Topics in Non-Commutative Geometry

Topics in Non-Commutative Geometry

YURI I. MANIN
Series: Porter Lectures
Copyright Date: 1991
Pages: 172
https://www.jstor.org/stable/j.ctt7ztvn8
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    Topics in Non-Commutative Geometry
    Book Description:

    There is a well-known correspondence between the objects of algebra and geometry: a space gives rise to a function algebra; a vector bundle over the space corresponds to a projective module over this algebra; cohomology can be read off the de Rham complex; and so on. In this book Yuri Manin addresses a variety of instances in which the application of commutative algebra cannot be used to describe geometric objects, emphasizing the recent upsurge of activity in studying noncommutative rings as if they were function rings on "noncommutative spaces." Manin begins by summarizing and giving examples of some of the ideas that led to the new concepts of noncommutative geometry, such as Connes' noncommutative de Rham complex, supergeometry, and quantum groups. He then discusses supersymmetric algebraic curves that arose in connection with superstring theory; examines superhomogeneous spaces, their Schubert cells, and superanalogues of Weyl groups; and provides an introduction to quantum groups. This book is intended for mathematicians and physicists with some background in Lie groups and complex geometry.

    Originally published in 1991.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-6251-1
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. PREFACE
    (pp. vii-2)
    Yuri I. Manin
  4. CHAPTER 1 An Overview
    (pp. 3-32)

    1.1. Commutative Geometry. The classical Euclidean geometry studies properties of some special subsets of plane and space: circles, triangles, pyramids, etc. Some of the crucial notions are those of a measure (of an angle, distance, surface, volume) and of “congruence” or equality of geometric objects.

    An implicit basic object that only a century ago started to become a subject of independent geometric study is the group of motions. In fact, measures can be introduced as various motion invariants, and equality can be defined in terms of orbits of this group.

    Since Descartes, this geometric picture became enriched with an essential...

  5. CHAPTER 2 Supersymmetric Algebraic Curves
    (pp. 33-95)

    1.1. Riemann Sphere. A Riemann sphere is the space of ℂ-points of a projective line ℙ¹. Its automorphism group is PGL(2), and we can identify GL(2) with the group CSp(2) of conformal automorphisms of a symplectic form of rank 2.

    In this section we shall define and study Riemann superspheres$ \mathbb{P}^{1|1} $and$ \mathbb{P}^{1|2} $as homogeneous spaces of conformal symplectic supergroups CSpO(2|1) and CSpO(2|2) respectively. Our main result is Theorem 1.12 that describes the geometric structures invariant with respect to these supergroups and in turn determining them. Riemann superspheres furnish the simplest examples of algebraic supercurves (those of genus zero) and...

  6. CHAPTER 3 Flag Superspaces and Schubert Supercells
    (pp. 96-123)

    1.1. Classical Supergroups. In this section, we shall first introduce classical supergroups and their flag spaces and then shall define combinatorial invariants that will be used for enumeration of Schubert supercells. The now standard approach based on the properties of Weyl groups has some drawbacks in our situation, and we therefore fall back upon the more classical notion of “relative position” of a pair of flags.

    We start with recalling and completing some constructions discussed in [Ma1]. A supergroup of the type SL, OSp, Π Sp, orQis defined by its standard representation space$ T \simeq \mathbb{C}^{m|n} $and an invariant. For...

  7. CHAPTER 4 Quantum Groups as Symmetries of Quantum Spaces
    (pp. 124-156)

    1.1. Hopf Superalgebras. In this chapter, we shall introduce and investigate a method of construction of quantum groups considered as “automorphisms objects” of noncommutative spaces (we continue to act in the framework of noncommutativealgebraicgeometry). A part of our results will be stated for quantum supergroups and superspaces, represented by$ \mathbb{Z}_{2} $-graded algebras and grade-preserving morphisms. This means that the definitions and results of Chapter 1, Section 3 should be slightly refined.

    We shall review here the main differences. Recall that the compatibility axiom of a bialgebra looks formally the same (see Chapter 1, Section 3.2(c)), but the multiplication...

  8. BIBLIOGRAPHY
    (pp. 157-163)
  9. INDEX
    (pp. 164-164)