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El. knyga: Quantum Isometry Groups

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This book offers an up-to-date overview of the recently proposed theory of quantum isometry groups. Written by the founders, it is the first book to present the research on the “quantum isometry group”, highlighting the interaction of noncommutative geometry and quantum groups, which is a noncommutative generalization of the notion of group of isometry of a classical Riemannian manifold. The motivation for this generalization is the importance of isometry groups in both mathematics and physics. The framework consists of Alain Connes’ “noncommutative geometry” and the operator-algebraic theory of “quantum groups”. The authors prove the existence of quantum isometry group for noncommutative manifolds given by spectral triples under mild conditions and discuss a number of methods for computing them. One of the most striking and profound findings is the non-existence of non-classical quantum isometry groups for arbitrary classical connected compact manifolds and, by using this, the authors explicitly describe quantum isometry groups of most of the noncommutative manifolds studied in the literature. Some physical motivations and possible applications are also discussed.

Recenzijos

Several examples and applications are provided, including an explicit description of the quantum isometry groups of most of the noncommutative Riemannian manifolds studied in literature. Introductory sections on the basics of Riemannian geometry noncommutative geometry and quantum groups make the book enjoyable to readers with no previous expertise in the subject. (Domenico Fiorenza, Mathematical Reviews, April 2018)

The book gives an up-to date of the results of an analogue of the group of isometries in the framework of noncommutative geometry and quantum groups. A unique feature of this book is the emphasis on the interaction of C-star algebraic compact quantum groups .The book is an interesting book for the researchers working in this area which is still hot enough. The physical motivations and possible applications given in this book are more interesting examples. (Ahmed Hegazi, zbMATH 1381.81004, 2018)

1 Preliminaries
1(36)
1.1 Operator Algebras and Hilbert Modules
1(7)
1.1.1 C* Algebras
1(3)
1.1.2 Von Neumann Algebras
4(1)
1.1.3 Free Product and Tensor Product
5(1)
1.1.4 Hilbert Modules
6(2)
1.2 Quantum Groups
8(13)
1.2.1 Hopf Algebras
8(2)
1.2.2 Compact Quantum Groups: Basic Definitions and Examples
10(7)
1.2.3 The CQG Uμ(2)
17(1)
1.2.4 The CQG SUμ(2)
18(1)
1.2.5 The Hopf *-algebras O(SUμ(2)) and Uμ(su(2))
19(1)
1.2.6 The CQG SOμ(3)
20(1)
1.3 Coaction of Compact Quantum Groups on a C* Algebra
21(8)
1.3.1 Coactions on Finite Quantum Spaces
22(3)
1.3.2 Free and Half-Liberated Quantum Groups
25(2)
1.3.3 The Coaction of SOμ(3) on the Podles' Spheres
27(2)
1.4 Dual of a Compact Quantum Group
29(1)
1.5 Coaction on von Neumann Algebras by Conjugation of Unitary Corepresentation
30(7)
References
33(4)
2 Classical and Noncommutative Geometry
37(32)
2.1 Classical Riemannian Geometry
37(13)
2.1.1 Forms and Connections
37(1)
2.1.2 The Hodge Laplacian of a Riemannian Manifold
38(1)
2.1.3 Spin Groups and Spin Manifolds
39(1)
2.1.4 Dirac Operators
40(1)
2.1.5 Isometry Groups of Classical Manifolds
41(9)
2.2 Noncommutative Geometry
50(8)
2.2.1 Spectral Triples: Definition and Examples
50(3)
2.2.2 The Noncommutative Space of Forms
53(3)
2.2.3 Laplacian in Noncommutative Geometry
56(2)
2.3 Quantum Group Equivariance in Noncommutative Geometry
58(11)
2.3.1 The Example of SUμ(2)
58(1)
2.3.2 The Example of the Podles' Spheres
58(2)
2.3.3 Constructions from Coactions by Quantum Isometries
60(3)
2.3.4 R-twisted Volume Form Coming from the Modularity of a Quantum Group
63(3)
References
66(3)
3 Definition and Existence of Quantum Isometry Groups
69(28)
3.1 The Approach Based on Laplacian
69(8)
3.1.1 The Definition and Existence of the Quantum Isometry Group
70(5)
3.1.2 Discussions on the Admissibility Conditions
75(2)
3.2 Definition and Existence of the Quantum Group of Orientation Preserving Isometries
77(13)
3.2.1 Motivation
77(1)
3.2.2 Quantum Group of Orientation-Preserving Isometries of an R-twisted Spectral Triple
78(5)
3.2.3 Stability and C* Coaction
83(3)
3.2.4 Comparison with the Approach Based on Laplacian
86(4)
3.3 The Case of J Preserving Quantum Isometries
90(2)
3.4 A Sufficient Condition for Existence of Quantum Isometry Groups Without Fixing the Volume Form
92(5)
References
95(2)
4 Quantum Isometry Groups of Classical and Quantum Spheres
97(32)
4.1 Classical Spheres: No Quantum Isometries
97(2)
4.2 Quantum Isometry Group of a Spectral Triple on Podles' Sphere
99(1)
4.3 Descriptions of the Podles' Spheres
100(2)
4.3.1 The Description as in [ 3]
100(1)
4.3.2 `Volume Form' on the Podles' Spheres
101(1)
4.4 Computation of the Quantum Isometry Groups
102(12)
4.4.1 Affineness of the Coaction
104(4)
4.4.2 Homomorphism Conditions
108(1)
4.4.3 Relations Coming from the Antipode
109(2)
4.4.4 Identification of SOμ(3) as the Quantum Isometry Group
111(3)
4.5 Another Spectral Triple on the Podles' Sphere: A Counterexample
114(15)
4.5.1 The Spectral Triple
115(2)
4.5.2 Computation of the Quantum Isometry Group
117(10)
References
127(2)
5 Quantum Isometry Groups of Discrete Quantum Spaces
129(20)
5.1 Quantum Isometry Groups of Finite Metric Spaces and Finite Graphs
130(6)
5.1.1 The Works of Banica [ 3] and Bichon [ 2]
130(1)
5.1.2 Noncommutative Geometry on Finite Metric Spaces
131(2)
5.1.3 Quantum Symmetry Groups of Banica and Bichon as Quantum Isometry Groups
133(3)
5.2 Quantum Isometry Groups for Inductive Limits
136(13)
5.2.1 Examples Coming from AF Algebras
138(5)
5.2.2 The Example of the Middle-Third Cantor Set
143(3)
References
146(3)
6 Nonexistence of Genuine Smooth CQG Coactions on Classical Connected Manifolds
149(14)
6.1 Smooth Coaction of a Compact Quantum Group and the No-Go Conjecture
149(6)
6.1.1 Definition of Smooth Coaction
149(1)
6.1.2 Statement of the Conjecture and Some Positive Evidence
150(3)
6.1.3 Defining the `Differential' of the Coaction
153(2)
6.2 Brief Sketch of Proof of Nonexistence of Genuine Quantum Isometries
155(3)
6.3 An Example of No-Go Result Without Quadratic Independence
158(5)
References
161(2)
7 Deformation of Spectral Triples and Their Quantum Isometry Groups
163(16)
7.1 Cocycle Twisting
163(5)
7.1.1 Cocycle Twist of a Compact Quantum Group
164(2)
7.1.2 Unitary Corepresentations of a Twisted Compact Quantum Group
166(1)
7.1.3 Deformation of a von Neumann Algebra by Dual Unitary 2-Cocycles
167(1)
7.2 Deformation of Spectral Triples by Unitary Dual Cocycles
168(1)
7.3 Quantum Isometry Groups of Deformed Spectral Triples
169(3)
7.4 Examples and Computations
172(7)
References
177(2)
8 Spectral Triples and Quantum Isometry Groups on Group C*-Algebras
179(20)
8.1 Connes' Spectral Triple on Group C*-Algebras and Their Quantum Isometry Groups
180(3)
8.1.1 Quantum Isometry Groups of (C[ Γ], l2(Γ), DΓ)
181(2)
8.2 The Case of Finitely Generated Abelian Groups
183(7)
8.2.1 Computation for the Groups Zn and Z
183(5)
8.2.2 Results for the General Case
188(2)
8.3 The Case of Free Products of Groups
190(3)
8.3.1 Some Quantum Groups
190(2)
8.3.2 Results for the Free Groups Fn
192(1)
8.3.3 Quantum Isometry Groups of Free Product of Finite Cyclic Groups
192(1)
8.4 Quantum Isometry Groups as Doublings
193(6)
8.4.1 Result for a Generating Set of Transpositions
194(3)
8.4.2 The Case When S Has a Cycle
197(1)
References
197(2)
9 An Example of Physical Interest
199(22)
9.1 Notations and Preliminaries
201(2)
9.1.1 Generalities on Real C* Algebras
201(1)
9.1.2 Quantum Isometries
202(1)
9.2 The Finite Noncommutative Space F
203(3)
9.2.1 The Elementary Particles and the Hilbert Space of Fermions
203(1)
9.2.2 The Spectral Triple
204(1)
9.2.3 A Hypothesis on the Υ Matrices
205(1)
9.3 Quantum Isometries of F
206(5)
9.3.1 QISO+j in two special cases
209(1)
9.3.2 Quantum Isometries for the Real C* Algebra AF
210(1)
9.4 Quantum Isometries of M × F
211(1)
9.5 Physical Significance of the Results
211(3)
9.5.1 Analysis of the Result for the Minimal Standard Model
213(1)
9.6 Invariance of the Spectral Action
214(7)
References
217(4)
10 More Examples and Open Questions
221
10.1 Free and Twisted Quantum Spheres and Their Projective Spaces
221(5)
10.1.1 Quantum Isometry Groups of Free and Half-Liberated Quantum Spheres
223(2)
10.1.2 Quantum Group of Affine Isometries a la Banica
225(1)
10.2 Easy Quantum Groups as Quantum Isometry Groups
226(1)
10.3 Quantum Symmetry Groups of Orthogonal Filtrations
227(1)
10.4 Equivariant Spectral Triples on the Drinfeld---Jimbo q-Deformation of Compact Lie Groups and Their Homogeneous Spaces
228(3)
10.5 Quantum Isometry Groups for Metric Spaces
231
References
234
DEBASHISH GOSWAMI is professor at the Stat-Math Unit of the Indian Statistical Institute, Kolkata, since 2011. Earlier, he held the positions of associate professor (200611) and assistant professor (200206) at the same department. He received his PhD degree from the Indian Statistical Institute, Kolkata, in 2000. After that, he spent couple of years in Germany and Italy for his post-doctoral research. His areas of interest include operator algebras, quantum groups, noncommutative geometry and noncommutative probability. He has published 40 research articles in several premier journals including Communications in Mathematical Physics (Springer), Advances in Mathematics (Springer), Journal of Functional Analysis, Physics Letters A, the Journal of Operator Theory, Transactions of the American Mathematical Society, Mathematische Annalen (Springer). In addition, he has published one book, Quantum Stochastic Processes and Noncommutative Geometry (Cambridge University Press).





He has received numerous awards and academic recognitions for his work, which include Shanti Swarup Bhatnagar Award (2012), J.C. Bose National Fellowship (2016), Swarnajayanthi Fellowship (2009), B.M. Birla Science Prize (2006), the INSA Medal for Young Scientists (2004), the Junior Associateship of I.C.T.P. (Italy) for 200307 and the Alexander von Humboldt Fellowship for 200001. He has been elected Fellow of the Indian Academy of Sciences (Bangalore) in the year 2015. He has visited several universities and institutes in India and abroad, including Massachusetts Institute of Technology (Cambridge), University of California (Berkeley), Max Planck Institute for Mathematics (Bonn), The Fields Institute (Canada), The Instituto Nacional de Matemįtica Pura e Aplicada (Brazil), University of Lancaster (England), and The Institute for Research in Fundamental Science (Iran), and delivered invited lectures at many conferences and workshops all over the world.







JYOTISHMAN BHOWMICK is assistant professor at the Stat-Math Unit of the Indian Statistical Institute, Kolkata. He obtained his PhD from the Indian Statistical Institute, Kolkata, in 2010 followed by post-doctoral positions at the Abdus Salam International Centre for Theoretical Physics, Trieste, and the University of Oslo. His research interests include noncommutative geometry and compact quantum groups. He has published several articles in several journals of repute including Communications in Mathematical Physics (Springer), Journal of Noncommutative Geometry, Journal of Functional Analysis, Transactions of the American Mathematical Society, Proceedings of the American Mathematical Society, and so on. He has received the INSA medal for Young Scientists in the year 2015. He is an associate of the Indian Academy of Sciences (Bangalore). His academic visits to foreign universities and institutes include Institut des hautes études scientifiques  (Paris), The Fields Institute (Canada), University of Glasgow (Scotland), etc.
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