Introduction to the Theory of Schemes and Quantum Groups

The language of «sheaves with nilpotents» is an integral part of the baggage of a modern mathematical physicist, especially one studying or using applications of supersymmetries. The book contains a processed record of a two-year lecture course by Yu.I. Manin on Grothendieck's theory of schemes—geometrization of commutative algebra. The exposition is exceptionally clear and accessible to second-year students of mathematical faculties and slightly older students of physical faculties. The chapter added in this edition (the third) contains fragments of another course by Yu.I. Manin on algebraic geometry, demonstrating how the theory of schemes and cohomology of sheaves «works» in more complex situations. The currently nonexistent noncommutative geometry is a science that studies noncommutative algebras of «functions on what we cannot yet define». The fourth chapter of the book presents an introduction to the theory of quadratic algebras and quantum groups—a section of noncommutative geometry that arose from examples and the theory of integrable dynamical systems. Quantum groups describe (unknown before these lectures) symmetries of ordinary spaces, much broader than those described by Lie groups. The first edition of the book was published in 2012.