Holomorphic functions of a matrix - definition and properties
The exponential function, Lie-Trotter formula
Properties of the matrix logarithm and exponential function, the problem of uniqueness of the square root of a matrix
One-parameter subgroups in \(\mathrm{GL}(n,\mathbb{C})\) - automatic smoothness
Positive matrices and polar decomposition of an invertible matrix, special cases
Lie algebras, examples, elementary properties, subalgebras, ideals, the center, direct sum
Homomorphisms of Lie algebras, the adjoint representation
Complexification of areal Lie algebra - examples
Simple, solvable and nilpotent Lie algebras - examples
The Lie algebra of a matrix Lie group
Lie algebra homomorphism associated to a homomorphism of matrix Lie groups
The formula \(\mathrm{Ad}_{\mathrm{e}^X}=\mathrm{e}^{\mathrm{ad}_X}\)
The exponential mapping - local invertibility
Corollaries: matrix Lie groups are manifolds, the Lie algebra of a matrix Lie group is its tangent space at the unit
The range of the exponential map, generation of the connected component of the unit, Lie group homomorphisms from a connected Lie group and Lie algebra homomorphisms
Exponential coordinates and automatic smoothness of a Lie group homomorphism
The Heisenberg group and the passage from a Lie algebra homomorphis to a Lie group homomorphism
The derivative of the exponential map
The Baker-Campbell-Hausdorff formula
Local Lie group homomorphisms, passage from a Lie algebra homomorphism to a local Lie group homomorphism
Extension of a local homomorphism from \(G\) to \(H\) to a global homomorphism \(G\to H\) for simply connected \(G\)
Lie subalgebras and analytic subgroups
Representations of Lie groups and associated representations of Lie algebras, intertwining operators, equivalence, irreducibility for representations of Lie algebras
Representations of \(\mathfrak{su}(2)\) and \(\mathfrak{sl}(2,\mathbb{C})\)
Direct sums and tenmsor products of representations of Lie groups and algebras
Complete reducibility of representations, Schur's lemma
Universal covers, examples
The Lie algebra \(\mathfrak{sl}(3,\mathbb{C})\)
Weights, weight vectors, roots of the algebra \(\mathfrak{sl}(3,\mathbb{C})\)
Simple roots and order on the set of weights
Cyclic representations with a highest weight - irreducibility
Classification of representations of \(\mathfrak{sl}(3,\mathbb{C})\)
