Quantum groups and noncommutative geometry

By Shahn Majid. 2000

304 citations recorded by the source corpus.

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalization of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of space–time, in particular, amounts to a postulated new force or physical effect called cogravity.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8e158f557c16648363ef6c956cbcc45d3c7f9450

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Julien Bichon
2. T. Banica
3. L. Mesref
4. G. Giachetta
5. V. Ginzburg
6. Thomas Ernst
7. G. Sardanashvily
8. Moritz Weber

In Mathematical Physics

1. Julien Bichon
2. T. Banica
3. L. Mesref
4. G. Giachetta
5. V. Ginzburg
6. Thomas Ernst
7. G. Sardanashvily
8. I. Losev

In Statistical and Nonlinear Physics

1. G. Giachetta
2. L. Mesref
3. V. Ginzburg
4. T. Banica
5. Julien Bichon
6. Thomas Ernst
7. G. Sardanashvily
8. Osamu Iyama

Citations in the corpus

• What is geometry in quantum theory (2004, background): “(iv) Hopf algebras and, in particular, quantum groups make a contribution to many quantum theories [63, 156, 157].”
• QUANTUM FIELD THEORIES (2004, methodology): “where C (q) is a constant and where the quantum hypergeometric function (17)We use Manin’s notation [26].”
• Algebraic quantum permutation groups (2007, background): “In the case of finite-dimensional algebras, it is not difficult to show that given a finite-dimensional algebra A, there exists a universal bialgebra coacting on A, and hence, through Manin’s Hopf envelope construction [13], a universal Hopf algebra coacting on A.”
• The history of q-calculus and a new method (2000, background): “[601] Manin Y. I.: Quantum groups and noncommutative geometry.”
• Quantum groups acting on 4 points (2007, methodology): “Therefore A(k,−1) is isomorphic with the algebra B4k considered by Masuoka (Definition 3.3 in [28]).”
• The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry (2004, unknown): “It appears in many other generalizations as well, for example in the Koszul duality associated with quantum groups (see [Manin 1988].”
• A Short survey of noncommutative geometry (2000, unknown): “The non-triviality of this homotopy class shows up in the intersection form K∗(A)×K∗(A) → Z which is obtained from the Fredholm index of D with coefficients in K∗(A⊗A0).”
• Lectures on Noncommutative Geometry (2005, methodology): “We refer to [77, 83, 94, 96, 97, 118] for the general theory of Hopf algebras and quantum groups.”