# Topics in Non-Commutative Geometry

YURI I. MANIN
Series: Porter Lectures
Pages: 172
https://www.jstor.org/stable/j.ctt7ztvn8

1. Front Matter
(pp. i-iv)
(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)