Lie Groups and Lie Algebras
- 2025
- Book
- 1. edition
- Author
- M. S. Raghunathan
- Book Series
- Texts and Readings in Mathematics
- Publisher
- Springer Nature Singapore
About this book
This is an essential textbook the advanced undergraduate and graduate students of mathematics. Organized into seven chapters, the book treats real and p-adic groups in a unified manner. Chapter 1 outlines the preliminary material that will be used in the rest of the book. Chapter 2 is on analytic functions and is of an elementary nature. This material is included to cater to students who may not be familiar with p-adic fields. Chapter 3 introduces analytic manifolds and contains standard material. The only notable feature is that it covers both real and p-adic analytic manifolds. All the standard results on Lie groups are proved in Chaps. 4 and 5. Some of the proofs are, however, different from those in the earlier literature. Some results are not found in the literature, though they are kind of folklore among the experts in Lie theory. The last two chapters (Chaps. 6 and 7) are on Lie algebras and cover the structure theory as found in the first of the Bourbaki volumes on the subject. In these chapters, some proofs are new.
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Table of Contents
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Frontmatter
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Chapter 1. Notational Conventions and Other Preliminaries
M. S. RaghunathanThe chapter begins by setting down standard notational conventions that will be used throughout the book, ensuring clarity and consistency in mathematical expressions. It covers fundamental concepts from algebra, including the definition and properties of sets, relations, and order relations, providing a solid foundation for understanding more complex algebraic structures. The chapter also delves into the theory of groups, rings, and modules, discussing their properties and interactions in detail. It explores the structure of Lie algebras, their representations, and the role of derivations, offering insights into the deeper mathematical structures that underpin many areas of advanced mathematics. Additionally, the chapter provides an introduction to the theory of locally compact groups and fields, discussing their topological properties and the role of Haar measures. It concludes with a discussion on covering spaces and the fundamental group, highlighting their importance in topology and algebraic geometry. Throughout, the chapter emphasizes the interconnectedness of these topics, providing a holistic view of the mathematical landscape.AI Generated
This summary of the content was generated with the help of AI.
AbstractWe 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. -
Chapter 2. Analytic Functions
M. S. RaghunathanThe chapter introduces the concept of k-valued analytic functions on open sets in kn, where k is a local field. It begins by defining these functions and discussing their properties, particularly when k is the field of complex numbers or real numbers. For non-archimedean fields, the chapter adopts the approach of convergent power series, providing a detailed exploration of formal power series and their operations. The text delves into the structure of the set of formal power series, including the definition of the augmentation map and the maximal ideal. It also covers substitution in formal power series and the definitions of key series such as the exponential, logarithmic, and binomial series. The chapter further discusses the convergence of power series, the radius of convergence, and the stability of convergent power series under differentiation. It concludes with a discussion on analytic functions, their properties, and the principle of analytic continuation. The chapter is rich with mathematical rigor and provides a deep dive into the theory of analytic functions, making it an essential read for those interested in advanced mathematical concepts.AI Generated
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AbstractIn 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. -
Chapter 3. Analytic Manifolds
M. S. RaghunathanThe chapter begins by outlining the theory of analytic manifolds over local fields, drawing parallels with real analytic manifolds to facilitate understanding. It discusses the fundamental concepts of charts, atlases, and the topology of these manifolds, highlighting the differences between archimedean and non-archimedean fields. The text delves into the definition of analytic diffeomorphisms and the compatibility of charts, providing a clear framework for understanding the structure of analytic manifolds. It also explores the implications of Serre's theorem, which states that every paracompact analytic manifold over a non-archimedean field is analytically isomorphic to a disjoint union of discs. The chapter further examines the definition of analytic functions and germs, the tangent space, and the differential of maps, offering a comprehensive overview of the key topics in the study of analytic manifolds. Additionally, it covers vector fields, differential forms, and integration on manifolds, providing a thorough foundation for advanced studies in this area.AI Generated
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AbstractIn 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. -
Chapter 4. Lie Groups
M. S. RaghunathanThe chapter begins by introducing Lie groups, named after the Norwegian mathematician Sophus Lie, and their fundamental role in the theory of continuous transformation groups. It explores the correspondence between Lie groups and Lie algebras, as established by the Fundamental Theorem of Lie Theory, which reduces many problems about Lie groups to algebraic questions about Lie algebras. The text delves into the contributions of key figures such as Wilhelm Killing and Elie Cartan, and the modern treatment of Lie groups as presented by Claude Chevalley. It provides a detailed definition of a Lie group over a locally compact field and discusses the analyticity of the group structure and the inverse map. The chapter also covers the differential of the group operation at the identity, the analyticity of the inverse map, and the implications for the structure of Lie groups. Furthermore, it examines the exponential map, its properties, and its role in defining the Lie algebra of a Lie group. The text includes numerous examples of Lie groups, such as the general linear group, orthogonal groups, and the special linear group, illustrating their natural analytic structures and group operations. It also discusses the exponential map in the context of the general linear group and the conditions for its convergence in both archimedean and non-archimedean fields. The chapter concludes with a discussion on the functorial nature of the assignment of Lie algebras to Lie groups and the properties of the exponential map in relation to group homomorphisms.AI Generated
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AbstractLie 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. -
Chapter 5. Lie Groups: The Theorems of Cartan and Lie
M. S. RaghunathanThe chapter begins by establishing a crucial lemma and a theorem due to Cartan, which are instrumental in proving the converse of a central result in Lie theory. It explores the Lie algebra of a Lie subgroup and its natural relationship with the Lie algebra of the ambient group. The text delves into the characterization of Lie subalgebras corresponding to Lie subgroups, providing detailed proofs and lemmas that elucidate these relationships. A significant portion of the chapter is dedicated to the Fundamental Theorem of Lie Theory, which asserts the existence of a Lie subgroup corresponding to any Lie subalgebra. The chapter also discusses the structure of Lie subgroups in both archimedean and non-archimedean fields, highlighting the differences and similarities in their behaviors. Additionally, it covers the properties of Lie subgroups in the context of GL(V) and the concept of admissible representations, offering a comprehensive view of the interplay between Lie groups and their algebras. The chapter concludes with a discussion on the derived series and central series of Lie groups and algebras, providing insights into their solvability and nilpotency.AI Generated
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AbstractThe 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. -
Chapter 6. Lie Algebras: Theorems of Engel, Lie and Cartan
M. S. RaghunathanThe chapter begins by establishing the foundational role of Lie algebras in understanding Lie groups, setting the stage for an in-depth exploration of Lie algebras over an arbitrary field of characteristic zero. It introduces key theorems by Engel and Lie, which are crucial for understanding nilpotent and solvable Lie algebras, and Cartan's theorem, which provides a criterion for the solvability of Lie subalgebras. The text delves into the structure of Lie algebras, discussing representations, the adjoint representation, and the Killing form, which are essential for grasping the deeper properties of these algebraic structures. The chapter also covers derivations, semi-direct products, and the implications of these concepts for Lie groups, particularly when the ground field is a local field. It concludes with a discussion on semi-simple Lie algebras, their characterization, and the role of the Killing form in determining their structure. Throughout, the chapter provides numerous examples and proofs, making it a comprehensive resource for advanced study in Lie theory.AI Generated
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AbstractAs 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 \(\mathfrak {gl}(V)\). -
Chapter 7. Lie Algebras: Structure Theory
M. S. RaghunathanThe chapter begins by establishing the complete reducibility of finite-dimensional representations of semi-simple Lie algebras, a result first proven by Hermann Weyl. It then presents a purely algebraic proof by J. H. C. Whitehead, adapted by Bourbaki, which also demonstrates that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple subalgebra and its radical. A key result is the proof of Ado's theorem, which asserts that every finite-dimensional Lie algebra admits a faithful finite-dimensional representation. The text delves into the structure of semi-simple Lie algebras, exploring the properties of their representations and the implications of complete reducibility. It also discusses the decomposition of Lie algebras into semi-simple and solvable components, providing a foundational understanding of their structural properties. The chapter concludes with a proof of Ado's theorem, which is crucial for the representation theory of Lie algebras, and discusses the classification of Lie algebras of low dimensions, offering insights into their unique structures and representations.AI Generated
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AbstractIn 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. -
Backmatter
- Title
- Lie Groups and Lie Algebras
- Author
-
M. S. Raghunathan
- Copyright Year
- 2025
- Publisher
- Springer Nature Singapore
- Electronic ISBN
- 978-981-9718-32-0
- DOI
- https://doi.org/10.1007/978-981-97-1832-0
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