The standard books on Lie theory begin immediately with the general case: a smooth manifold that is also a group. The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. This approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time. The PeterWeyl Theorem.

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Lie Algebras and the Exponential Mapping. The BakerCampbellHausdorff Formula. Basic Representation Theory. The Representations of SU3. Highest Weight Semisimple Lie Algebras. Representations of Complex Semisimple Lie Algebras. The BorelWeil Construction.

## Introduction to Lie Algebras and Representation Theory | The Nile | TheMarket NZ

More on Roots and Weights. Quick Introduction to Groups.

A2 Examples of Groups. A21 The trivial group. A26 Complex numbers of absolute value 1 under multiplication.

A3 Subgroups the Center and Direct Products. A4 Homomorphisms and Isomorphisms. A5 Quotient Groups. Linear Algebra Review. B2 Diagonalization. B4 The Jordan Canonical Form. B6 Inner Products. More on Lie Groups. C12 Tangent space.

## Introduction to Lie algebras

C13 Differentials of smooth mappings. A3 Subgroups the Center and Direct Products. A4 Homomorphisms and Isomorphisms. A5 Quotient Groups. Linear Algebra Review.

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B2 Diagonalization. B4 The Jordan Canonical Form. B6 Inner Products. More on Lie Groups. C12 Tangent space. C13 Differentials of smooth mappings. C14 Vector fields. C15 The flow along a vector field. C16 Submanifolds of vector spaces. C17 Complex manifolds.

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C22 The Lie algebra. C23 The exponential mapping. C25 Quotient groups and covering groups. C26 Matrix Lie groups as Lie groups. C3 Examples of Nonmatrix Lie Groups. C4 Differential Forms and Haar Measure. D2 The WignerEckart Theorem. D3 More on Vector Operators.

E2 The Universal Cover. Matrix Lie Groups. Verma Modules. SU2 and SO3. X is nilpotent.