Theory of smooth Bol loops. Texts of lectures. (Teoriya gladkikh lup Bola. Teksty lektsij).

*(Russian, English)*Zbl 0584.53001
Moskva: Izdatel’stvo Universiteta Druzhby Narodov. 80 p. R. 0.15 (1985).

The book under review consists of lecture notes. The aim of the authors is to construct a theory for smooth Bol loops in analogy to the theory of Lie groups.

A set B with a fixed element e and binary operations (*) and (\(\setminus)\) satisfying the following conditions is said to be a left loop with a two-sided identity element: 1) \(e*a=a*e=a\), \(a\in B\), 2) \(a*(a\setminus b)=b\), and 3) \(a\setminus (a*b)=b\), a,b\(\in B\). A left loop (B,*,\(\setminus,e)\) with two-sided identity element e is called a Bol loop if the left Bol identity \((a*(b*a))*c=a*(b*(a*c))\) \((B_ 1)\) holds for all a,b,c\(\in B\). A set (Q,*,\(\setminus,e)\) is said to be a left local loop with two-sided identity element e if Q is a topological space with fixed element e and for some neighborhood U of e two continuous mappings \(U\times U\to Q\quad ((a,b)\to a*b)\) and \(U\times U\to Q\quad ((a,b)\to a\setminus b)\) are defined and satisfy the three conditions mentioned above for all a,b\(\in U\). A left local loop (B,*,\(\setminus,e)\) with two-sided identity element is called a (left) local Bol loop if \((B_ 1)\) holds for every a,b,c\(\in B\) sufficiently close to e.

To construct a Lie theory for smooth Bol loops, the authors consider as a tangent algebra not a binary Lie algebra but a binary-ternary Bol algebra. As a particular case of this theory, the theory of smooth Moufang loops can be obtained. In this case the Bol algebra becomes a Mal’cev algebra. The authors’ interest in this topic was stimulated by applications to the theory of affine connections. The binary-ternary Bol algebra mentioned above is a particular case of a so-called W-algebra. A W-algebra is a finite-dimensional vector space where a bilinear operation \(\xi\) \(\cdot \eta\) and trilinear operation \(<\xi,\eta,\tau >\) are defined and they satisfy the following identities: (i) \(\xi \cdot \xi =0\), (ii) \(<\xi,\xi,\xi >=0\), (iii) \((\xi \cdot \eta)\cdot \tau +(\eta \cdot \tau)\cdot \xi +(\tau \cdot \xi)\cdot \eta =<\xi,\eta,\tau >+<\eta,\tau,\xi >+<\tau,\xi,\eta >-<\eta,\xi,\tau >-\tau,\eta,\xi >- <\xi,\tau,\eta >\). Note that W-algebras were introduced and systematically used in web geometry by M. A. Akivis [see Sib. Mat. Zh. 15, 3-15 (1974; Zbl 0288.53021); 17, 5-11 (1976; Zbl 0337.53018), and 19, 243-253 (1978; Zbl 0388.53007)]. Because of this, K. H. Hofmann and K. Strambach in their recent papers [”Akivis algebra of a homogeneous Lie loop” (to appear); ”Lie’s fundamental theorems for analytical loops”, preprint no. 837, TH Darmstadt (1985); ”Topological quasigroups” in Theory and Applications of Quasigroups and Loops (to appear)] call them Akivis algebras. Note also that a so-called Lie triple algebra is a particular case of an Akivis algebra satisfying some additional relations.

The authors hope that the loops, in particular Bol loops, will be a tool of further research in natural sciences. In this connection they point out that a symmetric space is practically a Bol loop, and that the category of left loops is equivalent to the category of equipped homogeneous spaces.

The book consists of four chapters: 1. Introduction: basic definitions and examples. 2. On locally symmetric spaces associated with local analytic Bol loops. Some applications to the theory of analytic Moufang loops. 3. Theory of local analytic Bol loops. Differential geometric approach. 4. Embedding of a local analytic Bol loop into a local Lie group.

The book is based on papers of the authors. The necessary prerequisite knowledge in differential geometry can be found in the first chapters of the book of S. Helgason [Differential geometry and symmetric spaces (1962; Zbl 0111.181)]. One of the authors’ results of local nature is outlined as a consequence of the last theorem of the book and is worth to be mentioned here: every Bol algebra is a tangent Akivis algebra of some local analytic loop with left Bol property. The notes give a good up-to- date introduction into the subject and can be useful for both specialists in the field and graduate students studying the subject. At the end it should be noticed that the notes are bilingual: on even-numbered pages the original Russian text is presented while odd-numbered pages contain its English translation.

A set B with a fixed element e and binary operations (*) and (\(\setminus)\) satisfying the following conditions is said to be a left loop with a two-sided identity element: 1) \(e*a=a*e=a\), \(a\in B\), 2) \(a*(a\setminus b)=b\), and 3) \(a\setminus (a*b)=b\), a,b\(\in B\). A left loop (B,*,\(\setminus,e)\) with two-sided identity element e is called a Bol loop if the left Bol identity \((a*(b*a))*c=a*(b*(a*c))\) \((B_ 1)\) holds for all a,b,c\(\in B\). A set (Q,*,\(\setminus,e)\) is said to be a left local loop with two-sided identity element e if Q is a topological space with fixed element e and for some neighborhood U of e two continuous mappings \(U\times U\to Q\quad ((a,b)\to a*b)\) and \(U\times U\to Q\quad ((a,b)\to a\setminus b)\) are defined and satisfy the three conditions mentioned above for all a,b\(\in U\). A left local loop (B,*,\(\setminus,e)\) with two-sided identity element is called a (left) local Bol loop if \((B_ 1)\) holds for every a,b,c\(\in B\) sufficiently close to e.

To construct a Lie theory for smooth Bol loops, the authors consider as a tangent algebra not a binary Lie algebra but a binary-ternary Bol algebra. As a particular case of this theory, the theory of smooth Moufang loops can be obtained. In this case the Bol algebra becomes a Mal’cev algebra. The authors’ interest in this topic was stimulated by applications to the theory of affine connections. The binary-ternary Bol algebra mentioned above is a particular case of a so-called W-algebra. A W-algebra is a finite-dimensional vector space where a bilinear operation \(\xi\) \(\cdot \eta\) and trilinear operation \(<\xi,\eta,\tau >\) are defined and they satisfy the following identities: (i) \(\xi \cdot \xi =0\), (ii) \(<\xi,\xi,\xi >=0\), (iii) \((\xi \cdot \eta)\cdot \tau +(\eta \cdot \tau)\cdot \xi +(\tau \cdot \xi)\cdot \eta =<\xi,\eta,\tau >+<\eta,\tau,\xi >+<\tau,\xi,\eta >-<\eta,\xi,\tau >-\tau,\eta,\xi >- <\xi,\tau,\eta >\). Note that W-algebras were introduced and systematically used in web geometry by M. A. Akivis [see Sib. Mat. Zh. 15, 3-15 (1974; Zbl 0288.53021); 17, 5-11 (1976; Zbl 0337.53018), and 19, 243-253 (1978; Zbl 0388.53007)]. Because of this, K. H. Hofmann and K. Strambach in their recent papers [”Akivis algebra of a homogeneous Lie loop” (to appear); ”Lie’s fundamental theorems for analytical loops”, preprint no. 837, TH Darmstadt (1985); ”Topological quasigroups” in Theory and Applications of Quasigroups and Loops (to appear)] call them Akivis algebras. Note also that a so-called Lie triple algebra is a particular case of an Akivis algebra satisfying some additional relations.

The authors hope that the loops, in particular Bol loops, will be a tool of further research in natural sciences. In this connection they point out that a symmetric space is practically a Bol loop, and that the category of left loops is equivalent to the category of equipped homogeneous spaces.

The book consists of four chapters: 1. Introduction: basic definitions and examples. 2. On locally symmetric spaces associated with local analytic Bol loops. Some applications to the theory of analytic Moufang loops. 3. Theory of local analytic Bol loops. Differential geometric approach. 4. Embedding of a local analytic Bol loop into a local Lie group.

The book is based on papers of the authors. The necessary prerequisite knowledge in differential geometry can be found in the first chapters of the book of S. Helgason [Differential geometry and symmetric spaces (1962; Zbl 0111.181)]. One of the authors’ results of local nature is outlined as a consequence of the last theorem of the book and is worth to be mentioned here: every Bol algebra is a tangent Akivis algebra of some local analytic loop with left Bol property. The notes give a good up-to- date introduction into the subject and can be useful for both specialists in the field and graduate students studying the subject. At the end it should be noticed that the notes are bilingual: on even-numbered pages the original Russian text is presented while odd-numbered pages contain its English translation.

Reviewer: V.V.Goldberg

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

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

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

53A60 | Differential geometry of webs |

20N05 | Loops, quasigroups |

22A30 | Other topological algebraic systems and their representations |