Lectures related to Vector Fields

Exponential Maps: Properties and Applications in Lie Groups

Covers the properties of the exponential map in Lie groups and their algebras, including smoothness and the relationship between subgroups and algebras.

Lie Algebra: Group Theory

Explores Lie Algebra's connection to Group Theory through associative operations and Jacobi identities.

Representation Theory: Algebras and Homomorphisms

Covers the goals and motivations of representation theory, focusing on associative algebras and homomorphisms.

Integral Curves and Exponential Map

Explores integral curves on manifolds and the significance of the exponential map in Lie groups.

Group Algebra: Maschke's Theorem

Explores Wedderburn's theorem, group algebras, and Maschke's theorem in the context of finite dimensional simple algebras and their endomorphisms.

Properties of the Exponential Map

Explores the properties of the exponential map in Lie groups and algebras.

Linear Lie Groups: Definitions and Theorems

Discusses linear Lie groups, their definitions, properties, and the relationship between integral curves and vector fields.

Decomposition of Group Algebras

Covers examples of group algebra decomposition and simple infinite-dimensional algebras.

Lie Algebras: Introduction and Structure

Introduces Lie algebras, vector spaces with a special bracket operation.

Complete Reducibility of Complex Representations

Covers the complete reducibility of complex representations and the relation between Lie algebras and Lie groups.

Lie Algebra: Vector Space and Multiplication Law

Covers Lie Algebra, focusing on vector space and multiplication law.

Lie Theorems and Group Algebra

Covers Lie theorems, group algebra, Ado's theorem, and spacetime symmetries.

Lie Algebras and Representations

Explores Lie algebras, representations, tensor products, and commutation relations in mathematics.

Macdonald identities

Delves into Macdonald identities, covering affine root systems, modular forms, and Lie algebras.

Jacobi Identity in Lie Algebra

Explores the significance of the Jacobi identity in Lie algebra and its impact on linear vector spaces.

Lie Algebra: Casimirs and Poincaré

Explores Lie algebra, Casimirs, and Poincaré in SO(3) transformations.

Symmetry in Quantum Field Theory

Explores associativity, Lie algebra, Lie groups, relativity, and symmetry preservation in quantum field theory.

Hilpot Liegr: Understanding Nilpotent Lie Groups

Explores nilpotent Lie groups, their orbits, and adjoint actions, illustrating simplicity and form.

Lie Algebra: Representations

Explores Lie algebra representations, emphasizing SU(2) and traceless matrices, explained by Alfredo Glioti.

Lie Algebra Homomorphisms

Explores the association of Lie algebras to algebraic groups and homomorphisms.