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The quadratic Casimir element
Definition
Let be a complex semisimple Lie algebra, and let be the Killing form from The Killing form of a semisimple Lie algebra. Choose dual bases and with respect to , so . The quadratic Casimir element is
By The quadratic Casimir element is independent of the choice of dual bases, this element is independent of the chosen dual bases.
Depends on
Used by
- Smooth and K-finite vectors for SL2(R), and the (g,K)-module Definition
- The Casimir comparison on a weight space Lemma
- The central reduction of U(sl2) is simple away from the finite-dimensional central characters Lemma
- The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
- The quadratic Casimir element is central Proposition
- Generic irreducibility and the exceptional parameter lattice Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Pavel Etingof, Representations of Lie Groups (standard reference, not scraped)
- Alexander Kleshchev, Lectures on Infinite Dimensional Lie Algebras (standard reference, not scraped)