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**Topics in noncommutative geometry.**
*(English)*
Zbl 0724.17007

M. B. Porter Lectures. Princeton etc.: Princeton University Press. 164 p. $ 35.00 (1991).

The lectures are divided into four rather incoherent topics in noncommutative geometry according to the four chapters of the book. The overview given in chapter 1 reports sources of noncommutative geometry from functional analysis and from algebraic geometry. The cyclic cohomology is described following A. Connes, Hopf algebras are introduced in the spirit of G. Hochschild, and a category-theoretic approach is given following the Tannaka-Krein philosophy of tensor categories. The chapters 2 and 3 are devoted to special questions of algebraic supergeometry. The reader should consult the author’s book on “Gauge field theory and complex geometry” [Moskva, Nauka (1984; Zbl 0576.53002); English translation: Springer (1988)] for several definitions and results used in the text.

Chapter 2 contains a detailed discussion of the Hopf superalgebras corresponding to conformal symplectic supergroups. Basic considerations of two-dimensional conformal field theories are carried out on an algebraic geometric level. Finally the Ph.D. Thesis of A. Levin on elliptic SUSY curves is reported.

In chapter 3 the author deals with flag spaces of the classical supergroups \(\text{SL}\), \(\text{OSp}\), \(\Pi\text{Sp}\), and \(\text{Q}\). The special properties of Weyl groups and root systems, of the Bott-Samelson desingularization, and of Borel subgroups are discussed in the super case.

The last chapter contains some aspects of the theory of quantum supergroups. The quantum general linear group is defined on a quantum superspace by universal properties of the noncommutative Hopf superalgebra in question. The examples as well as some of the considerations are rather sketchy reporting results from earlier papers of the author.

The detailed content looks as follows: Chapter 1. An overview. 1. Sources of noncommutative geometry, 2. Non-commutative de Rham complex and cyclic cohomology, 3. Quantum groups and Yang-Baxter equations, 4. Monoidal and tensor categories as a unifying machine.

Chapter 2. Supersymmetric algebraic curves. 1. A superextension of the Riemann sphere, 2. SUSY-families and Schottky groups, 3. Automorphic Jacobi-Schottky superfunctions, 4. Superprojective structures, 5. Sheaves of the Virasoro and Neveu-Schwarz algebras, 6. The second construction of the Neveu-Schwarz sheaves, 7. Elliptic SUSY-families, 8. Super-theta- functions.

Chapter 3. Flag superspaces and Schubert supercells. 1. Classical supergroups and flag superspaces, 2. Schubert supercells, 3. Superlength in flag Weyl groups, 4. Order in flag Weyl groups and closure of Schubert supercells, 5. Singularities of Schubert supercells, 6. Root systems and parabolic subgroups.

Chapter 4. Quantum groups as symmetries of quantum spaces. 1. Quantum supergroups, 2. Automorphisms of quantum spaces, 3. General linear supergroups, 4. Regular quantum spaces, 5. \(\text{GL}_ q(n)\) at the roots of unity: Frobenius at characteristic zero and the Hopf fundamental group, 6. Quantum tori and quantum theta-functions.

Bibliography. Index.

Chapter 2 contains a detailed discussion of the Hopf superalgebras corresponding to conformal symplectic supergroups. Basic considerations of two-dimensional conformal field theories are carried out on an algebraic geometric level. Finally the Ph.D. Thesis of A. Levin on elliptic SUSY curves is reported.

In chapter 3 the author deals with flag spaces of the classical supergroups \(\text{SL}\), \(\text{OSp}\), \(\Pi\text{Sp}\), and \(\text{Q}\). The special properties of Weyl groups and root systems, of the Bott-Samelson desingularization, and of Borel subgroups are discussed in the super case.

The last chapter contains some aspects of the theory of quantum supergroups. The quantum general linear group is defined on a quantum superspace by universal properties of the noncommutative Hopf superalgebra in question. The examples as well as some of the considerations are rather sketchy reporting results from earlier papers of the author.

The detailed content looks as follows: Chapter 1. An overview. 1. Sources of noncommutative geometry, 2. Non-commutative de Rham complex and cyclic cohomology, 3. Quantum groups and Yang-Baxter equations, 4. Monoidal and tensor categories as a unifying machine.

Chapter 2. Supersymmetric algebraic curves. 1. A superextension of the Riemann sphere, 2. SUSY-families and Schottky groups, 3. Automorphic Jacobi-Schottky superfunctions, 4. Superprojective structures, 5. Sheaves of the Virasoro and Neveu-Schwarz algebras, 6. The second construction of the Neveu-Schwarz sheaves, 7. Elliptic SUSY-families, 8. Super-theta- functions.

Chapter 3. Flag superspaces and Schubert supercells. 1. Classical supergroups and flag superspaces, 2. Schubert supercells, 3. Superlength in flag Weyl groups, 4. Order in flag Weyl groups and closure of Schubert supercells, 5. Singularities of Schubert supercells, 6. Root systems and parabolic subgroups.

Chapter 4. Quantum groups as symmetries of quantum spaces. 1. Quantum supergroups, 2. Automorphisms of quantum spaces, 3. General linear supergroups, 4. Regular quantum spaces, 5. \(\text{GL}_ q(n)\) at the roots of unity: Frobenius at characteristic zero and the Hopf fundamental group, 6. Quantum tori and quantum theta-functions.

Bibliography. Index.

Reviewer: Helmut Boseck (Greifswald)

### MSC:

58B34 | Noncommutative geometry (à la Connes) |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

16T05 | Hopf algebras and their applications |

16T20 | Ring-theoretic aspects of quantum groups |

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

58A50 | Supermanifolds and graded manifolds |

46L87 | Noncommutative differential geometry |

16W55 | “Super” (or “skew”) structure |

14A22 | Noncommutative algebraic geometry |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

81Q60 | Supersymmetry and quantum mechanics |

46L85 | Noncommutative topology |