Discrete, finite and Lie groups. Comprehensive group theory in geometry and analysis (Q6165723)

The book under review is a self-contained and exhaustive treatise on group theory dedicated in particular to physics scholars. The text is divided into eighteen chapters of which the reviewer reports the title and a brief summary. (1) Groups: the intuitive notion. (2) Fundamental notions of algebra. (3) Groups: noticeable examples and some developments. (Groups of matrices: general linear, special linear, unitary, special unitary, orthogonal, special orthogonal, symplectic.) (4) Basic elements of finite group theory. (Cayley's theorem, Lagrange's theorem, conjugacy classes, derived subgroup, Schur's lemmas, characters, irreducible representations.) (5) Finite subgroups of \(\mathrm{SO}(3)\): the ADE classification. (The A-D-E classification of finite subgroups of \(\mathrm{SO}(3)\), based on the Diophantine equations \(\frac{1}{k_{1}}+\frac{1}{k_{2}}=\frac{2}{n}\) and \(\frac{1}{k_{1}}+\frac{1}{k_{2}}+\frac{1}{k_{3}}=1+\frac{2}{n}\).) (6) Manifolds and Lie groups. (Differentiable manifolds, germs of smooth functions, tangent and cotangent spaces, differential \(k\)-forms, homotopy, homology, cohomology, the Green-Riemann formula, Cauchy theorem.) (7) The relation between Lie groups and Lie algebras. (Maurer-Cartan forms on Lie group manifolds, Maurer-Cartan equations, linear Lie groups.) (8) Crystallographic groups and group extensions. (Lattices and crystallographic groups, Bravais lattices for \(n = 3\) and \(n = 2\), group extensions and space groups, general theory of group extensions, finite group cohomology, Frobenius congruences.) (9) Monodromy groups of differential equations. (Ordinary differential equations, second order differential equations with singular points, solutions at regular singular points, Fuchsian equations with three regular singular points and the \(P\)-symbol, the hypergeometric equation and its solutions, an algebraic representation of the torus by means of a cubic.) (10) Structure of Lie algebras. (Types of Lie algebras and Levi decomposition, solvable Lie algebras, semi-simple Lie algebras, the adjoint representation and Cartan's criteria.) (11) Root systems and their classification. (Cartan subalgebras, root systems, the Weyl group, the Cartan matrix, Dynkin diagrams, identification of the classical Lie algebras, the exceptional Lie algebras.) (12) Lie algebra representation theory. (Linear representations, weights of a representation, tensor products, the Lie algebra \(\mathfrak{a}_{2}\), the Lie algebra \(\mathfrak{sp}(4, \mathbb{R}) \simeq \mathfrak{so}(2,3)\), its fundamental representation, and its Weyl group.) (13) Exceptional Lie algebras. (The exceptional Lie algebras \(\mathfrak{g}_{2}\), \(\mathfrak{f}_{4}\) and \(\mathfrak{e}_{8}\).) (14) In depth study of a simple group. (The simple group \(L_{168} \simeq \mathrm{PSL}(2, 7)\), this seven-dimensional irreducible representation, this three-dimensional complex representations and this eight-dimensional representation.) (15) A primary on the theory of connections and metrics. (Connections on principal bundles, Ehresmann connections, the magnetic monopole and the Hopf fibration of \(\mathbb{S}^{3}\), Riemannian and pseudo-Riemannian metrics, the Levi-Civita connection, affine connections, curvature and torsion of an affine connection, geodesics, Lorentzian and Riemannian manifolds, the Riemannian example of the Lobachevsky-Poincaré plane.) (16) Isometries and the geometry of coset manifolds. (Symmetric spaces and Élie Cartan, isometries and Killing vector fields, solvable group representation of non-compact coset manifolds, The Tits-Satake projection.) (17) Functional spaces and non-compact Lie algebras. (A very short introduction to measure theory and the Lebesgue integral, space of square summable functions, Hilbert space, the Weierstrass theorem, the Schmidt orthogonalization algorithm, orthogonal polynomials, the Heisenberg group and its Lie algebra, self-adjoint operators.) (18) Harmonic analysis and conclusive remarks. (Harmonics on coset spaces, differential operators on \(H\)-harmonics.) Unfortunately, misleading statements are also made in this book: e.g. (p. 14): ``The number of elements of a group \(G\) can be finite, infinite but denumerable, or continuously infinite [\dots]'', while it is well known that groups of any cardinality exist.