Orbital Integrals on Reductive Lie Groups and Their Algebras

Reductive Groups

This note contains below topics, Review of algebraic schemes over a field, Algebraic groups over a field geometric properties, Examples of algebraic groups, Group theory, Representations, Some basic constructions, Tannaka duality and applications, The Lie algebra of an algebraic group, Groups of multiplicative type, Algebraic groups acting on schemes, Homogeneous spaces, Tori acting on schemes, Unipotent groups, Solvable algebraic groups, Borel and Cartan subgroups, Isogenies and universal covers, Semisimple and reductive groups, Split semisimple groups and their root systems.

Author(s): J.S. Milne

139 Pages

Lie groups and Lie algebras

This book contains below topics, Manifolds, Lie groups, Lie groups, Homogeneous spaces Lie group actions, Tensor fields, Classical Lie groups, The exponential map of a Lie group, Lie algebras, Fundamental theorems of Lie theory, Proofs of the fundamental theorems of Lie theory, Representations of Lie groups and Lie algebras, The universal enveloping algebra of a Lie algebra, Free Lie algebras the BakerCampbellHausdorff formula, Solvable and nilpotent Lie algebras theorems of Lie and Engel, Semisimple and reductive Lie algebras the Cartan criteria.

Author(s): Pavel Etingof

268 Pages

Notes on G2 The Lie algebra and the Lie group

This note contains below topics, A summary of the KillingCartan classification via roots, A first linear model of split G2, Understanding G2 better, Generic 3 forms in F7, G2 and the spheres in dimension 6, G2 and spinors.

Author(s): Cristina Draper

50 Pages

Lecture notes on Cherednik algebras

This note contains below topics, Classical and quantum OlshanetskyPerelomov systems for finite Coxeter groups, The rational Cherednik algebra, The MacdonaldMehta integral, Parabolic induction and restriction functors for rational Cherednik algebras, The KnizhnikZamolodchikov functor, Rational Cherednik algebras and Hecke algebras for varieties with group actions, Symplectic reflection algebras, CalogeroMoser spaces, Quantization of ClaogeroMoser spaces.

Author(s): Pavel Etingof, Xiaoguang Ma

92 Pages

Lie Algebras and Their Representations

This note explains the following topics, Representations of sl2, Structure and classification of simple lie algebras, Structure theory of semisimple lie algebras and root systems.

Author(s): University of Cambridge

34 Pages

Introduction to Lie Groups by Alistair Savage

This note focus on the socalled matrix Lie groups since this allows us to cover the most common examples of Lie groups in the most direct manner and with the minimum amount of background knowledge. Topics covered includes, Matrix Lie groups, Topology of Lie groups, Maximal tori and centres, Lie algebras and the exponential map, Covering groups.

Author(s): Alistair Savage

111 Pages

Lecture notes in Lie Algebras

This note covers the following topics, Universal envelopping algebras, Levis theorem, Serres theorem, KacMoody Lie algebra, The Kostants form of the envelopping algebra and A beginning of a proof of the Chevalleys theorem.

Author(s): David Kazhdan

NA Pages

Orbital Integrals on Reductive Lie Groups and Their Algebras

This is an open source book written by Francisco Bulnes. The purpose of this book is to present a complete course on global analysis topics and establish some orbital applications of the integration on topological groups and their algebras to harmonic analysis and induced representations in representation theory.

Author(s): Francisco Bulnes

194 Pages