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Topics in Cyclic Theory [Mīkstie vāki]

(Lancaster University), (University of Oxford)
  • Formāts: Paperback / softback, 328 pages, height x width x depth: 227x152x20 mm, weight: 480 g, Worked examples or Exercises
  • Sērija : London Mathematical Society Student Texts
  • Izdošanas datums: 09-Jul-2020
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 1108790445
  • ISBN-13: 9781108790444
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Topics in Cyclic TheoryPalielināt
  • Formāts: Paperback / softback, 328 pages, height x width x depth: 227x152x20 mm, weight: 480 g, Worked examples or Exercises
  • Sērija : London Mathematical Society Student Texts
  • Izdošanas datums: 09-Jul-2020
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 1108790445
  • ISBN-13: 9781108790444
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781108790444.html
Keywords:
Noncommutative geometry combines themes from algebra, analysis and geometry and has significant applications to physics. This book focuses on cyclic theory, and is based upon the lecture courses by Daniel G. Quillen at the University of Oxford from 198892, which developed his own approach to the subject. The basic definitions, examples and exercises provided here allow non-specialists and students with a background in elementary functional analysis, commutative algebra and differential geometry to get to grips with the subject. Quillen's development of cyclic theory emphasizes analogies between commutative and noncommutative theories, in which he reinterpreted classical results of Hamiltonian mechanics, operator algebras and differential graded algebras into a new formalism. In this book, cyclic theory is developed from motivating examples and background towards general results. Themes covered are relevant to current research, including homomorphisms modulo powers of ideals, traces on noncommutative differential forms, quasi-free algebras and Chern characters on connections.

Recenzijas

'The monograph is an excellent introduction to cyclic theory and an absolute must to any academic library, let alone a superb first-hand account and a selfless tribute to the late Daniel G. Quillen.' Igor V. Nikolaev, zbMATH 'These lectures reveal the breadth of Quillen's interests and the depth of the ideas developed. The lectures are clear and careful, rich in detail. The book is an opportunity to be lectured anew by this extraordinary mathematician.' John McCleary, Mathematical Association of America

Papildus informācija

This accessible introduction for Ph.D. students and non-specialists provides Quillen's unique development of cyclic theory.
Introduction 1(3)
1 Background Results
4(26)
1.1 Graded Algebras
4(3)
1.2 Derivations
7(2)
1.3 Commutators and Traces
9(3)
1.4 Tensor Algebras
12(4)
1.5 Real Clifford Algebras
16(2)
1.6 Lie Bracket
18(3)
1.7 The Poisson Bracket
21(4)
1.8 Extensions of Algebras via Modules
25(2)
1.9 Deformations of the Standard Product
27(3)
2 Cyclic Cocycles And Basic Operators
30(25)
2.1 The Chain Complex
30(4)
2.2 The A and b Operators
34(1)
2.3 Cyclic Cocycles on a Manifold
34(3)
2.4 Double Complexes
37(1)
2.5 The b' and N Operators
38(1)
2.6 Hochschild Cohomology
39(2)
2.7 Vector Traces
41(1)
2.8 Bianchi's Identity
42(1)
2.9 Projective Modules
43(9)
2.10 Singular Homology
52(3)
3 Algebras Of Operators
55(32)
3.1 The Gelfand Transform
55(4)
3.2 Ideals of Compact Operators on Hilbert Space
59(5)
3.3 Algebras of Operators on Hilbert Space
64(5)
3.4 Fredholm Operators
69(2)
3.5 Index Theory on the Circle via Toeplitz Operators
71(5)
3.6 The Index Formula for Toeplitz Operators
76(2)
3.7 Wallach's Formula
78(4)
3.8 Extensions of Commutative C*-Algebras
82(4)
3.9 Idempotents and Generalized Toeplitz Operators
86(1)
4 Gns Algebra
87(20)
4.1 Idempotents and Dilations
87(1)
4.2 GNS Theorem for States on a C*-Algebra
88(2)
4.3 GNS Algebra
90(1)
4.4 Stinespring's Theorem
91(3)
4.5 The Generalized Stinespring Theorem
94(2)
4.6 Uniqueness of GNS(p)
96(1)
4.7 Projective Hilbert Modules
96(6)
4.8 Algebras Associated with the Continuous Functions on the Circle
102(1)
4.9 Algebras Described by Universal Mapping Properties
102(1)
4.10 The Universal GNS Algebra of the Tensor Algebra
103(1)
4.11 The Cuntz Algebra
104(1)
4.12 Fredholm Modules
105(2)
5 Geometrical Examples
107(30)
5.1 Fredholm Modules over the Circle
107(1)
5.2 Heat Kernels on Riemannian Manifolds
108(6)
5.3 Green's Function
114(2)
5.4 Maxwell's Equation
116(3)
5.5 Dirac Operators
119(4)
5.6 Theta Summable Fredholm Modules
123(4)
5.7 Duhamel's Formula
127(3)
5.8 Quantum Harmonic Oscillator
130(4)
5.9 Chern Polynomials and Generating Functions
134(3)
6 The Algebra Of Noncommutative Differential Forms
137(34)
6.1 Kahler Differentials on an Algebraic Curve
138(6)
6.2 Homology of Kahler Differential Forms
144(6)
6.3 Noncommutative Differential Forms QA
150(5)
6.4 Ω1 A as an A-bimodule
155(6)
6.5 The Cuntz Algebra with Fedosov's Product
161(3)
6.6 Cyclic Cochains on the Cuntz Algebra
164(1)
6.7 Tensor Algebra with the Fedosov Product
165(4)
6.8 Completions
169(2)
7 Hodge Decomposition And The Karoubi Operator
171(30)
7.1 Hodge Decomposition on a Compact Riemann Surface
172(3)
7.2 The b Operator and Hochschild Homology
175(3)
7.3 The Karoubi Operator
178(3)
7.4 Connes's B-operator
181(2)
7.5 The Hodge Decomposition
183(5)
7.6 Harmonic Forms
188(1)
7.7 Mixed Complexes in the Homology Setting
189(5)
7.8 Homology of the Reduced Differential Forms
194(1)
7.9 Cyclic Cohomology
195(1)
7.10 Traces on RA and Cyclic Cocycles on A
196(5)
8 Connections
201(28)
8.1 Connections and Curvature on Manifolds
201(5)
8.2 The Chern Character
206(7)
8.3 Deforming Flat Connections
213(2)
8.4 Universal Differentials
215(2)
8.5 Connections on Modules over an Algebra
217(4)
8.6 Derivations and Automorphisms
221(2)
8.7 Lifting and Automorphisms of QA
223(6)
9 Cocycles For A Commutative Algebra Over A Manifold
229(23)
9.1 Poisson Structures on a Manifold
229(2)
9.2 Weyl Algebras
231(5)
9.3 Representations of the Heisenberg Group
236(3)
9.4 Quantum Trace Formula
239(3)
9.5 The Poisson Bracket and Symbols
242(6)
9.6 Cocycles Generated by Commutator Products
248(4)
10 Cyclic Cochains
252(17)
10.1 Traces Modulo Powers of an Ideal
252(2)
10.2 Coalgebra
254(2)
10.3 Quotienting by the Commutator Subspace
256(2)
10.4 Bar Construction
258(3)
10.5 Cochains with Values in an Algebra
261(3)
10.6 Analogue of Ω1 R for the Bar Construction
264(2)
10.7 Traces Modulo an Ideal
266(1)
10.8 Analogue of the Quotient by Commutators
266(2)
10.9 Connes's Chain and Cochain Bicomplexes
268(1)
11 Cyclic Cohomology
269(24)
11.1 Connes's Double Cochain Complex
269(1)
11.2 Connes's S Operator
270(2)
11.3 Connes's Long Exact Sequence
272(1)
11.4 A Homotopy Formula for Cocycles Associated with Traces
273(2)
11.5 Universal Graded and Ungraded Cocycles
275(3)
11.6 Deformations of Fredholm Modules
278(3)
11.7 Homotopy Formulas
281(2)
11.8 Cyclic Cocycles over the Circle
283(1)
11.9 Connections over a Compact Manifold
284(1)
11.10 The Trivial Bundle
285(1)
11.11 Cocycles Arising from the Connection
286(3)
11.12 Super Connections and Twisted Dirac Operators
289(4)
12 Periodic Cyclic Homology
293(14)
12.1 The X Complex and Periodic Cyclic Homology
293(2)
12.2 X(A) for Commutative Differential Graded Algebras
295(2)
12.3 The Canonical Filtration
297(4)
12.4 The Hodge Approximation to Cyclic Theory
301(6)
References 307(5)
List of Symbols 312(5)
Index of Subjects 317
Daniel G. Quillen proved Adam's conjecture in topological K-theory, and Serre's conjecture that all projective modules over a polynomial ring are free. He was awarded the Cole Prize in Algebra and the Fields Medal in 1978. He was Waynflete Professor of Pure Mathematics at the University of Oxford, where he lectured on K-theory and cyclic homology. Gordon Blower is Professor of Mathematical Analysis at Lancaster University, with interests in random matrices and applications of operator theory. He attended Quillen's lectures on cyclic theory when he was a junior researcher in Oxford.