Loops in group theory and Lie theory.

*(English)*Zbl 1050.22001
de Gruyter Expositions in Mathematics 35. Berlin: de Gruyter (ISBN 3-11-017010-8/hbk). xi, 361 p. (2002).

The present book is a comprehensive treatise devoted to the theory of topological, differentiable and algebraic loops. Since any loop can be regarded as a “sharply transitive section in a group”, the authors discuss the theory of loops as part of group theory and thus employ group theoretical methods rather than genuinely loop theoretical (i.e., nonassociative) ones. Sharply transitive sections in Lie groups give rise to Lie theory for smooth loops, which are studied via differential geometry.

The book consists of two parts. In the first part, after reviewing basic notions concerning loops, the authors examine the structure of proper loops which are extensions of groups by groups. This special notion of extension, studied in [H. Scheerer, Math. Ann. 206, 149–155 (1973; Zbl 0247.22010)], and called Scheerer extension, does not meet the obstruction coming from the general theory of loop extensions and admits a transparent description in terms of group theory. In Sections 3 and 4, 3-nets associated to loops are introduced and studied; in particular the relationship between configurations in 3-nets and corresponding identities in the loop are discussed, together with the local versions of these concepts in the case of differentiable objects. Some sections are devoted to the study of Bruck loops (also called \(K\)-loops, cf. [H. Kiechle, Theory of \(K\)-loops (Lecture Notes in Mathematics 1778, Springer, Berlin) (2002; Zbl 0997.20059)], [A. Kreuzer, Math. Proc. Cambr. Philos. Soc. 123, No. 1, 53–57 (1998; Zbl 0895.20052)]). This investigation is motivated by the increasing importance of this class of loops in algebra and geometry, being connected, e.g., with the classification problem of 2-transitive permutation groups which can be represented by means of the so-called near-domains first introduced by H. Karzel in [Abh. Math. Semin. Univ. Hamb. 32, 191–206 (1968; Zbl 0162.24101)]. (It is still an open problem to find examples of near-domains which are not near-fields.)

The second part of the book is devoted to the application of the theory to topological and differentiable loops in manifolds of small dimension (from one to four); in particular the authors describe topological loops in 1- and 2-dimensional manifolds such that the group topologically generated by their left translations is a Lie group and classify connected smooth Bol loops as well as strongly left alternative left A-loops of dimension less than or equal to 2. A whole section is dedicated to the hyperbolic plane loops and their isotopism classes. For a completely different approach to this particular subject see also [H. Karzel and H. Wefelscheid, Groups with an involutory antiautomorphism and \(K\)-loops; application to space-time-world and hyperbolic geometry. I: Result. Math. 23, No. 3–4, 338–354 (1993; Zbl 0788.20034)].

The book consists of two parts. In the first part, after reviewing basic notions concerning loops, the authors examine the structure of proper loops which are extensions of groups by groups. This special notion of extension, studied in [H. Scheerer, Math. Ann. 206, 149–155 (1973; Zbl 0247.22010)], and called Scheerer extension, does not meet the obstruction coming from the general theory of loop extensions and admits a transparent description in terms of group theory. In Sections 3 and 4, 3-nets associated to loops are introduced and studied; in particular the relationship between configurations in 3-nets and corresponding identities in the loop are discussed, together with the local versions of these concepts in the case of differentiable objects. Some sections are devoted to the study of Bruck loops (also called \(K\)-loops, cf. [H. Kiechle, Theory of \(K\)-loops (Lecture Notes in Mathematics 1778, Springer, Berlin) (2002; Zbl 0997.20059)], [A. Kreuzer, Math. Proc. Cambr. Philos. Soc. 123, No. 1, 53–57 (1998; Zbl 0895.20052)]). This investigation is motivated by the increasing importance of this class of loops in algebra and geometry, being connected, e.g., with the classification problem of 2-transitive permutation groups which can be represented by means of the so-called near-domains first introduced by H. Karzel in [Abh. Math. Semin. Univ. Hamb. 32, 191–206 (1968; Zbl 0162.24101)]. (It is still an open problem to find examples of near-domains which are not near-fields.)

The second part of the book is devoted to the application of the theory to topological and differentiable loops in manifolds of small dimension (from one to four); in particular the authors describe topological loops in 1- and 2-dimensional manifolds such that the group topologically generated by their left translations is a Lie group and classify connected smooth Bol loops as well as strongly left alternative left A-loops of dimension less than or equal to 2. A whole section is dedicated to the hyperbolic plane loops and their isotopism classes. For a completely different approach to this particular subject see also [H. Karzel and H. Wefelscheid, Groups with an involutory antiautomorphism and \(K\)-loops; application to space-time-world and hyperbolic geometry. I: Result. Math. 23, No. 3–4, 338–354 (1993; Zbl 0788.20034)].

Reviewer: Elena Zizioli (Brescia)

##### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20N05 | Loops, quasigroups |

22E60 | Lie algebras of Lie groups |

51H20 | Topological geometries on manifolds |

57S10 | Compact groups of homeomorphisms |