##
**Polynomial identities and asymptotic methods.**
*(English)*
Zbl 1105.16001

Mathematical Surveys and Monographs 122. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3829-6/hbk). xiii, 352 p. (2005).

We start with a brief summary of results of the theory of PI-algebras, or algebras with polynomial identities, related to the main topics of the present book. Recall that a PI-algebra is an associative algebra \(A\) over a field \(F\) which satisfies a nontrivial polynomial identity. This means that there exists a nonzero polynomial \(f(x_1,\dots,x_n)\) in the free associative algebra \(F\langle X\rangle=F\langle x_1,x_2,\dots\rangle\) such that \(f(a_1,\dots,a_n)=0\) for all \(a_1,\dots,a_n\in A\). Hence PI-algebras may be considered as a generalization of commutative algebras (satisfying the polynomial identity \(x_1x_2-x_2x_1\)) and finite dimensional algebras (which satisfy the standard identity \(St_n(x_1,\dots,x_n)\), \(n>\dim A\)).

The origin of PI-algebras can be traced back in the papers by Dehn, 1922, and Wagner, 1936, in their study of projective geometry with noncommutative coordinates. Later, starting with the pioneer paper by Kaplansky, 1948, and further developed in the 1960’s and 1970’s, it turned out that PI-algebras have a rich structure theory, in the spirit of the structure theory of finite dimensional algebras and commutative algebras. In 1950, Amitsur and Levitzki proved, in a purely combinatorial way, that the \(k\times k\) matrix algebra over a field satisfies the standard identity of degree \(2k\), and this is the only identity of minimal degree for this algebra. This was the beginning of a new approach to PI-algebras, to study the polynomial identities satisfied by a concrete important algebra. The discovery, by purely combinatorial methods, of central polynomials for matrices, independently due to Formanek and Razmyslov, 1972-73, gave rise to a new development of the structure theory of PI-algebras. Later, the combinatorial theory of PI-algebras was related with the more general theory of trace identities developed in the 1970’s by Procesi via invariant theory and independently by Razmyslov.

The description of the T-ideal \(\text{Id}(A)\) of a given algebra \(A\), or the ideal of the free associative algebra \(F\langle X\rangle=F\langle x_1,x_2,\dots\rangle\) consisting of the polynomial identities of the algebra \(A\), is a difficult problem completely solved in few cases only. Specht, 1950, conjectured that when \(F\) is a field of characteristic 0, every T-ideal is finitely generated, and the Specht problem was one of the main driving forces in PI-theory for more than 25 years. It was solved completely in a series of papers by Kemer, who developed the structure theory of T-ideals, involving superidentities of superalgebras and graded tensor products with the Grassmann, or exterior, algebra. In recent years the technique of Kemer has become one of the basic tools in the study of polynomial identities of a given algebra.

In characteristic 0, every polynomial identity is equivalent to a set of multilinear identities. Therefore, it is natural to slice any T-ideal into the subspaces \(P_n\subset K\langle X\rangle\) of multilinear polynomials of degree \(n\). This allows to measure how big is \(\text{Id}(A)\) considering the sequence \(\dim(P_n\cap\text{Id}(A))\), \(n=1,2,\dots\). In a seminal paper in 1972, Regev proved that the vector space \(P_n\cap\text{Id}(A)\) is very big and is very close to \(P_n\) itself. So, he suggested to introduce the codimension sequence \(c_n(A)=\dim P_n/(P_n\cap\text{Id}(A))\), \(n=1,2,\dots\). The main quantitative result in the paper by Regev was that \(c_n(A)\leq a^n\) for some \(a\geq 0\), i.e. the codimensions are exponentially bounded. From here Regev derived the important structural result that the tensor product of PI-algebras is PI again. Since \(P_n\) has the natural structure of an \(S_n\)-module and \(\text{Id}(A)\) is invariant under endomorphisms, \(P_n/(P_n\cap\text{Id}(A))\) inherits the \(S_n\)-module structure of \(P_n\). Nowadays the sequence of \(S_n\)-characters of \(P_n/(P_n\cap\text{Id}(A))\), \(n=1,2,\dots\), (the cocharacter sequence of \(A\)), is one of the main objects in the modern quatitative study of the T-ideals as well as one of the main tools in the theory.

The present book is devoted to the study of the asymptotic behaviour of the main numerical invariants of polynomial identities, mostly over a field of characteristic 0. The theorem of Regev for the exponential growth of the codimensions implies that for any PI-algebra \(A\) one has that \(\limsup_{n\to\infty}\root n \of{c_n(A)}\) exists. In the few cases when the behaviour of the codimensions was known it turned out that \(\lim_{n\to\infty}\root n\of{c_n(A)}\) also exists and is an integer. In the early 1980’s, Amitsur conjectured that this is a general phenomenon. A more precise conjecture of Regev states that \(c_n(A)\approx\alpha\cdot n^\beta\cdot\gamma^n\), where \(\alpha\in\mathbb{R}\), \(\alpha\geq 0\), \(\beta\in\mathbb{Z}/2\) and \(\gamma\in\mathbb{Z}\), \(\gamma\geq 0\). The core of the book is the highly nontrivial proof of the conjecture of Amitsur given by the authors in a series of papers in the end of the 1990’s. In their approach they combined the combinatorial methods whose development was started by Regev with the structure theory of T-ideals due to Kemer.

The first chapters of the book are introductory. Chapter 1 contains the basic definitions and an account of the main results of the structure theory of PI-algebras. Chapter 2 deals with representation theory of symmetric groups because this is one of the main tools in the quantitative study of polynomial identities. Chapter 3 is devoted to group graded algebras and their graded identities and especially superalgebras and polynomial superidentities. This is a fast growing subject. More importantly, there is a well understood connection between superidentities and ordinary identities that allows to reduce many of the considerations to the finite dimensional case, and this is one of the basic reductions in the whole book. Chapter 4 studies codimensions and colengths and the most important properties of their asymptotic behaviour. Chapter 5 considers invariant theory of matrices and trace polynomial identities. This subject is interesting of its own and is an important area of modern mathematics. Its concrete goal in the book is to prove the existence of suitable central polynomials for matrix algebras used in Chapter 6 to find the precise lower bound of the codimension growth of a given algebra.

Chapter 6 is central for the book. It contains the positive solution of the conjecture of Amitsur for the existence and the integrality of the exponent \(\exp(A)=\lim_{n\to\infty}\root n \of{c_n(A)}\) of a PI-algebra \(A\). It also gives a way to calculate explicitly the exponent in terms of the dimensions of the diagonal subalgebras of the block triangular matrices satisfying the polynomial identities of \(A\). Chapter 7 describes the PI-algebras with polynomial codimension growth, i.e. of exponent \(\leq 1\). Chapter 8 contains another important recent contribution by the authors. They consider T-ideals \(\text{Id}(A)\) with the property that the exponent of \(A\) is bigger than the exponent of any T-ideal \(\text{Id}(B)\) strictly containing \(\text{Id}(A)\). The main result is that such ideals are exactly the products of the verbally prime ideals from the theory of Kemer. Hence for each integer \(d>1\) the number of maximal T-ideals of exponent \(d\) is finite. Chapter 9 deals with exponents of concrete polynomial identities. The next two chapters extend the approach of the authors to graded algebras and algebras with involution. In this case, instead of representation theory of the symmetric group the authors use representation theory of wreath products. Finally, Chapter 12 deals with numerical invariants and their asymptotics in the case of some classes of nonassociative algebras, as Lie algebras and superalgebras, alternative and Jordan algebras. The Appendix contains the proof of a theorem of Cohen and Regev which generalizes the classical four squares theorem of Lagrange to the case of hyperbolic (or super) integers. Mysteriously, such kind of results is related with the exponent of some concrete polynomial identities.

Recently, several books devoted to the theory of PI-algebras have appeared. Nevertheless, the present book has a minor intersection with the others. Written by two of the leading experts in the theory of PI-algebras, the book is interesting and useful. I believe that the algebraic community will share my opinion. The text is suitable both for beginners and experts. The book (or parts of it) may serve as a graduate course on PI-algebras. It can be used as a good source of references.

The origin of PI-algebras can be traced back in the papers by Dehn, 1922, and Wagner, 1936, in their study of projective geometry with noncommutative coordinates. Later, starting with the pioneer paper by Kaplansky, 1948, and further developed in the 1960’s and 1970’s, it turned out that PI-algebras have a rich structure theory, in the spirit of the structure theory of finite dimensional algebras and commutative algebras. In 1950, Amitsur and Levitzki proved, in a purely combinatorial way, that the \(k\times k\) matrix algebra over a field satisfies the standard identity of degree \(2k\), and this is the only identity of minimal degree for this algebra. This was the beginning of a new approach to PI-algebras, to study the polynomial identities satisfied by a concrete important algebra. The discovery, by purely combinatorial methods, of central polynomials for matrices, independently due to Formanek and Razmyslov, 1972-73, gave rise to a new development of the structure theory of PI-algebras. Later, the combinatorial theory of PI-algebras was related with the more general theory of trace identities developed in the 1970’s by Procesi via invariant theory and independently by Razmyslov.

The description of the T-ideal \(\text{Id}(A)\) of a given algebra \(A\), or the ideal of the free associative algebra \(F\langle X\rangle=F\langle x_1,x_2,\dots\rangle\) consisting of the polynomial identities of the algebra \(A\), is a difficult problem completely solved in few cases only. Specht, 1950, conjectured that when \(F\) is a field of characteristic 0, every T-ideal is finitely generated, and the Specht problem was one of the main driving forces in PI-theory for more than 25 years. It was solved completely in a series of papers by Kemer, who developed the structure theory of T-ideals, involving superidentities of superalgebras and graded tensor products with the Grassmann, or exterior, algebra. In recent years the technique of Kemer has become one of the basic tools in the study of polynomial identities of a given algebra.

In characteristic 0, every polynomial identity is equivalent to a set of multilinear identities. Therefore, it is natural to slice any T-ideal into the subspaces \(P_n\subset K\langle X\rangle\) of multilinear polynomials of degree \(n\). This allows to measure how big is \(\text{Id}(A)\) considering the sequence \(\dim(P_n\cap\text{Id}(A))\), \(n=1,2,\dots\). In a seminal paper in 1972, Regev proved that the vector space \(P_n\cap\text{Id}(A)\) is very big and is very close to \(P_n\) itself. So, he suggested to introduce the codimension sequence \(c_n(A)=\dim P_n/(P_n\cap\text{Id}(A))\), \(n=1,2,\dots\). The main quantitative result in the paper by Regev was that \(c_n(A)\leq a^n\) for some \(a\geq 0\), i.e. the codimensions are exponentially bounded. From here Regev derived the important structural result that the tensor product of PI-algebras is PI again. Since \(P_n\) has the natural structure of an \(S_n\)-module and \(\text{Id}(A)\) is invariant under endomorphisms, \(P_n/(P_n\cap\text{Id}(A))\) inherits the \(S_n\)-module structure of \(P_n\). Nowadays the sequence of \(S_n\)-characters of \(P_n/(P_n\cap\text{Id}(A))\), \(n=1,2,\dots\), (the cocharacter sequence of \(A\)), is one of the main objects in the modern quatitative study of the T-ideals as well as one of the main tools in the theory.

The present book is devoted to the study of the asymptotic behaviour of the main numerical invariants of polynomial identities, mostly over a field of characteristic 0. The theorem of Regev for the exponential growth of the codimensions implies that for any PI-algebra \(A\) one has that \(\limsup_{n\to\infty}\root n \of{c_n(A)}\) exists. In the few cases when the behaviour of the codimensions was known it turned out that \(\lim_{n\to\infty}\root n\of{c_n(A)}\) also exists and is an integer. In the early 1980’s, Amitsur conjectured that this is a general phenomenon. A more precise conjecture of Regev states that \(c_n(A)\approx\alpha\cdot n^\beta\cdot\gamma^n\), where \(\alpha\in\mathbb{R}\), \(\alpha\geq 0\), \(\beta\in\mathbb{Z}/2\) and \(\gamma\in\mathbb{Z}\), \(\gamma\geq 0\). The core of the book is the highly nontrivial proof of the conjecture of Amitsur given by the authors in a series of papers in the end of the 1990’s. In their approach they combined the combinatorial methods whose development was started by Regev with the structure theory of T-ideals due to Kemer.

The first chapters of the book are introductory. Chapter 1 contains the basic definitions and an account of the main results of the structure theory of PI-algebras. Chapter 2 deals with representation theory of symmetric groups because this is one of the main tools in the quantitative study of polynomial identities. Chapter 3 is devoted to group graded algebras and their graded identities and especially superalgebras and polynomial superidentities. This is a fast growing subject. More importantly, there is a well understood connection between superidentities and ordinary identities that allows to reduce many of the considerations to the finite dimensional case, and this is one of the basic reductions in the whole book. Chapter 4 studies codimensions and colengths and the most important properties of their asymptotic behaviour. Chapter 5 considers invariant theory of matrices and trace polynomial identities. This subject is interesting of its own and is an important area of modern mathematics. Its concrete goal in the book is to prove the existence of suitable central polynomials for matrix algebras used in Chapter 6 to find the precise lower bound of the codimension growth of a given algebra.

Chapter 6 is central for the book. It contains the positive solution of the conjecture of Amitsur for the existence and the integrality of the exponent \(\exp(A)=\lim_{n\to\infty}\root n \of{c_n(A)}\) of a PI-algebra \(A\). It also gives a way to calculate explicitly the exponent in terms of the dimensions of the diagonal subalgebras of the block triangular matrices satisfying the polynomial identities of \(A\). Chapter 7 describes the PI-algebras with polynomial codimension growth, i.e. of exponent \(\leq 1\). Chapter 8 contains another important recent contribution by the authors. They consider T-ideals \(\text{Id}(A)\) with the property that the exponent of \(A\) is bigger than the exponent of any T-ideal \(\text{Id}(B)\) strictly containing \(\text{Id}(A)\). The main result is that such ideals are exactly the products of the verbally prime ideals from the theory of Kemer. Hence for each integer \(d>1\) the number of maximal T-ideals of exponent \(d\) is finite. Chapter 9 deals with exponents of concrete polynomial identities. The next two chapters extend the approach of the authors to graded algebras and algebras with involution. In this case, instead of representation theory of the symmetric group the authors use representation theory of wreath products. Finally, Chapter 12 deals with numerical invariants and their asymptotics in the case of some classes of nonassociative algebras, as Lie algebras and superalgebras, alternative and Jordan algebras. The Appendix contains the proof of a theorem of Cohen and Regev which generalizes the classical four squares theorem of Lagrange to the case of hyperbolic (or super) integers. Mysteriously, such kind of results is related with the exponent of some concrete polynomial identities.

Recently, several books devoted to the theory of PI-algebras have appeared. Nevertheless, the present book has a minor intersection with the others. Written by two of the leading experts in the theory of PI-algebras, the book is interesting and useful. I believe that the algebraic community will share my opinion. The text is suitable both for beginners and experts. The book (or parts of it) may serve as a graduate course on PI-algebras. It can be used as a good source of references.

Reviewer: Vesselin Drensky (Sofia)

### MSC:

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

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

16R20 | Semiprime p.i. rings, rings embeddable in matrices over commutative rings |

16R30 | Trace rings and invariant theory (associative rings and algebras) |

16R40 | Identities other than those of matrices over commutative rings |

16R50 | Other kinds of identities (generalized polynomial, rational, involution) |

16P90 | Growth rate, Gelfand-Kirillov dimension |

16W22 | Actions of groups and semigroups; invariant theory (associative rings and algebras) |

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

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

20C30 | Representations of finite symmetric groups |