Abstract
In this chapter we review the main concepts from the theory of topological groups and Lie groups that we need for the main part of the book. We define topological groups and present some important of Mathematics and Statisticstant examples, including the classical groups. We briefly describe the role of covering groups and investigate properties of Haar measure on locally compact, Hausdorff groups. We define Lie groups and their Lie algebras, and introduce the important concepts of the exponential map and the adjoint representation. In the last part of the chapter we consider the Laplacian as an algebraic object belonging to the universal enveloping algebra of the group.
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Notes
- 1.
There is one additional classical group, the complex symplectic group \(Sp(n, \mathbb {C})\), which we will not need to discuss in this volume.
- 2.
It is interesting that these are the only real finite-dimensional division algebras. There is one further nonassociative real finite-dimensional division algebra and this is the octonions. The collection of unit vectors therein may be identified with the sphere \(S^{7}\). Another fascinating fact is that the only spheres that have the geometric property of being parallisable are \(S^{0}, S^{1}, S^{3}\) and \(S^{7}\).
- 3.
The use of the inverse in left but not right translation makes life easier later on when we want the homomorphism property to hold for the pullback to various function spaces.
- 4.
This means that every point \(p\) in \(X\) has a neighbourhood \(U\) so that any closed loop in \(U\) that is based at \(p\) may be continuously deformed to \(p\). Any connected finite-dimensional manifold has this property.
- 5.
In this case \(\natural \) must also be a group homomorphism.
- 6.
See Appendix A.5 for background on regular Borel measures.
- 7.
An alternative approach (which is more in the spirit of Haar’s pioneering 1933 paper) in which the measure is constructed directly, may be found in e.g. Cohn [50], Theorem 9.2.1, pp. 305–309 or Folland [68], Sect. 2.2.
- 8.
We use the same notation for norms in \(C_{0}(G)\) and \(L^{\infty }(G, m_{L})\), but it should be clear from the context which space we are in.
- 9.
See the section “Guide to Notation and a Few Useful Facts”, if you need background on the complexification of a real vector space.
- 10.
In some books, both instances of \(C^{\infty }\) in the definition of a Lie group are replaced by the stronger condition of “real analyticity”. In fact, these two seemingly different ways of defining a Lie group give rise to exactly the same objects. This is a consequence of the solution of Hilbert’s Fifth Problem; see the classic work by Montomery and Zippin [151], and for more modern treatments, Chapter H in Stroppel [199] or Chapter II of Kaplansky [112].
- 11.
The group \(\Pi ^{\infty }\) is compact, and this is a consequence of Tychonoff’s theorem (see Appendix A.1). The \(p\)-adic number fields are locally compact, abelian groups (see Folland [68], pp. 34–36).
- 12.
The reason for the word “representation” here will become clear in Chap. 2.
- 13.
“Geometry is the science which studies the properties of figures preserved under the transformations of a certain group of transformations, or, as one also says, the science which studies the invariants of a group of transformations,” see Yaglom [218], p. 115.
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Applebaum, D. (2014). Lie Groups. In: Probability on Compact Lie Groups. Probability Theory and Stochastic Modelling, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-319-07842-7_1
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