Quantum Groups and Noncommutative Geometry

Yuri I. Manin is a Professor at the Max Planck Institute for Mathematics in Bonn. Personal distinctions include: Principal Researcher, Steklov Mathematical Institute, 1960-1993; since 1993 Principal Researcher in absentia. Professor (Algebra Chair), University of Moscow 1965-1992. Professor, M.I.T. 1992-1993. Scientific Member, MPI for Mathematics since 1993. Director, MPI for Mathematics 1995-2005, now Professor Emeritus. Board of Trustees Professor, Northwestern University (Evanston, USA) 2002-2011, now Professor Emeritus. Lenin Prize 1967. Brouwer Medal 1987. Frederic Esser Nemmers Prize 1994. Rolf Schock Prize in Mathematics 1999. King Faisal International Prize in Mathematics 2002. Georg Cantor Medal 2002. Order pour le Mérite for Science and Art, Germany, 2007. Great Cross of Merit with Star, Germany, 2008. János Bolyai International Mathematical Prize, Hungarian Academy of Sciences, 2010. Member of nine Academies of Sciences. Honorary degrees at Sorbonne, Oslo, Warwick. Honorary Member of the London Math. Society.

1. The Quantum Group GL(2).- 2. Bialgebras and Hopf Algebras.- 3. Quadratic Algebras as Quantum Linear Spaces.- 4. Quantum Matrix Spaces. I. Categorical Viewpoint.- 5. Quantum Matrix Spaces. II. Coordinate Approach.- 6. Adding Missing Relations.- 7. From Semigroups to Groups.- 8. Frobenius Algebras and the Quantum Determinant.- 9. Koszul Complexes and the Growth Rate of Quadratic Algebras.- 10. Hopf *-Algebras and Compact Matrix Pseudogroups.- 11. Yang-Baxter Equations.- 12. Algebras in Tensor Categories and Yang-Baxter Functors.- 13. Some Open Problems.- 14. The Tannaka-Krein Formalism and (Re)Presentations of Universal Quantum Groups.- Bibliography.- Index.