Lie Groups and Lie Algebras

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Notational Conventions and Other Preliminaries

   M. S. Raghunathan

   Abstract

   We set down in this chapter some notational conventions (much of it standard) for the entire book. We also record here some facts and results from algebra and topology and about locally compact groups and fields without proofs which we will be using freely. The material described can be found in standard under-graduate/graduate texts.

3. Chapter 2. Analytic Functions

   M. S. Raghunathan

   Abstract

   In this chapter we introduce the definition of a k-valued analytic function on an open set \(\Omega \) in \(k^n\) where k is a local field k and prove the main results about them. When \(k\simeq \mathbb {C}\) this is a familiar concept defined via the existence of the first derivative with respect to a complex variable. When \(k=\mathbb {R}\) one may define a \(\mathbb {R}\)-valued analytic function on \(\Omega \) as the restriction to \(\Omega \) of a \(\mathbb {C}\)-valued analytic function on an open set \(\tilde{\Omega }\) in \(\mathbb {C}^n\) containing \(\Omega \). An alternative approach (following Weierstrass) is via convergent power series over \(\mathbb {R}\). When k is not archimedean, only the second approach is available and is adopted in this chapter.

4. Chapter 3. Analytic Manifolds

   M. S. Raghunathan

   Abstract

   In this chapter we outline some of the theory of analytic manifolds over a local field k. Anyone familiar with real analytic manifolds will see that this outline is a simple carry over of basic concepts and results on real analytic manifolds to cover the non-archimedean case. We do not touch upon the deeper results on real or complex analytic manifolds. In particular we do not deal with the topology of real or complex manifolds which is a fascinating subject. The topology of analytic manifolds over a non-archimedean field, on the other hand, is far from interesting: by a theorem of Serre, every paracompact analytic manifold over a non-archimedean field is analytically isomorphic to a disjoint union of discs.

5. Chapter 4. Lie Groups

   M. S. Raghunathan

   Abstract

   Lie groups are named after the Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups. They were christened Lie groups by Arthur Tress, a student of Lie. Lie proved the basic result which sets up a correspondence between Lie Groups and Lie Algebras which we call the Fundamental Theorem of Lie Theory. This reduces most problems about Lie groups to (essentially algebraic) questions about Lie algebras. Wilhelm Killing and Elie Cartan made giant strides in developing the theory. The modern treatment of Lie groups is due to Claude Chevalley whose 1950 book on Lie groups is a classic. All text-books on Lie groups (including this one) essentially follow his treatment.

6. Chapter 5. Lie Groups: The Theorems of Cartan and Lie

   M. S. Raghunathan

   Abstract

   The Lie algebra of a Lie subgroup, as we saw, is in a natural fashion a Lie subalgebra of the Lie algebra of the ambient group. A central result of Lie theory is a converse of this assertion, due to Lie, and is known as the Fundamental Theorem of Lie Theory. Before proceeding to formulate and prove such a converse, we will establish a Lemma and a theorem due to Cartan, which we will use in the proof of the converse.

7. Chapter 6. Lie Algebras: Theorems of Engel, Lie and Cartan

   M. S. Raghunathan

   Abstract

   As we saw in Chap. 5, properties of Lie groups are controlled to a considerable extent by their Lie algebras. So a study of Lie Groups is inevitably tied up with the study of Lie algebras. In this and the next chapter we study the structure of Lie algebras. This is done over an arbitrary field k of characteristic zero—the theorems we prove do not require the assumption that the ground field is a local field. The theorems have implications for Lie groups over k when k is a local field. We draw attention to some of these implications, but this is not done exhaustively. We prove theorems due to Engel and Lie which deal with nilpotent and solvable Lie algebras and a theorem of Cartan’s giving a criterion for the solvability of a Lie subalgebra of \(\mathfrak {gl}(V)\).

8. Chapter 7. Lie Algebras: Structure Theory

   M. S. Raghunathan

   Abstract

   In this chapter we prove some basic results about semi-simple Lie algebras. The first of these is that every finite dimensional representation of a semi-simple Lie algebra is completely reducible.

9. Backmatter