Preface

Summary

This book stems from notes of a master's course given at the Université Pierre et Marie Curie. This is an introduction to the theory of Lie groups and to the study of their representations, with applications to analysis. In this introductory text we do not present the general theory of Lie groups, which assumes a knowledge of differential manifolds. We restrict ourself to linear Lie groups, that is groups of matrices. The tools used to study these groups come mainly from linear algebra and differential calculus. A linear Lie group is defined as a closed subgroup of the linear group GL(n, ℝ). The exponential map makes it possible to associate to a linear Lie group its Lie algebra, which is a subalgebra of the algebra of square matrices M(n, ℝ) endowed with the bracket [X, Y] = XY − YX. Then one can show that every linear Lie group is a manifold embedded in the finite dimensional vector space M(n, ℝ). This is an advantage of the definition we give of a linear Lie group, but it is worth noticing that, according to this definition, not every Lie subalgebra of M(n, ℝ) is the Lie algebra of a linear Lie group, that is a closed subgroup of GL(n, ℝ). The Haar measure of a linear Lie group is built in terms of differential forms, and these are used to establish several integration formulae, linking geometry and analysis.