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Central character of a Lie algebra module
Definition
Let be a complex Lie algebra and let be a complex vector space carrying a -module structure. Let be the induced action from Lie algebra actions extend to unital actions of the enveloping algebra. A central character of is a unital complex-algebra homomorphism
such that every central element acts by
When such a map exists, one says that has central character .
Depends on
Used by
- Central characters are dot-Weyl orbits Corollary
- Primitive ideals are partitioned by their dot-orbit central character Corollary
- An intersection of two primitive ideals need not be primitive Counterexample
- Generalized central-character subcategories Definition
- The central reduction of the enveloping algebra at a central character Definition
- Primitive ideals of U(sl2) at a generic central character Example
- The annihilator of the trivial sl(2)-module Example
- A Verma composition factor has the same central character Lemma
- Central elements act by scalars on cyclic highest-weight modules Lemma
- Every central character of a semisimple enveloping algebra arises from a weight Lemma
- The central reduction of U(sl2) is simple away from the finite-dimensional central characters Lemma
- A primitive ideal determines a central character Proposition
- The Verma annihilator contains the central-character ideal Proposition
- Generic irreducibility and the exceptional parameter lattice Theorem
- The Casselman–Osborne constraint on weights of nilradical cohomology Theorem
Dependency tree · one level
2 results within one dependency step 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)
- Yiannis Sakellaridis, Verma Modules and the Category O (standard reference, not scraped)