Lie algebras and global groups
Different levels of information
Section titled “Different levels of information”A Lie algebra describes infinitesimal generators and their commutators. A Lie group adds global information. A physical model adds an action on fields or states and further theory data.
Use the universal cover
Section titled “Use the universal cover”A connected Lie group has a simply connected covering Lie group with the same Lie algebra. Its covering kernel is discrete and central. In the compact semisimple setting, possible global forms are organized by subgroups of a finite center.
Highest weights remember the lattice
Section titled “Highest weights remember the lattice”Dominant integral highest weights parameterize irreducibles of a simply connected compact semisimple group. Passing to a quotient restricts the permitted weights to those with trivial action of the quotient kernel.
An immediate example
Section titled “An immediate example”SU(2) and SO(3) share a Lie algebra. Their ordinary representations differ: the first admits half-integer spin, while the second admits integer spin only. The workbench makes this global restriction visible without changing the local algebra.
Decode the labels and quotient
Section titled “Decode the labels and quotient”Roots and weights derives Dynkin labels and dimension formulas. Covers and global forms tests the center action explicitly, and real forms separates compact, split and Lorentz examples. Continue along the Lie and topology path for worked branching and bundle questions.
Continue in the atlas
Section titled “Continue in the atlas”Sources
Section titled “Sources”Source curation and automated validation are separate from independent expert review.
First public edition · mathematical baseline 6.0.0. Source curation and automated validation are distinct from independent expert review.
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