This note covers the
following topics: Geometric preliminaries, The Lie algebra of a Lie group, Lie
algebras, Geometry of Lie groups, The Universal Enveloping Algebra,
Representations of Lie groups, Compact Lie groups, Root systems, Classificiation
of compact Lie groups, Representations of compact Lie groups.
The primary aim of this note
is the introduction and discussion of the finite dimensional semisimple Lie
algebras over algebraically closed fields of characteristic and their
representations. Topics covered includes: Types of algebras, Jordan algebras,
Lie algebras and representation, Matrix algebras, Lie groups, Basic structure
theory and Basic representation theory, Nilpotent representations, Killing forms
and semisimple Lie algebras, Semisimple Lie algebras, Representations of
semisimple algebras
This note focus on
the so-called 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.
This
note covers the following topics: Fundamentals of Lie Groups, A Potpourri of
Examples, Basic Structure Theorems, Complex Semisimple Lie algebras,
Representation Theory, Symmetric Spaces.
This note covers the following topics:
Universal envelopping algebras, Levi's theorem, Serre's theorem, Kac-Moody Lie
algebra, The Kostant's form of the envelopping algebra and A beginning of a
proof of the Chevalley's theorem.
This note covers the following
topics: Ideals and homomorphism, Nilpotent and solvable Lie algebras , Jordan
decomposition and Cartan's criterion, Semisimple Lie algebras and the Killing
form, Abstract root systems, Weyl group and Weyl chambers, Classification of
semisimple Lie algebras , Exceptional Lie algebras and automorphisms,
Isomorphism Theorem, Conjugacy theorem.
This note covers the
following topics: Matrix and Lie Groups, Dynamics and Control on Matrix Groups,
Optimality and Riccati Equations, Geometric Control.
This is a lecture note for beginners on representation theory of
semisimple finite dimensional Lie algebras. It is shown how to use infinite
dimensional representations to derive the Weyl character formula.
This note explains the
following topics: Basic definitions and examples, Theorems of Engel and Lie, The
Killing form and Cartan’s criteria, Cartan subalgebras, Semisimple Lie algebras,
Root systems, Classification and examples of semisimple Lie algebras.
This
note explains the following topics: Lie groups, Lie algebra associated to a group, Correspondence between groups
and algebras, classification of connected compact Lie groups, theory of Cartan Weyl.
This book presents a simple straightforward introduction, for the
general mathematical reader, to the theory of Lie algebras, specifically to
the structure and the (finite dimensional) representations of the semisimple
Lie algebras.
This note covers the following topics: Numerical analysts in Plato’s
temple, Theory and background, Runge–Kutta on manifolds and RK-MK, Magnus and
Fer expansions, Quadrature and graded algebras, Alternative coordinates,
Adjoint methods, Computation of exponentials, Stability and backward error
analysis, Implementation, Applications.
This note covers the following topics: Applications of the Cartan calculus, category of split orthogonal vector
spaces, Super Poison algebras and Gerstenhaber algebras, Lie groupoids and Lie
algebroids, Friedmann-Robertson-Walker metrics in general relativity, Clifford
algebras.