Lie Groups: Structure and Translations

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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.

Group Actions: Theory and Examples

Explores concrete examples of group actions on sets, focusing on actions that do not change the set.

Lie Groups: Representations and Transformations

Explores Lie groups, scalar fields, and vector spaces transformations.

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

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.

Chinese Remainder Theorem: Rings and Fields

Covers the Chinese remainder theorem for commutative rings and integers, polynomial rings, and Euclidean domains.

Recap of Group Theory

Provides a recap of group theory, defining a group as a set with a multiplication operation.

Symmetry: Noether Theorem

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

Elements of Lie Groups and Algebras

Explores the transformation of vectors and tensors in quantum physics, emphasizing Lie groups and algebras.