|Statement||Matilde Marcolli / Deepak Parashar (eds.).|
|Series||Aspects of mathematics. E -- v. 41|
|The Physical Object|
|Pagination||viii, 240 p. ;|
|Number of Pages||240|
|ISBN 10||9783834898319, 9783834814425|
This book aims to present different methods and perspectives in the theory of quantum groups and to provide a bridge between the algebraic, representation-theoretic, analytic, and differential-geometric approaches. It also covers recent developments in noncommutative geometry which have close Price: $ This book aims to present different methods and perspectives in the theory of quantum groups and to provide a bridge between the algebraic, representation-theoretic, analytic, and differential-geometric approaches. It also covers recent developments in noncommutative geometry which have close relations to quantization and quantum group symmetries. This textbook presents an expanded write-up of Manin's celebrated Montreal author systematically develops an approach to quantum groups as symmetry objects in noncommutative geometry in contrast to the more deformation-oriented approach due to Faddeev, Drinfeld, and others. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative by:
In this expanded write-up of those lectures, Manin systematically develops an approach to quantum groups as symmetry objects in noncommutative geometry in contrast to the more deformation-oriented approach due to Faddeev, Drinfeld, and others. Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups GL[sub q](N), SO[sub q](N), and Sp[sub q](N). Journal of Mathematical Physics, Vol. 44, Issue. 2, p. gebra of functions on a noncommutative locally compact space, with C-algebras play-ing the role of continuous functions and von Neumann algebras playing the role of measurable functions. From this perspective a quantum group is a C-/von Neumann algebra with some additional structure making the noncommutative space a group-like object. space in quantum mechanics but there are many others, such as the leaf spaces of foliations, duals of nonabelian discrete groups, the space of Penrose tilings, the noncommutative torus which plays a role in M-theory compactiﬁcation, and ﬁnally the space of Q-lattices which is a natural geometric space .
This book provides a self-contained introduction to quantum groups as algebraic objects. Based on the author's lecture notes from a Part III pure mathematics course at Cambridge University, it is suitable for use as a textbook for graduate courses in quantum groups or as a supplement to modern courses in advanced by: Quantum vector spaces were basically invented by Manin in his book, " Noncommutative Geometry and Quantum Groups," and I will only be talking about a subclass of his quantum vector spaces. To get at the concept, you have to think like an algebraic geometer -- or a quantum field theorist. 5. The metric aspect of noncommutative geometry 34 Chapter 1. Noncommutative Spaces and Measure Theory 39 1. Heisenberg and the Noncommutative Algebra of Physical Quantities 40 2. Statistical State of a Macroscopic System and Quantum Statistical Mechanics 45 3. Modular Theory and the Classiﬂcation of Factors 48 4. This book provides a treatment of the theory of quantum groups (quantized universal enveloping algebras and quantized algebras of functions) and q-deformed algebras (q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus.