Lectures related to Lie group–Lie algebra correspondence

Lie Algebra: Group Theory

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

Nilpotent Lie Groups

Covers the properties of nilpotent Lie groups and the construction of non-degenerate alternating 2-forms.

Symmetry in Quantum Field Theory

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

Lie Algebra: Vector Space and Multiplication Law

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

Lie Algebra: Casimirs and Poincaré

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

Lie Theorems and Group Algebra

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

Symmetry: Noether Theorem

Explores the consequences of symmetry, focusing on the Noether Theorem and coordinated charges in Lie groups.

Dynamics on Homogeneous Spaces and Number Theory

Covers dynamical systems on homogeneous spaces and their interactions with number theory.

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.

General Fields: Lorentz Representations

Covers the representation of Lorentz transformations through general fields and the consequences of symmetry.

Complete Reducibility of Complex Representations

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

Simple Lie Algebras: Classification and Properties

Explores the classification and properties of simple complex Lie algebras, emphasizing their connection with 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.

Lie Algebra: Representations and Symmetry Groups

Covers Lie algebra, group representations, symmetry groups, and Schur's lemma in the context of symmetry and group operations.

Weil Representation and Heis Operators

Covers the Weil representation, Heis operators, Stone-Neumann theorem, unitary operators, Lie algebra structure, and symplectic form.

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 Theorem and Irreducible Representations

Explores the Lie theorem and irreducible representations in group theory.

Quantum Field Theory: Gauge Symmetries

Explores gauge symmetries in quantum field theory, emphasizing invariance under transformations and the role of ghost fields.

Quantum Field Theory: Poincaré Group

Explores Einsteinian relativity, the Lorentz group, and Poincaré transformations, emphasizing proper and non-orthochronous components.