##
**Analysis of Dirac systems and computational algebra.**
*(English)*
Zbl 1064.30049

Progress in Mathematical Physics 39. Boston, MA: Birkhäuser (ISBN 0-8176-4255-2/hbk). xiv, 332 p. (2004).

The book contains six chapters and the bibliography. Chapter 1, Background material, provides some necessary material and the most important tools to be used throughout the book. In order to make the book accessible to the readers who are not experts in either the algebraic or analytic aspects, there is given the minimum of exposition of the various needed tools which are usually spread over several books, thus the chapter is divided into three parts: algebraic tools, analytical tools and elements of hyperfunctions theory. The first headings refers to the fundamental notions of commutative algebra that underlie both algebraic geometry and algebraic analysis, the basic notions of sheaf theory, and the foundations of the theory of Gröbner bases. The part on analytical tools assembles the fundamental ideas on the space of distributions. The last section provides the reader with the fundamental notions about hyperfunctions as specific generalized functions. An appendix contains the basic definitions of category theory. It is worth mentioning that although the chapter is intended to be just a collection of auxiliary, for book’s main purposes, facts it is nicely written, expresses the author’s vision of them from a certain optics and can serve also as a good survey.

An analogous observation could be made about Chapter 2, Computational algebraic analysis for systems of linear constant coefficients differential equations: it is a pleasantly readable although in no way trivial overview of those aspects of algebraic analysis which is necessary for understanding the original sections of the book making the latter sufficiently self-closed. The authors note that historically Euler was the first major mathematician to use the term algebraic analysis in connection with his important work on general solutions to linear ordinary differential equations with constant coefficients. Currently the term refers to the work of the Japanese school of Kyoto (Sato, Kashiwara, Kawai and their co-workers) which founded and developed methods to analyze algebraically systems of linear partial differential equations with real analytic coefficients. Their results, however, rest on some preliminary work, in which algebra was used to study general properties of systems of linear differential equations with constant coefficients. The titles of the sections of the chapter give a good idea of its contents: Section 2.1, A primer of algebraic analysis; Section 2.2, The Ehrenpreis-Palamodov fundamental principle, where there is discussed the culminating and most important result of this algebraic theory, which allows to write the solutions of a homogeneous system of partial differential equations with constant coefficients in integral form thus representing a very far reaching generalization of Euler’s Fundamental principle; Section 2.3, The Fundamental Principle for hyperfunctions, where it is shown that the Fundamental Principle holds also for hyperfunction solutions of homogeneous systems of p.d.e. with constant coefficients; Section 2.4, Using computational algebra software: a section in experimental mathematics, were there are considered the systems which have an arbitrarily large number of equations and unknowns thus making it impossible to compute directly the Gröbner bases for the modules involved as well as their syzygies; in order to make reasonable guesses, it is necessary to use some computer software; how to do this, is explained in the section.

Chapter 3 deals with quaternion-valued functions. First of all, in Section 3.1, Regular functions of one quaternionic variable, there are introduced the main results for those functions which, by the way, bear also the names monogenic, hyperholomorphic, \(\mathbb H\)-regular, etc. in the literature. The material of the section is mostly instrumental serving for developing the theory of quaternionic hyperfunctions in one variable. The latter is being done in Section 3.2, Quaternionic hyperfunctions in one variable, providing the foundation for a quaternionic theory of hyperfunctions, at least for one quaternionic variable. There are considered here: the sheaf of regular functions; an analogue of the Mittag-Leffler theorem; the fact that the sheaf of \(\mathbb H\)-hyperfunctions defines a flabby sheaf, but this is only a small part of what the reader can find here. Next Section 3.3, Several quaternionic variables: an analytic approach, summatrizes a series of results obtained by D. Pertici for regular quaternionic functions of several quaternionic variables such as a quaternionic analogue of the Bochner-Martinelli formula and Hartog’s theorem for regular functions in \(\mathbb {H}^n\). In a natural way, it is followed by Section 3.4, Several quaternionic variables: an algebraic approach, which is aimed at comprehending of certain phenomena from the previous section in terms of algebraic analysis. The authors begin with a careful analysis of the case of two quaternionic variables showing in particular how the non-commutativity of the quaternionic multiplication causes difficulties which do not allow one to use Ehrenpreis’s approach to the removability of compact singularities of regular functions. Using efficiently analytic tools such as, for instance, the right-hand side invertibility of the Cauchy-Fueter operator, there is found a set of syzygies to which the computational means are, then, applied. This is complemented with some reasonings about the case of three variables leading, then, to some conjectures about the general case of \(n\) variables. A special attention is paid to an explanation of the role of the Gröbner bases and the CoCoA method.

The above conjectures deal with the length of the resolutions, the degree of the syzygies and the Betti numbers of the resolution. The proofs of them constitute the rest of the section; the theorems formulations require mostly unwieldy formulas for being exposed so that they are omitted here. Section 3.5, The Moisil-Theodorescu system, studies a variation of the Cauchy-Fueter system both in one and several short quaternionic variables. As the authors remark, there is no reason to suppose that they share the same algebraic analysis so that the section discusses both common features and differences; for instance it is shown that the two theories have characteristic varieties of different dimensions; at the same time, the Moisil-Theodorescu regular functions keep having Hartogs’ phenomenon.

Chapter 4, Special first order systems in Clifford analysis, is intended for extending the ideas and results of Chapter 3 onto the Clifford analysis setting by which it is meant the study of the Dirac operator and some of its variations. Since an arbitrary Clifford algebra is a much more subtle and sophisticated object than the skew-field of quaternions, the whole Section 4.1, Introduction to Clifford algebras, comments on many aspects of it. The reader finds here a description of standard Clifford algebras, sub-sections about their endomorphisms and spinor spaces, and about classification of real Clifford algebras. Section 4.2, Introduction to Clifford analysis, introduces, first, the Dirac and the Weyl operators, explains why Dirac operator is Spin\((m)\)-invariant and why the Cauchy-Fueter equation plays a special role. After that, there is recalled that the resolutions associated to the Cauchy-Fueter system cannot, in general, be described only in terms of the Cauchy-Fueter operators and it is noted that this is, in a sense, a rather unpleasant characteristic of the quaternionic setting; in the general Clifford analysis case, the theory of Dirac complexes sometimes leads to a much simpler structure, which can be described in terms of what the author call radical algebra. The section explains the notion, as well as that of Fischer decomposition. In Section 4.3, The Dirac complex for two, three and four operators, it is carried out an analysis similar to that of Chapter 3, now for the case of Dirac operators acting on Clifford algebra-valued functions. The main feature is the complete description of all the maps in the complex, thus giving the possibility to apply the duality theorems in Section 2.1. The title of the section refers to the number of the Dirac operators involved, that is, to the number of hypercomplex variables, but the dimension of each variable is also of importance.

In particular, it’s turned out that the Dirac complexes depend on both parameters: for two operators there is one for \(m=2\) and another for \(m>2\), \(m\) being the number of variables; for three operators the situation is much more complicated; for four operators only a partial description has been given. The case of more than four operators is not mentioned.

Section 4.4, Special systems in Clifford analysis, consists of the three subsections: Subsection 4.4.1, Generalized systems, meaning some variations of the Dirac operators obtained by replacing \(\sum^m_{i=1} e_i\partial_{x_i}\) with \(\sum^m_{i=1} \b{u}_i \partial_{x_i}\) where \(\b{u}_i\) are arbitrary elements in a Clifford algebra \(\mathbb{C}_M\) although the treatment is restricted to the case in which the coefficients of the operators are Clifford 1-vectors; Subsection 4.4.2, Systems using the Witt basis, meaning the usage of the Hermitian setting, Subsection 4.4.3, Combinatorial systems.

For generalized systems, there is introduced a concept of being Dirac-like when the resolution of a system behaves as the resolution of a system of \(n\) Dirac operators in \(\mathbb{C}_M\); that is, if it has the same length, the same degrees of the mappings, and Betti numbers proportional to the corresponding invariants of the resolution of \(n\) Dirac operators. There are obtained several sufficient conditions ensuring a system to be Dirac-like. The other two subsections contain an ample gamma of systems which originate from other reasons; in particular, the combinatorial systems are constructed starting from some incidence structures that correspond to finite geometries, which allowed to have interpreted quite nicely the results of algebraic analysis of those systems.

Chapter 5, Some first order linear operators in physics, is a continuation, in a sense, of the previous one since it keeps considering special systems, now those having a strong flavor of physics. In Section 5.1, Physics and algebra of Maxwell and Proca fields, an algebraic analysis is realized directly, without appealing to Clifford algebras, whilst in Section 5.2, Variations on Maxwell system in the space of biquaternions, the Maxwell system is imbedded into the regularity condition in the sense of biquaternionic, or complex quaternionic, analysis called in the book the \(\mathfrak{D}_Z\)-regularity; the latter is studied in Section 5.3, Properties of \(\mathfrak{D}_Z\)-regular functions. Sections 5.4, The Dirac equation and the linearization problem, and 5.5, Octonionic Dirac equation, contain not too much of algebraic analysis.

Short Chapter 6, Open problems and avenues for further research, and Bibliography of 211 titles complete the book.

Altogether the book is a pioneering, and quite successful attempt to apply computational and algebraic techniques to several branches of hypercomplex analysis but not only to it. This is even more surprising since Clifford analysis, or study of Dirac and some other related systems, has really undergone a renaissance in the last 15–20 years, but the book provides a very different way to look at some important questions which arise when one tries to develop multidimensional theories.

An analogous observation could be made about Chapter 2, Computational algebraic analysis for systems of linear constant coefficients differential equations: it is a pleasantly readable although in no way trivial overview of those aspects of algebraic analysis which is necessary for understanding the original sections of the book making the latter sufficiently self-closed. The authors note that historically Euler was the first major mathematician to use the term algebraic analysis in connection with his important work on general solutions to linear ordinary differential equations with constant coefficients. Currently the term refers to the work of the Japanese school of Kyoto (Sato, Kashiwara, Kawai and their co-workers) which founded and developed methods to analyze algebraically systems of linear partial differential equations with real analytic coefficients. Their results, however, rest on some preliminary work, in which algebra was used to study general properties of systems of linear differential equations with constant coefficients. The titles of the sections of the chapter give a good idea of its contents: Section 2.1, A primer of algebraic analysis; Section 2.2, The Ehrenpreis-Palamodov fundamental principle, where there is discussed the culminating and most important result of this algebraic theory, which allows to write the solutions of a homogeneous system of partial differential equations with constant coefficients in integral form thus representing a very far reaching generalization of Euler’s Fundamental principle; Section 2.3, The Fundamental Principle for hyperfunctions, where it is shown that the Fundamental Principle holds also for hyperfunction solutions of homogeneous systems of p.d.e. with constant coefficients; Section 2.4, Using computational algebra software: a section in experimental mathematics, were there are considered the systems which have an arbitrarily large number of equations and unknowns thus making it impossible to compute directly the Gröbner bases for the modules involved as well as their syzygies; in order to make reasonable guesses, it is necessary to use some computer software; how to do this, is explained in the section.

Chapter 3 deals with quaternion-valued functions. First of all, in Section 3.1, Regular functions of one quaternionic variable, there are introduced the main results for those functions which, by the way, bear also the names monogenic, hyperholomorphic, \(\mathbb H\)-regular, etc. in the literature. The material of the section is mostly instrumental serving for developing the theory of quaternionic hyperfunctions in one variable. The latter is being done in Section 3.2, Quaternionic hyperfunctions in one variable, providing the foundation for a quaternionic theory of hyperfunctions, at least for one quaternionic variable. There are considered here: the sheaf of regular functions; an analogue of the Mittag-Leffler theorem; the fact that the sheaf of \(\mathbb H\)-hyperfunctions defines a flabby sheaf, but this is only a small part of what the reader can find here. Next Section 3.3, Several quaternionic variables: an analytic approach, summatrizes a series of results obtained by D. Pertici for regular quaternionic functions of several quaternionic variables such as a quaternionic analogue of the Bochner-Martinelli formula and Hartog’s theorem for regular functions in \(\mathbb {H}^n\). In a natural way, it is followed by Section 3.4, Several quaternionic variables: an algebraic approach, which is aimed at comprehending of certain phenomena from the previous section in terms of algebraic analysis. The authors begin with a careful analysis of the case of two quaternionic variables showing in particular how the non-commutativity of the quaternionic multiplication causes difficulties which do not allow one to use Ehrenpreis’s approach to the removability of compact singularities of regular functions. Using efficiently analytic tools such as, for instance, the right-hand side invertibility of the Cauchy-Fueter operator, there is found a set of syzygies to which the computational means are, then, applied. This is complemented with some reasonings about the case of three variables leading, then, to some conjectures about the general case of \(n\) variables. A special attention is paid to an explanation of the role of the Gröbner bases and the CoCoA method.

The above conjectures deal with the length of the resolutions, the degree of the syzygies and the Betti numbers of the resolution. The proofs of them constitute the rest of the section; the theorems formulations require mostly unwieldy formulas for being exposed so that they are omitted here. Section 3.5, The Moisil-Theodorescu system, studies a variation of the Cauchy-Fueter system both in one and several short quaternionic variables. As the authors remark, there is no reason to suppose that they share the same algebraic analysis so that the section discusses both common features and differences; for instance it is shown that the two theories have characteristic varieties of different dimensions; at the same time, the Moisil-Theodorescu regular functions keep having Hartogs’ phenomenon.

Chapter 4, Special first order systems in Clifford analysis, is intended for extending the ideas and results of Chapter 3 onto the Clifford analysis setting by which it is meant the study of the Dirac operator and some of its variations. Since an arbitrary Clifford algebra is a much more subtle and sophisticated object than the skew-field of quaternions, the whole Section 4.1, Introduction to Clifford algebras, comments on many aspects of it. The reader finds here a description of standard Clifford algebras, sub-sections about their endomorphisms and spinor spaces, and about classification of real Clifford algebras. Section 4.2, Introduction to Clifford analysis, introduces, first, the Dirac and the Weyl operators, explains why Dirac operator is Spin\((m)\)-invariant and why the Cauchy-Fueter equation plays a special role. After that, there is recalled that the resolutions associated to the Cauchy-Fueter system cannot, in general, be described only in terms of the Cauchy-Fueter operators and it is noted that this is, in a sense, a rather unpleasant characteristic of the quaternionic setting; in the general Clifford analysis case, the theory of Dirac complexes sometimes leads to a much simpler structure, which can be described in terms of what the author call radical algebra. The section explains the notion, as well as that of Fischer decomposition. In Section 4.3, The Dirac complex for two, three and four operators, it is carried out an analysis similar to that of Chapter 3, now for the case of Dirac operators acting on Clifford algebra-valued functions. The main feature is the complete description of all the maps in the complex, thus giving the possibility to apply the duality theorems in Section 2.1. The title of the section refers to the number of the Dirac operators involved, that is, to the number of hypercomplex variables, but the dimension of each variable is also of importance.

In particular, it’s turned out that the Dirac complexes depend on both parameters: for two operators there is one for \(m=2\) and another for \(m>2\), \(m\) being the number of variables; for three operators the situation is much more complicated; for four operators only a partial description has been given. The case of more than four operators is not mentioned.

Section 4.4, Special systems in Clifford analysis, consists of the three subsections: Subsection 4.4.1, Generalized systems, meaning some variations of the Dirac operators obtained by replacing \(\sum^m_{i=1} e_i\partial_{x_i}\) with \(\sum^m_{i=1} \b{u}_i \partial_{x_i}\) where \(\b{u}_i\) are arbitrary elements in a Clifford algebra \(\mathbb{C}_M\) although the treatment is restricted to the case in which the coefficients of the operators are Clifford 1-vectors; Subsection 4.4.2, Systems using the Witt basis, meaning the usage of the Hermitian setting, Subsection 4.4.3, Combinatorial systems.

For generalized systems, there is introduced a concept of being Dirac-like when the resolution of a system behaves as the resolution of a system of \(n\) Dirac operators in \(\mathbb{C}_M\); that is, if it has the same length, the same degrees of the mappings, and Betti numbers proportional to the corresponding invariants of the resolution of \(n\) Dirac operators. There are obtained several sufficient conditions ensuring a system to be Dirac-like. The other two subsections contain an ample gamma of systems which originate from other reasons; in particular, the combinatorial systems are constructed starting from some incidence structures that correspond to finite geometries, which allowed to have interpreted quite nicely the results of algebraic analysis of those systems.

Chapter 5, Some first order linear operators in physics, is a continuation, in a sense, of the previous one since it keeps considering special systems, now those having a strong flavor of physics. In Section 5.1, Physics and algebra of Maxwell and Proca fields, an algebraic analysis is realized directly, without appealing to Clifford algebras, whilst in Section 5.2, Variations on Maxwell system in the space of biquaternions, the Maxwell system is imbedded into the regularity condition in the sense of biquaternionic, or complex quaternionic, analysis called in the book the \(\mathfrak{D}_Z\)-regularity; the latter is studied in Section 5.3, Properties of \(\mathfrak{D}_Z\)-regular functions. Sections 5.4, The Dirac equation and the linearization problem, and 5.5, Octonionic Dirac equation, contain not too much of algebraic analysis.

Short Chapter 6, Open problems and avenues for further research, and Bibliography of 211 titles complete the book.

Altogether the book is a pioneering, and quite successful attempt to apply computational and algebraic techniques to several branches of hypercomplex analysis but not only to it. This is even more surprising since Clifford analysis, or study of Dirac and some other related systems, has really undergone a renaissance in the last 15–20 years, but the book provides a very different way to look at some important questions which arise when one tries to develop multidimensional theories.

Reviewer: Michael Shapiro (Mexico)

### MSC:

30G35 | Functions of hypercomplex variables and generalized variables |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

35Q40 | PDEs in connection with quantum mechanics |