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The endomorphism division algebra of an irreducible representation
Definition
For an irreducible finite-dimensional -representation of , put Schur's lemma says that every nonzero member of this endomorphism ring is invertible, so is a division ring (Schur's lemma for irreducible representations: a nonzero intertwiner is an isomorphism, and is a division ring). We call it the endomorphism division algebra of . Scalars give a central embedding .
Depends on
- Intertwiners, the spaces $\operatorname{Hom}_G(V,W)$ and $\operatorname{End}_G(V)$, equivalent representations, and faithful representations
- Subrepresentations, direct sums of representations, and irreducibility
- Schur's lemma for irreducible representations: a nonzero intertwiner is an isomorphism, and $\operatorname{End}_G(V)$ is a division ring
- Division ring: a ring with $1 \ne 0$ in which every nonzero element is a unit
Used by
Dependency tree · two levels
15 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
- Gabor Wiese, Galois Representations, Section 2.5 (standard reference, not scraped)
- Weizhe Zheng, Lectures on Algebra, Section 4.2 (standard reference, not scraped)