Free Lie algebras.

*(English)*Zbl 0798.17001
London Mathematical Society Monographs. New Series. 7. Oxford: Clarendon Press (ISBN 0-19-853679-8/hbk). xvii, 269 p. (1993).

Let \(A\) be an alphabet and let \(K\) be a ring. The free Lie algebra \(L(A)\) is then the Lie \(K\)-algebra freely generated by \(A\). The theory of free Lie algebras grew from the need of computing formally with commutators \([a,b]= ab- ba\) (using only Jacobi identity and antisymmetry). It can be traced back to the turn of the century in the works of Campbell (1898), Poincaré (1899), Baker (1904) or Hausdorff (1906).

The book under review is devoted to an extensive presentation of free Lie algebras (and related results). One can very roughly divide it into three main parts: first a purely algebraic study of free Lie algebras; secondly an overview of the combinatorial theory of bases of \(L(A)\); finally the modern point of view of algebraic combinatorics on free Lie algebras. Formally the book consists of the ten following chapters:

0. Introduction; 1. Lie polynomials; 2. Algebraic properties; 3. Logarithms and exponentials; 4. Hall bases; 5. Applications of Hall sets; 6. Shuffle algebra and subwords; 7. Circular words; 8. The action of the symmetric group; 9. The Solomon descent algebra.

Every chapter consists of a main body followed by an appendix and by some historical and bibliographical notes. The appendices contain several more specialized or more recent results connected with the subject of the corresponding chapter.

The book begins with an introduction (Chapter 0) that deals first with the general Poincaré-Birkhoff-Witt theorem. The author presents then Lazard’s elimination theorem and defines the concept of Lazard set (it is a set which is constructed by successive applications of Lazard’s elimination process). These sets will be shown in Chapter 4 to be exactly the Hall sets (defined in a completely different manner). Using these notions, it is then shown that \(L(A)\) is a free \(K\)-module and that it can be realized as the Lie subalgebra of the free associative algebra generated by \(A\).

Chapter 1 (Lie polynomials) is devoted to some Hopf algebraic aspects of the theory. Four characterizations of Lie polynomials are given. Among them, one may mention both Friedrich’s criterion (\(P\) is a Lie polynomial iff \(P\) is primitive for the coproduct \(\delta\) defined on \(K\langle A\rangle\) by requiring \(\delta(a)= 1\otimes a+a\otimes 1\) for all letters \(a\in A\)) and invariance under Dynkin’s projection. The author defines then a new law on \(K\langle A\rangle\): the shuffle product, denoted \(\sqcup \sqcup\), which is in duality with the canonical coproduct \(\delta\) of \(K\langle A\rangle= {\mathcal U} (L(A))\). He shows then how to construct two Hopf algebra structures on \(K\langle A\rangle\) (the first has the ordinary concatenation operation as product and \(\delta\) as coproduct and the second has \(\sqcup \sqcup\) as product and is equipped with the Cauchy coproduct). A duality between these two structures is presented. Finally the convolution product denoted \(*\) on \(\text{End}(K \langle A\rangle)\) is defined. Recall that one has \(f*g= \text{conc}\circ (f\otimes g)\circ \delta\), for all \(f,g\in \text{End}(K\langle A\rangle)\) (where conc denotes the concatenation product). Some connections between this convolution algebra and the two previous Hopf structures are presented.

Chapter 2 (Algebraic properties) deals essentially with the study of Lie subalgebras and of automorphisms of free Lie algebras. The author gives a simple proof of Shirshov’s theorem (that says that every Lie subalgebra of a free Lie algebra is itself a free Lie algebra). A characterization of automorphisms of \(L(A)\) in terms of Jacobians is then given. Finally a defect theorem for free Lie algebras is presented.

Chapter 3 (Logarithms and exponentials) is devoted to the study of the relations existing between Lie series and the Hausdorff group (i.e. the group of exponentials of Lie series). Several characterizations of Lie series are then given. They lead naturally to the Campbell-Baker- Hausdorff formula. Another interesting consequence of these criteria is the fact that every Chen series is always the exponential of a Lie series.

The author introduces then an important decomposition of \(K\langle A\rangle\). Recall first that the symmetrized product of \(n\) polynomials \(P_ 1,\dots, P_ n\) of \(K\langle A\rangle\) is defined by setting \[ (P_ 1 P_ 2\dots P_ n)= {\textstyle {1\over {n!}}} \sum_{\sigma\in {\mathfrak S}_ n} P_{\sigma(1)} P_{\sigma(2)} \dots P_{\sigma(n)}. \] One can then show that one has \(K\langle A\rangle= \bigoplus_{n\geq 0} U_ n\) where \(U_ n\) is the submodule of \(K\langle A\rangle\) generated by the symmetrized products of \(n\) Lie polynomials. The author introduces then the canonical projections \(\pi_n\) which are just the projections of \(K\langle A\rangle\) on \(U_n\) relative to the previous (canonical) decomposition. It turns out that each projection can be identified with an idempotent of \(K[{\mathfrak S}_n]\). The use of \(\pi_1\) is in particular of main use in the proof of a theorem of Goldberg that gives an explicit expression for the coefficient of a word in the Hausdorff series \[ H(a_1 a_2\cdots a_p)= \log(e^{a_1} e^{a_2}\dots e^{a_p}). \] The chapter ends finally with some connections between derivation and exponentiation.

Chapters 4 and 5 are devoted to the construction and study of an important class of (linear) bases of \(L(A)\): the so-called Hall bases. The author presents first some properties of Hall words (i.e. the words obtained by taking the foliages of Hall trees). He proves in particular that every word \(w\) can be uniquely decomposed as \(w= h_1 h_2\cdots h_n\) where \(h_1\geq h_2\geq\dots \geq h_n\) are Hall words. The exposition chosen is based on the construction of a rewriting system on Hall trees (this allows the author to obtain directly decompositions on the Poincaré-Birkhoff-Witt basis of \(K\langle A\rangle\) associated with a Hall basis of \(L(A)\)). The author ends Chapter 4 by proving the identity between Lazard and Hall basis, a classical result due to Viennot.

Chapter 5 begins with the presentation of the Lyndon basis which is a Hall basis of particular interest. It continues by a study of its dual basis. The author uses also his techniques in order to provide simple bases for the factors of the derived series of \(L(A)\). Finally he shows how to use Hall sets in order to construct variable length codes (a code is a free submonoid of the free monoid \(A^*\)) with interesting synchronization properties.

Chapter 6 (Shuffle algebra and subwords) studies first \(K\langle A\rangle\) equipped with the shuffle product \(\sqcup \sqcup\). One shows in particular that this algebra is in fact an algebra of polynomials with the Lyndon words as a transcendence basis. The author introduces then the notion of subword function: it is a function from the free group \(F(A)\) into \(\mathbb{Z}\) of the form \(g\to {g\choose u}\) where this binomial notation is defined by \[ \mu(w)= \sum_{v\in A^*} {w\choose v} v, \] \(\mu\) denoting the Magnus transformation defined by setting \(\mu(a)= 1+a\) for every \(a\in A\). An interesting result is the algebraic independence of all subword functions on \(F(A)\). The end of the chapter is devoted to the study of the central lower series of \(F(A)\) and especially to the presentation of the calculus of basic commutators. The appendix presents among other results some applications of the material of this chapter to control theory.

Chapter 7 (Circular words) is devoted to the study of the connection between free Lie algebras and circular words. Recall that a circular word (or a necklace) is just a conjugacy class in the free monoid. This chapter begins therefore with the enumeration of primitive circular words (Witt’s formula). It is also shown that Hall and Lyndon words form sections of circular words. An optimal algorithm, due to Duval, for generating Lyndon words is then presented. The author also gives explicit bijections between multisets of primitive necklaces, multisets of Hall words and single words.

The two last chapters of the book deal with the representation theoretical aspects of the theory. Chapter 8 (The action of the symmetric group) begins by presenting the natural actions of the symmetric and the linear groups on \(L(A)\). The second subsection of this chapter is then devoted to the computation of the characteristic of the action of \(S_n\) on the multilinear part of degree \(n\) of \(L(A)\). The irreducible components of this representation are then studied and made explicit: one finds in particular a combinatorial interpretation of the multiplicity of these irreducible components. The two last subsections of Chapter 8 are devoted to Lie idempotents. Recall that a Lie idempotent is an idempotent of \(K[{\mathfrak S}_n]\) that encodes a projection of \(K\langle A\rangle\) onto \(L(A)\). The author presents the connections between three classical Lie idempotents due to Dynkin, Klyachko and Solomon.

The last chapter of the book is finally devoted to the study of the connections that exist between Solomon’s descent algebra \(\Sigma_ n\) and \(L(A)\). Recall that \(\Sigma_n\) is the subalgebra of \(K[{\mathfrak S}_n]\) which has the descent sets \(D_S= \sum_{D(\sigma) =S}\sigma\) as basis (where \(D(\sigma)= \{i\), \(\sigma(i)> \sigma(i+1)\}\)). All the three Lie idempotents mentioned above are for instance elements of \(\Sigma_n\). The author studies first the connections between Solomon’s descent algebra and the canonical decomposition of \(K\langle A\rangle\). Several homomorphisms of \(\Sigma_n\) are also presented: the author introduces in particular an homomorphism from \(\Sigma_n\) into the ring of homogeneous symmetric functions of degree \(n\) equipped with the internal product. This chapter ends with the introduction of the algebra of quasisymmetric functions over a totally ordered alphabet \((A,<)\), a construction due to Gessel. This algebra is generated by the commutative functions \[ M_C= \sum_{x_1< x_2< \ldots< x_k} x_1^{i_1} x_ ^{i_2} \dots x_k^{i_k}, \] where \(C=(i_1, i_2,\ldots, i_k)\) is a composition. These quasisymmetric functions are used to count several classes of permutations defined by descent conditions.

The book under review is very well written. It can be considered as a first class reference on free Lie algebras. It can be highly recommended to everyone interested in this subject.

The book under review is devoted to an extensive presentation of free Lie algebras (and related results). One can very roughly divide it into three main parts: first a purely algebraic study of free Lie algebras; secondly an overview of the combinatorial theory of bases of \(L(A)\); finally the modern point of view of algebraic combinatorics on free Lie algebras. Formally the book consists of the ten following chapters:

0. Introduction; 1. Lie polynomials; 2. Algebraic properties; 3. Logarithms and exponentials; 4. Hall bases; 5. Applications of Hall sets; 6. Shuffle algebra and subwords; 7. Circular words; 8. The action of the symmetric group; 9. The Solomon descent algebra.

Every chapter consists of a main body followed by an appendix and by some historical and bibliographical notes. The appendices contain several more specialized or more recent results connected with the subject of the corresponding chapter.

The book begins with an introduction (Chapter 0) that deals first with the general Poincaré-Birkhoff-Witt theorem. The author presents then Lazard’s elimination theorem and defines the concept of Lazard set (it is a set which is constructed by successive applications of Lazard’s elimination process). These sets will be shown in Chapter 4 to be exactly the Hall sets (defined in a completely different manner). Using these notions, it is then shown that \(L(A)\) is a free \(K\)-module and that it can be realized as the Lie subalgebra of the free associative algebra generated by \(A\).

Chapter 1 (Lie polynomials) is devoted to some Hopf algebraic aspects of the theory. Four characterizations of Lie polynomials are given. Among them, one may mention both Friedrich’s criterion (\(P\) is a Lie polynomial iff \(P\) is primitive for the coproduct \(\delta\) defined on \(K\langle A\rangle\) by requiring \(\delta(a)= 1\otimes a+a\otimes 1\) for all letters \(a\in A\)) and invariance under Dynkin’s projection. The author defines then a new law on \(K\langle A\rangle\): the shuffle product, denoted \(\sqcup \sqcup\), which is in duality with the canonical coproduct \(\delta\) of \(K\langle A\rangle= {\mathcal U} (L(A))\). He shows then how to construct two Hopf algebra structures on \(K\langle A\rangle\) (the first has the ordinary concatenation operation as product and \(\delta\) as coproduct and the second has \(\sqcup \sqcup\) as product and is equipped with the Cauchy coproduct). A duality between these two structures is presented. Finally the convolution product denoted \(*\) on \(\text{End}(K \langle A\rangle)\) is defined. Recall that one has \(f*g= \text{conc}\circ (f\otimes g)\circ \delta\), for all \(f,g\in \text{End}(K\langle A\rangle)\) (where conc denotes the concatenation product). Some connections between this convolution algebra and the two previous Hopf structures are presented.

Chapter 2 (Algebraic properties) deals essentially with the study of Lie subalgebras and of automorphisms of free Lie algebras. The author gives a simple proof of Shirshov’s theorem (that says that every Lie subalgebra of a free Lie algebra is itself a free Lie algebra). A characterization of automorphisms of \(L(A)\) in terms of Jacobians is then given. Finally a defect theorem for free Lie algebras is presented.

Chapter 3 (Logarithms and exponentials) is devoted to the study of the relations existing between Lie series and the Hausdorff group (i.e. the group of exponentials of Lie series). Several characterizations of Lie series are then given. They lead naturally to the Campbell-Baker- Hausdorff formula. Another interesting consequence of these criteria is the fact that every Chen series is always the exponential of a Lie series.

The author introduces then an important decomposition of \(K\langle A\rangle\). Recall first that the symmetrized product of \(n\) polynomials \(P_ 1,\dots, P_ n\) of \(K\langle A\rangle\) is defined by setting \[ (P_ 1 P_ 2\dots P_ n)= {\textstyle {1\over {n!}}} \sum_{\sigma\in {\mathfrak S}_ n} P_{\sigma(1)} P_{\sigma(2)} \dots P_{\sigma(n)}. \] One can then show that one has \(K\langle A\rangle= \bigoplus_{n\geq 0} U_ n\) where \(U_ n\) is the submodule of \(K\langle A\rangle\) generated by the symmetrized products of \(n\) Lie polynomials. The author introduces then the canonical projections \(\pi_n\) which are just the projections of \(K\langle A\rangle\) on \(U_n\) relative to the previous (canonical) decomposition. It turns out that each projection can be identified with an idempotent of \(K[{\mathfrak S}_n]\). The use of \(\pi_1\) is in particular of main use in the proof of a theorem of Goldberg that gives an explicit expression for the coefficient of a word in the Hausdorff series \[ H(a_1 a_2\cdots a_p)= \log(e^{a_1} e^{a_2}\dots e^{a_p}). \] The chapter ends finally with some connections between derivation and exponentiation.

Chapters 4 and 5 are devoted to the construction and study of an important class of (linear) bases of \(L(A)\): the so-called Hall bases. The author presents first some properties of Hall words (i.e. the words obtained by taking the foliages of Hall trees). He proves in particular that every word \(w\) can be uniquely decomposed as \(w= h_1 h_2\cdots h_n\) where \(h_1\geq h_2\geq\dots \geq h_n\) are Hall words. The exposition chosen is based on the construction of a rewriting system on Hall trees (this allows the author to obtain directly decompositions on the Poincaré-Birkhoff-Witt basis of \(K\langle A\rangle\) associated with a Hall basis of \(L(A)\)). The author ends Chapter 4 by proving the identity between Lazard and Hall basis, a classical result due to Viennot.

Chapter 5 begins with the presentation of the Lyndon basis which is a Hall basis of particular interest. It continues by a study of its dual basis. The author uses also his techniques in order to provide simple bases for the factors of the derived series of \(L(A)\). Finally he shows how to use Hall sets in order to construct variable length codes (a code is a free submonoid of the free monoid \(A^*\)) with interesting synchronization properties.

Chapter 6 (Shuffle algebra and subwords) studies first \(K\langle A\rangle\) equipped with the shuffle product \(\sqcup \sqcup\). One shows in particular that this algebra is in fact an algebra of polynomials with the Lyndon words as a transcendence basis. The author introduces then the notion of subword function: it is a function from the free group \(F(A)\) into \(\mathbb{Z}\) of the form \(g\to {g\choose u}\) where this binomial notation is defined by \[ \mu(w)= \sum_{v\in A^*} {w\choose v} v, \] \(\mu\) denoting the Magnus transformation defined by setting \(\mu(a)= 1+a\) for every \(a\in A\). An interesting result is the algebraic independence of all subword functions on \(F(A)\). The end of the chapter is devoted to the study of the central lower series of \(F(A)\) and especially to the presentation of the calculus of basic commutators. The appendix presents among other results some applications of the material of this chapter to control theory.

Chapter 7 (Circular words) is devoted to the study of the connection between free Lie algebras and circular words. Recall that a circular word (or a necklace) is just a conjugacy class in the free monoid. This chapter begins therefore with the enumeration of primitive circular words (Witt’s formula). It is also shown that Hall and Lyndon words form sections of circular words. An optimal algorithm, due to Duval, for generating Lyndon words is then presented. The author also gives explicit bijections between multisets of primitive necklaces, multisets of Hall words and single words.

The two last chapters of the book deal with the representation theoretical aspects of the theory. Chapter 8 (The action of the symmetric group) begins by presenting the natural actions of the symmetric and the linear groups on \(L(A)\). The second subsection of this chapter is then devoted to the computation of the characteristic of the action of \(S_n\) on the multilinear part of degree \(n\) of \(L(A)\). The irreducible components of this representation are then studied and made explicit: one finds in particular a combinatorial interpretation of the multiplicity of these irreducible components. The two last subsections of Chapter 8 are devoted to Lie idempotents. Recall that a Lie idempotent is an idempotent of \(K[{\mathfrak S}_n]\) that encodes a projection of \(K\langle A\rangle\) onto \(L(A)\). The author presents the connections between three classical Lie idempotents due to Dynkin, Klyachko and Solomon.

The last chapter of the book is finally devoted to the study of the connections that exist between Solomon’s descent algebra \(\Sigma_ n\) and \(L(A)\). Recall that \(\Sigma_n\) is the subalgebra of \(K[{\mathfrak S}_n]\) which has the descent sets \(D_S= \sum_{D(\sigma) =S}\sigma\) as basis (where \(D(\sigma)= \{i\), \(\sigma(i)> \sigma(i+1)\}\)). All the three Lie idempotents mentioned above are for instance elements of \(\Sigma_n\). The author studies first the connections between Solomon’s descent algebra and the canonical decomposition of \(K\langle A\rangle\). Several homomorphisms of \(\Sigma_n\) are also presented: the author introduces in particular an homomorphism from \(\Sigma_n\) into the ring of homogeneous symmetric functions of degree \(n\) equipped with the internal product. This chapter ends with the introduction of the algebra of quasisymmetric functions over a totally ordered alphabet \((A,<)\), a construction due to Gessel. This algebra is generated by the commutative functions \[ M_C= \sum_{x_1< x_2< \ldots< x_k} x_1^{i_1} x_ ^{i_2} \dots x_k^{i_k}, \] where \(C=(i_1, i_2,\ldots, i_k)\) is a composition. These quasisymmetric functions are used to count several classes of permutations defined by descent conditions.

The book under review is very well written. It can be considered as a first class reference on free Lie algebras. It can be highly recommended to everyone interested in this subject.

Reviewer: Daniel Krob (Paris)

##### MSC:

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

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

05E05 | Symmetric functions and generalizations |

20C15 | Ordinary representations and characters |

20F12 | Commutator calculus |

05E10 | Combinatorial aspects of representation theory |

17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |

20F14 | Derived series, central series, and generalizations for groups |

68R15 | Combinatorics on words |

17B35 | Universal enveloping (super)algebras |

17A50 | Free nonassociative algebras |

94A45 | Prefix, length-variable, comma-free codes |