Lie Groups and Lie Algebras

Year 4 · 3 sections · 16 concepts · 60 practice questions

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What's covered

free1 · Matrix Lie groups4 concepts

Definition 1.1 · Matrix Lie groups
Definition 1.2 · The classical groups
Proposition 1.3 · Compactness and connectedness of the classical groups
Example 1.4 · SU(2) is the 3-sphere of unit quaternions

2 · The exponential and the logarithm5 concepts

Theorem 2.1 · The matrix exponential
Technique 2.2 · Computing matrix exponentials
Proposition 2.3 · The matrix logarithm and exp as a local homeomorphism
Theorem 2.4 · One-parameter subgroups
Theorem 2.5 · The Lie product formula

3 · The Lie algebra of a matrix Lie group7 concepts

Definition 3.1 · Lie algebras
Theorem 3.2 · The Lie algebra of a matrix Lie group
Proposition 3.3 · The classical Lie algebras and their dimensions
Example 3.4 · The exponential need not be surjective
Theorem 3.5 · A connected group is generated by exp of its Lie algebra
Theorem 3.6 · Lie group homomorphisms induce Lie algebra homomorphisms
Definition 3.7 · The adjoint representations Ad and ad

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Definition 1.2 · The classical groups

Classical groups = symmetries of a form. $\mathrm{O}(n)$ preserves the dot product, $\mathrm{U}(n)$ the Hermitian inner product, $\mathrm{Sp}$ a skew form, $\mathrm{O}(1, 3)$ the Minkowski form. Condition: $A^TBA = B$ where $B$ is the matrix of the form. "S" means det 1.

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