8. Actions of Lie groups and Lie algebras 20 9. Universal covering groups 25 10. The universal enveloping algebra 27 11. Representation theory of sl(2, C). 31 12. Compact Lie groups 35 13. The maximal torus of a compact Lie group 42 ...
turns out that in many ways such groups can be described and studied as easily as finitely generated groups — or even easier. The key role is played by the notion of a Lie algebra, the tangent space to G at identity. It turns out that the group operation on G defines a certain...
1Matrix Lie Groups1.1 Definition of a Matrix Lie GroupWe begin with a very important class of groups, the general linear groups. Thegroups we will study in this book will all be subgroups (of a certain sort) ofone of the general linear groups. This chapter makes use of various standard...
Lie AlgebraComments on theG →GRelationshipVarious Kinds of and Operations with Lie AlgebrasExercises for Chapter 9 Local Coordinates in a Lie Group Analysis of Associativity One-parameter Subgroups and Canonical Coordinates Integrability Conditions and Structure Constants Definition of a (real) Lie ...
We providea much simple definition for a matrix Lie group in Section 4. Showing that a matrix Liegroup is in fact a Lie group is discussed in standard texts such as [2].We also discuss Lie algebras [1], and the computation of the Lie algebra of a Liegroup in Section 5. We will ...
The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic ...
a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the...
The aim is to introduce the reader to the "Lie dictionary": Lie algebras and Lie groups. Special features of the presentation are its emphasis on formal groups (in the Lie group part) and the use of analytic manifolds on p-adic fields. Some knowledge of algebra and calculus is required ...
In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map...
In finite dimensions, the relation between Lie groups and Lie algebras isalmost one-to-one, the “almost” being due to the fact that the universal covering groupof a finite-dimensional Lie algebra (see subsection 2.6.6) may have a nontrivial discretenormal subgroup.We end the chapter with ...