pavel Safronov. "A categorical approach to dynamical quantum groups." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.68.

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Abstract We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups $F(G)$ and $F_q(G)$ are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group.

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