Representations of Lie algebras. An introduction through \(\mathfrak{gl}_n\) (Q2900752)

This book is based on an introductory course on Lie algebras given by the author and contains an introduction to the basics of the representation theory of Lie algebra which requires from the reader knowledge of only standard courses in linear algebra and groups and rings.NEWLINENEWLINEThe book consists of eight chapters. The first chapter gives some motivation coming from the theory of Lie groups. The second chapter introduces basic notions related to Lie algebras. The third chapter addresses the structure of Lie subalgebras, ideals and quotients and defines simple Lie algebras. The fourth chapter defines the notion of a module over a Lie algebra and addresses basics of the structure theory, including homomorphisms, submodules, simple modules and semi-simple modules. Chapter~5 classifies simple finite dimensional \(\mathfrak{sl}_2\)-modules. Chapter~6 studies duals, tensor products, homomorphisms and the Killing form. Here one can also find Schur's lemma and the definition of a Casimir operator. Chapter~7 is devoted to the study of integrable \(\mathfrak{gl}_n\)-modules. Finally, Chapter 8 provides some guide to further reading addressing classification of simple Lie algebras and the character theory for weight modules.NEWLINENEWLINEThe book contains many exercises and the Appendix, which completes the book, consists of solutions.