Combinatorial methods. Free groups, polynomials, and free algebras.

*(English)*Zbl 1039.16024
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 19. New York, NY: Springer (ISBN 0-387-40562-3/hbk). xii, 315 p. (2004).

The book consists of three parts: groups, polynomial algebras and free Nielsen-Schreier algebras. In each of these related classes the authors consider problems of orbits of automorphism groups, test elements, tame and wild automorphisms and some other topics.

The first part starts with the Nielsen-Schreier theorem, Whitehead’s method of determining whether two elements of a free group belong to one automorphic orbit of the automorphism group of a free group and Tietze’s method of isomorphism of groups with given presentations. Also free differential Fox calculus in integral group rings of a free group is exposed.

An element \(g\) of a group \(G\) is a test element if for any endomorphism \(\phi\) of the group \(G\) the equality \(\phi(g)=g\) implies that \(\phi\) is an automorphism. Properties of test elements in free groups are considered in chapter 2 using the free differential calculus.

Let \(J\) be a right ideal of the integral group ring \(\mathbb Z F\) of a free group \(F\). An element \(u\in F\) is \(J\)-primitive if the Fox derivatives of \(u\) generate \(J\) as a right ideal. Properties of \(\Delta\)-primitive elements for the augmentation ideal \(\Delta\) in \(\mathbb Z F\) are discussed in chapter 3.

In chapter 4 the authors consider the following problems. Is an endomorphism of a free group an automorphism if it (i) preserves the set of primitive elements, (ii) preserves a nontrivial automorphism orbit?

The second part is concerned with the Jacobian conjecture and with the following problems. (i) Is it the case that a retract of the polynomial algebra \(k[X_1,\dots,X_n]\) is isomorphic to a polynomial algebra? (ii) Is every automorphism of the polynomial algebra \(k[X_1,\ldots,X_n]\) tame? (iii) Is every automorphism of a polynomial algebra \(k[X_1,\dots,X_n]\) at least stably tame? Note that the second problem was negatively solved recently by U. U. Umirbaev and I. P. Shestakov [J. Am. Math. Soc. 17, No. 1, 197–227 (2004; Zbl 1056.14085)]. It was shown that Nagata’s automorphism of \(k[X,Y,Z]\) having the form \(\exp D\) for some locally nilpotent derivation \(D\) is tame.

Similar problems are considered for free associative algebras as well as orbits of variables in \(k[X_1, \dots,X_n]\) (and in free associative algebras) under the action of the automorphism group.

Part 3 is devoted to the study of homogeneous varieties of algebras in which subalgebras of free algebras are free (Schreier property). The following varieties have this property: the variety of all non-associative algebras, the variety of all (skew)commutative non-associative algebras, the variety of Lie algebras. In a variety of algebras a universal multiplicative enveloping algebra for a free algebra is introduced. Using Schreier’s technique the authors prove Cohn’s theorem of freeness of some subalgebras of free associative algebras. An analog of the free differential calculus for Schreier varieties of algebras is developed. A criterion is given in terms of the free differential calculus for the universal enveloping algebra of a free algebra under which the variety has the Schreier property. Also properties of test elements, primitive elements, orbits of automorphism groups in free algebras of Schreier varieties are investigated. In the last chapter a new variety of Leibniz algebras is studied. It is shown that free Leibniz algebras are residually finite. An analog of the Jacobian conjecture is proved for these algebras. There is an algorithm recognizing tame automorphisms of free Leibniz algebras of rank 2.

The book contains very interesting material to which the authors have made a valuable contribution. The book includes many open and very important problems. Some of them, as it was mentioned above, are solved. Different methods are used in proofs in the present book such as differential calculus in some rings, Gröbner bases, combinatorial methods and algorithms, algebraic geometry and commutative algebra. The exposition of the material is made with care. So the book could be recommended for students even as a textbook.

The first part starts with the Nielsen-Schreier theorem, Whitehead’s method of determining whether two elements of a free group belong to one automorphic orbit of the automorphism group of a free group and Tietze’s method of isomorphism of groups with given presentations. Also free differential Fox calculus in integral group rings of a free group is exposed.

An element \(g\) of a group \(G\) is a test element if for any endomorphism \(\phi\) of the group \(G\) the equality \(\phi(g)=g\) implies that \(\phi\) is an automorphism. Properties of test elements in free groups are considered in chapter 2 using the free differential calculus.

Let \(J\) be a right ideal of the integral group ring \(\mathbb Z F\) of a free group \(F\). An element \(u\in F\) is \(J\)-primitive if the Fox derivatives of \(u\) generate \(J\) as a right ideal. Properties of \(\Delta\)-primitive elements for the augmentation ideal \(\Delta\) in \(\mathbb Z F\) are discussed in chapter 3.

In chapter 4 the authors consider the following problems. Is an endomorphism of a free group an automorphism if it (i) preserves the set of primitive elements, (ii) preserves a nontrivial automorphism orbit?

The second part is concerned with the Jacobian conjecture and with the following problems. (i) Is it the case that a retract of the polynomial algebra \(k[X_1,\dots,X_n]\) is isomorphic to a polynomial algebra? (ii) Is every automorphism of the polynomial algebra \(k[X_1,\ldots,X_n]\) tame? (iii) Is every automorphism of a polynomial algebra \(k[X_1,\dots,X_n]\) at least stably tame? Note that the second problem was negatively solved recently by U. U. Umirbaev and I. P. Shestakov [J. Am. Math. Soc. 17, No. 1, 197–227 (2004; Zbl 1056.14085)]. It was shown that Nagata’s automorphism of \(k[X,Y,Z]\) having the form \(\exp D\) for some locally nilpotent derivation \(D\) is tame.

Similar problems are considered for free associative algebras as well as orbits of variables in \(k[X_1, \dots,X_n]\) (and in free associative algebras) under the action of the automorphism group.

Part 3 is devoted to the study of homogeneous varieties of algebras in which subalgebras of free algebras are free (Schreier property). The following varieties have this property: the variety of all non-associative algebras, the variety of all (skew)commutative non-associative algebras, the variety of Lie algebras. In a variety of algebras a universal multiplicative enveloping algebra for a free algebra is introduced. Using Schreier’s technique the authors prove Cohn’s theorem of freeness of some subalgebras of free associative algebras. An analog of the free differential calculus for Schreier varieties of algebras is developed. A criterion is given in terms of the free differential calculus for the universal enveloping algebra of a free algebra under which the variety has the Schreier property. Also properties of test elements, primitive elements, orbits of automorphism groups in free algebras of Schreier varieties are investigated. In the last chapter a new variety of Leibniz algebras is studied. It is shown that free Leibniz algebras are residually finite. An analog of the Jacobian conjecture is proved for these algebras. There is an algorithm recognizing tame automorphisms of free Leibniz algebras of rank 2.

The book contains very interesting material to which the authors have made a valuable contribution. The book includes many open and very important problems. Some of them, as it was mentioned above, are solved. Different methods are used in proofs in the present book such as differential calculus in some rings, Gröbner bases, combinatorial methods and algorithms, algebraic geometry and commutative algebra. The exposition of the material is made with care. So the book could be recommended for students even as a textbook.

Reviewer: Vyacheslav A. Artamonov (Moskva)

##### MSC:

16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

17A32 | Leibniz algebras |

20E05 | Free nonabelian groups |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

20E36 | Automorphisms of infinite groups |

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

17B01 | Identities, free Lie (super)algebras |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |